Ben Yaacov

Alexander Berenstein

Françoise Tomás Ibarlucía

Keywords: Metric model theory, continuous logic, infinitary continuous logic, automorphism groups of metric structures, Polish group theory, locally Roelcke precompact groups, Urysohn metric space, diversities, Urysohn diversity, dense conjugacy class, ample generics

We study metric model theory and Polish groups as automorphism groups of separable metric structures.

 and give a new proof of the Omitting Types Theorem of infinitary continuous logic. We also find a new way of calculating the type distance of infinitary continuous logic. Furthermore, we show an infinitary version of the Ryll-Nardzewski Theorem.

We also study the Roelcke completion of a Polish group and give a model theoretical characterisation of locally Roelcke precompact Polish groups. We do this by showing that the Roelcke completion of a Polish group can be considered as a certain set of types in metric model theory.

Furthermore, we develop the model theory of the Urysohn metric space U. We show that its theory T U eliminates quantifiers, that U is a prime model and that any separable model of T U is a disjoint union of isomorphic copies of U. Moreover, we show that the isometry group of U is locally Roelcke precompact by applying our result above. This was already known, but our proof is new.

Finally, we study the Urysohn diversity U. Diversities are a natural generalisation of metric spaces, where positive values are assigned not just to pairs, but to all finite subsets. We develop the model theory of U and show, among other things, that its automorphism group Aut(U) is locally Roelcke precompact, again by applying our result above. We also show that Aut(U) is a universal Polish group and that it has a dense conjugacy class. Lastly, we study the automorphism group of the rational Urysohn diversity U Q and show that Aut(U Q ) has ample generics -a property with many strong implications.

Résumé

Nous étudions la théorie des modèles métriques et les groupes polonais comme groupes d'automorphismes de structures métriques séparables.

Nous développons la logique continue infinitaire traitée dans [START_REF] Ben | Model theoretic forcing in analysis[END_REF][START_REF] Ben Yaacov | Metric Scott analysis[END_REF][START_REF] Caicedo | Omitting uncountable types and the strength of [0, 1]-valued logics[END_REF][START_REF] Christopher | Omitting types for infinitary [0, 1]-valued logic[END_REF] et donnons une nouvelle preuve du théorème d'omission des types pour la logique continue infinitaire. Nous trouvons également une nouvelle facon de calculer la distance de type de la logique continue infinitaire. De plus, nous montrons une version infinitaire du théorème de Ryll-Nardzewski.

Nous étudions également le complété de Roelcke d'un groupe polonais et donnons une caractérisation des groupes polonais localement Roelcke précompacts en utilisant la logique continue. Nous le faisons en montrant que le complété de Roelcke d'un groupe polonais peut être considéré comme un certain ensemble de types dans la théorie des modèles métriques.

De plus, nous développons la théorie des modèles de l'espace métrique d'Urysohn U. Nous montrons que sa théorie T U élimine les quantificateurs, que U est un modèle premier et que tout modèle séparable de T U est une union disjointe de copies isomorphes à U. De plus, nous montrons que le groupe d'isométries de U est localement Roelcke précompact en appliquant notre résultat obtenu précédemment. Bien que le résultat soit déjà connu, nous en apportons une preuve nouvelle.

Enfin, nous étudions la diversité d'Urysohn U. Les diversités sont une généralisation naturelle des espaces métriques, les valeurs positives sont attribuées non seulement aux paires, mais à tous les sous-ensembles finis. Nous développons la théorie du modèle de U et montrons, entre autres, que son groupe d'automorphismes Aut(U) est localement Roelcke précompact, encore une fois en appliquant notre résultat ci-dessus. Nous montrons également que Aut(U) est un groupe polonais universel et qu'il a une classe de conjugaison dense. Enfin, nous étudions le groupe d'automorphisme de la diversité rationnelle d'Urysohn U Q et montrons que Aut(U Q ) a des génériques amples -une propriété riche de conséquences, certaines étant très importantes.

Mots clés : Théorie des modèles métriques, logique continue, logique continue infinitaire, groupes d'automorphismes de structures métriques séparables, théorie des groupes polonais, groupes locallement Roelcke précompacts, l'espace métrique d'Urysohn, diversités, diversité d'Urysohn, classe de conjugaison dense, génériques amples.

Introduction

This dissertation deals with two major fields of mathematical research and the connections between them: metric model theory and Polish group theory. As it turns out, these two seemingly quite different topics are intimately connected. In fact, as we shall see in this thesis, the study of Polish groups is in some sense the same as the study of the automorphism groups of separable atomic metric structures.

Metric model theory is also known as continuous logic. Continuous logic was introduced in some form or another already in the 1960's, where any compact set was allowed as truth values (cf. [START_REF] Chang | Continuous model theory[END_REF]). Unfortunately, the theory never really took off, perhaps because the set-up was too general to get a satisfactory analogy between discrete model theory and the continuous counterpart. In recent years the theory has, however, experienced quite a renaissance with the works of Henson, Berenstein, Ben Yaacov, Usvyatsov and many more (cf. for instance [START_REF] Ben | Fraïssé limits of metric structures[END_REF][START_REF] Ben | Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups[END_REF][START_REF] Ben | On d-finiteness in continuous structures[END_REF][START_REF] Ben | Continuous first order logic and local stability[END_REF][START_REF] Ben | Topometric spaces and perturbations of metric structures[END_REF][START_REF] Ben Yaacov | Model theory with applications to algebra and analysis[END_REF][START_REF] Ben Yaacov | Metric Scott analysis[END_REF][START_REF] Farah | Model theory for C * -algebras[END_REF]). The focus has now been on metric structures, where instead of allowing a general compact set as truth values, we only allow compact intervals of the real line as possible truth values. Although we do lose some generality with this restriction, most of the structures we want to study, such as Banach spaces, Hilbert spaces, measure algebras and Polish groups, all fit well into this set-up.

The basic object on which we can define our metric structures are complete metric spaces. For technical reasons we require the metric to be bounded. This does cause a few problems, since most of the structures we are interested in are not bounded. However, in practice, this obstacle is easy to work around, and it is possible to describe even unbounded structures as metric structures without much effort. Just as in classical logic, the structures contain predicates and functions that in the metric setting must be uniformly continuous maps defined on some power of the structure. As logical connectives we allow any real-valued continuous function defined on products of compact intervals. Furthermore, and this is where the crucial advantage over the continuous logic of the 1960's lies, there are two natural quantifiers, namely the infimum and the supremum, thought of as metric versions of the existential and univer-11 sal quantifiers, respectively. Equipped with these two quantifiers the analogy between metric model theory and discrete model theory becomes absolutely remarkable. Almost every single concept and result from the classical setting has a corresponding concept or result in the metric setting. For instance, there are completely natural metric versions of the Compactness Theorem, the Löwenheim-Skolem Theorem, the Omitting Types Theorem and the Ryll-metric model theory discussed above, it becomes natural to ask if the same thing is true for more general logics. In the discrete setting, one of the more successful logics is the infinitary logic denoted L ω 1 ω . In this logic we allow conjunctions and disjunctions over countably many formulas, which gives the language more expressive power. However, the price for this power is high: the Compactness Theorem no longer holds. Nevertheless, L ω 1 ω provides a powerful tool in the study of structures. The most prominent example of the expressive power of this logic is probably Scott's theorem, stating that for any (classical) model, there is an L ω 1 ω -sentence characterising it up to isomorphism (cf. e.g. [START_REF] Keisler | Model theory for infinitary logic. Logic with countable conjunctions and finite quantifiers[END_REF]). In this thesis, we give an exposition of the infinitary continuous logic developed in [START_REF] Ben | Model theoretic forcing in analysis[END_REF][START_REF] Ben Yaacov | Metric Scott analysis[END_REF][START_REF] Caicedo | Omitting uncountable types and the strength of [0, 1]-valued logics[END_REF][START_REF] Christopher | Omitting types for infinitary [0, 1]-valued logic[END_REF]. Moreover, we expand upon this logic and give a new proof of the Omitting Types Theorem for infinitary continuous logic. In short, this theorem states that for any non-principal type it is possible to find a model where it is not realised, i.e. the type is omitted. As a corollary, we obtain a version of the Ryll-Nardzewski Theorem for infinitary continuous logic. This theorem describes the ℵ 0 -categorical structures, i.e. those structures whose theory has a unique separable model, in terms of the types they realise. Furthermore, we discover a new way of calculating the type distance in infinitary continuous logic. Defining such a distance is a delicate task, since the usual type distance of finitary logic depends heavily on the Compactness Theorem. Nevertheless, in [START_REF] Ben Yaacov | Metric Scott analysis[END_REF] the authors do manage to define a type distance for infinitary types. However, the definition might seem rather artificial and complicated at first glance. Our new version of the type distance generalises directly one way to calculate the distance in finitary logic: it is the supremum over the so-called 1-Lipschitz formulas, i.e. those formulas that are interpreted as 1-Lipschitz maps. Hence, it simplifies the definition from [START_REF] Ben Yaacov | Metric Scott analysis[END_REF] and gives perhaps a more intuitive idea of how to calculate the distance.

The other major theme of the thesis is, as mentioned, Polish group theory. Polish groups are everywhere in mathematics. Examples include the group S ∞ of all permutations of N, the group of homeomorphisms of a compact metrisable space, the unitary group of the separable infinite dimensional Hilbert space, isometry groups of separable metric spaces and, more generally, automorphism groups of separable metric structures. In classical model theory, the group S ∞ plays a particularly important role. It turns out that the automorphism group of any countable structure can be viewed as a closed subgroup of S ∞ . Conversely, given any closed subgroup of S ∞ , it is possible to construct a countable structure, known as the canonical structure, with this subgroup as its automorphism group (cf. [START_REF] Becker | The descriptive set theory of Polish group actions[END_REF] for the details).

Given this correspondence between closed subgroups of S ∞ and automorphism groups of classical countable structures, it is natural to ask if there is a similar correspondence in the general metric framework. The answer turns out to be yes: any automorphism group of a separable metric structure is, as mentioned, a Polish group, because it is a closed subgroup of an isometry group of a complete separable metric space. Conversely, given any Polish group G, it is possible to define a metric structure, known as the canonical metric structure associated to G, such that G is the automorphism group of this structure. Usually, this construction is attributed to Julien Melleray. However, curiously enough, he himself writes in [START_REF] Melleray | A note on Hjorth's oscillation theorem[END_REF] that he is not sure who the construction should be attributed to. In any case, we will use this structure to give the aforementioned characterisation of locally Roelcke precompact Polish groups. Local Roelcke precompactness is a property of the so-called Roelcke completion of a Polish group. This completion is the completion of a certain uniformity we can impose on the group. Uniformities were originally introduced in the 1930's by André Weil in order to capture the intuitive idea of relative closeness. That is, in a uniform space it makes sense to say that a is closer to x than b is to y. On general topological groups it is always possible to find uniformities inducing the group topology. Among these uniformities, there are four canonical uniformities of special interest: the left, right, two-sided and Roelcke uniformities. For Polish groups, the two-sided uniformity is always complete. Hence, only the other three can provide new information when we consider the group completions. Furthermore, by a famous theorem of Birkhoff and Kakutani, it is always possible to find a left-invariant compatible metric inducing the left uniformity. Likewise, we can always find a right-invariant compatible metric inducing the right uniformity. Moreover, it is not hard to show that any left-, respectively, right-invariant metric will induce the left, respectively right uniformity. The Roelcke uniformity is then defined as the meet of the left and the right, i.e. it is the finest uniformity contained in both of them. Using any left-or right-invariant metric, it is possible to define a metric that induces the Roelcke uniformity. The completion of the group with respect to the Roelcke uniformity is known as the Roelcke completion, and it is this completion that we will study by using type spaces of continuous logic.

The connection between the Roelcke completion and type spaces of continuous logic was also studied in [START_REF] Ben | Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups[END_REF]. There the authors together with C. Rosendal (cf. [START_REF] Rosendal | A topological version of the Bergman property[END_REF]) obtained the theorem below that characterises the so-called Roelcke precompact groups, i.e. those groups whose Roelcke completion is compact, in terms of their actions. Roelcke precompact groups have been studied extensively in the literature (see for instance [START_REF] Ben | Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups[END_REF][START_REF] Roelcke | Uniform structures on topological groups and their quotients[END_REF][START_REF] Rosendal | Coarse geometry of topological groups[END_REF][START_REF] Uspenskij | The Roelcke compactification of unitary groups[END_REF][START_REF] Uspenskij | The Roelcke compactification of groups of homeomorphisms[END_REF][START_REF] Uspenskij | Compactifications of topological groups[END_REF][START_REF] Uspenskij | On subgroups of minimal topological groups[END_REF][START_REF] Zielinski | Locally roelcke precompact polish groups[END_REF] just to mention some). One reason for this, is that their actions tend to be somewhat nicely behaved and easy to understand, as the theorem below explains. Loosely speaking, they have a somewhat compact feel to them, even though there are, of course, many Roelcke precompact groups that are nowhere near compact. The theorem from [START_REF] Ben | Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups[END_REF] (cf. also [START_REF] Rosendal | A topological version of the Bergman property[END_REF]) reads:

Theorem ([13, Theorem 2.4], [START_REF] Rosendal | A topological version of the Bergman property[END_REF]). Let G be a Polish group. Then the following are equivalent:

(i) G is Roelcke precompact.

(ii) Whenever G acts continuously by isometries on a complete metric space X and the set of orbit closures X G is compact, then for all n, X n G is compact, where G acts on X n via the diagonal action.

(iii) There is a Polish metric space X and a homeomorphic group embedding G → Iso(X) such that for the induced action of G on X, the space X n G is compact for every n.

An action such as the one in (ii) of the theorem above, where X n G is compact for all n, is called an approximately oligomorphic action (compared to an oligomorphic action, where there is only finitely many orbits for each n). It turns out that by using the Ryll-Nardzewski Theorem for continuous logic (cf. [START_REF] Ben Yaacov | Model theory with applications to algebra and analysis[END_REF]Theorem 12.10]), one gets the following description of Roelcke precompact groups:

Corollary. Let G = Aut(M) be the automorphism group of a metric structure and suppose that M G is compact. Then the following are equivalent:

(i) G is Roelcke precompact.

(ii) Th(M) is ℵ 0 -categorical.

Armed with these results, it becomes much simpler to check if a given group is Roelcke precompact. For instance, S ∞ , the unitary group of a separable Hilbert space [START_REF] Uspenskij | The Roelcke compactification of unitary groups[END_REF], the automorphism group of a standard probability space [START_REF] Glasner | The group Aut(µ) is Roelcke precompact[END_REF], the automorphism group of the random graph [START_REF] Ben | Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups[END_REF] and the isometry group of the bounded Urysohn space [START_REF] Uspenskij | On subgroups of minimal topological groups[END_REF] are all Roelcke precompact, which may be seen using the result above. Just like locally compact spaces are a natural generalisation of the compact ones, Roelcke precompact groups can be generalised to the so-called locally Roelcke precompact (lRpc) groups. Here we do not require the group itself to be precompact in the Roelcke uniformity, but only that there is a precompact neighbourhood of the identity. In particular, any Roelcke precompact and any locally compact group is also lRpc. However, it is not immediate that the Roelcke completion of a locally Roelcke precompact group is locally compact. Nevertheless, due to a recent theorem of Zielinski, this is fortunately the case: Theorem (Zielinski, cf. [START_REF] Zielinski | Locally roelcke precompact polish groups[END_REF]). A Polish group G is locally Roelcke precompact if and only if the Roelcke completion of G is locally compact.

the Urysohn diversity are both locally Roelcke precompact. We will get back to diversities below.

Although lRpc groups have not received as much attention as the Roelcke precompact ones, they do posses interesting features e.g. from the viewpoint of geometric group theory. For instance, they can allow non-trivial so-called coarse geometry. This should be seen in opposition to the Roelcke precompact groups, because these are always coarsely bounded and hence have trivial largescale geometry. At the same time, lRpc groups are only a slight generalisation of the Roelcke precompact ones and hence share many of the same features. Moreover, lRpc groups are always locally bounded, so their large-scale geometry is somehow still tangible. Since we will not really be concerned with these geometrical considerations in this thesis, we will not indulge ourselves and digress from the matter at hand. Instead, we refer the reader to Rosendal's manuscript [START_REF] Rosendal | Coarse geometry of topological groups[END_REF] and Zielinski's paper [START_REF] Zielinski | Locally roelcke precompact polish groups[END_REF] for an introduction to coarse geometry and more on lRpc groups. However, what we will see in this thesis is a model theoretical characterisation of lRpc groups, inspired by the one of the Roelcke precompact ones above. More specifically, we will see that it is always possible to embed the Roelcke completion isometrically into a certain type space, in such a way that the Roelcke completion is homeomorphic to the image of this embedding equipped with the logic topology τ . Since τ is a nice compact topology, it becomes much easier to verify topological properties of the Roelcke completion. In particular, the group is lRpc if and only if the Roelcke completion equipped with τ is open inside its τ -closure. It turns out that this is rather straightforward to verify for both the isometry group of the Urysohn metric space and the automorphism group of the Urysohn diversity, once you understand the model theory of these structures. Therefore, before we can make this verification we will need to develop the model theory of the Urysohn metric space U. Recall that this space is the unique (up to isometry) universal ultrahomogeneous separable complete metric space. Here, universal means that it contains an isometric copy of any finite metric space. Ultrahomogeneous means that any isometry between finite subsets extends to a surjective isometry of the whole space. U was first constructed by Urysohn in the 1920's in the posthumously published paper [START_REF] Urysohn | Sur un espace métrique universel[END_REF] and was long almost completely forgotten. However, in the 1980's Katětov gave a new construction of U, where he "built" a version of the Urysohn space "around" any separable metric space. Not long after, Uspenskij realised that using this construction, it was rather easy to show that the isometry group of U was a universal Polish group, i.e. that any Polish group embeds as a closed subgroup of Iso(U). This renewed the interest in U and its isometry group, and today new interesting results keep appearing in the literature (cf. for instance [START_REF] Ben Yaacov | Polish topometric groups[END_REF][START_REF] Cameron | Some isometry groups of the Urysohn space[END_REF][START_REF] Malicki | Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures[END_REF][START_REF] Melleray | Some geometric and dynamical properties of the Urysohn space[END_REF][START_REF] Sabok | Automatic continuity for isometry groups[END_REF]).

As mentioned, we will develop the model theory of U in this thesis. A priori, U does not fit well into the set-up of metric model theory because it is an unbounded metric space. Therefore, most authors working in logic choose to consider the bounded version of U. However, there are several work arounds for the unboundedness problem, as we shall see in Chapter 5. There we will show an "almost categoricity" result concerning the theory of U, saying that any separable model of the theory is a disjoint union of at most countably many copies of U. From this result it will follow that the theory eliminates quantifiers and that U is an atomic model. Moreover, as we mentioned above, we can use our result to show that Iso(U) is lRpc. This was of course already known, but our approach is new.

We will also study the automorphism group of the Urysohn diversity U. Diversities were introduced in [START_REF] Bryant | Hyperconvexity and tight-span theory for diversities[END_REF] and [START_REF] Bryant | Diversities and the geometry of hypergraphs[END_REF] by Bryant and Tupper. Their motivation was to generalise applications of metric space theory to combinatorial optimisation and graph theory to the hypergraph setting. The idea is simple enough: a diversity assigns positive values to all finite subsets (and not just to pairs of points as for a metric). This assignment must, of course, uphold some axioms similar to those of a metric. In particular, if we only consider pairs of points we obtain a metric space. It turns out that this generalises metric spaces quite nicely. Moreover, and this is where the interest of model theory begins, it turns out that there is a unique ultrahomogeneous universal separable diversity, namely the Urysohn diversity U. In this thesis, we shall develop the model theory of U along the same lines as for the Urysohn metric space. In particular, we will show the same "almost categoricity" result for U, namely that any separable model of the theory of U is a disjoint union of at most countably many copies of U. As for the metric space, this result will imply that the theory of U eliminates quantifiers and that U is an atomic structure. Moreover, it implies, using our characterisation of the lRpc groups, that the automorphism group of U is locally Roelcke precompact.

U was first constructed in [START_REF] Bryant | A universal separable diversity[END_REF] by a construction similar to the Katětov construction of the Urysohn metric space. As mentioned, Uspenskij exploited Katětov's construction and showed that the isometry group the Urysohn metric space is a universal Polish group. In Chapter 6, we adapt Uspenskij's proof and show that the automorphism group of U is universal as well.

In [START_REF] Bryant | Fraïssé limits for relational metric structures[END_REF], the authors of [START_REF] Bryant | A universal separable diversity[END_REF] constructed U as a metric Fraïssé limit in the sense of Ben Yaacov in [START_REF] Ben | Fraïssé limits of metric structures[END_REF]. In this thesis, we will give yet another construction of U using classical Fraïssé theory. More precisely, we will show that the class of finite rational diversities, i.e. diversities whose map only takes rational values, is a so-called Fraïssé class. A Fraïssé class is a class with certain properties that ensure that there is a so-called limit for the class, i.e. the Fraïssé limit, containing all the structures of the class. Moreover, this limit is ultrahomogeneous, which in this more general setting means that any isomorphism between finite substructures extends to an automorphism of the whole structure. This way of approximating infinite structures by their finite substructures has proven extremely useful. Many well-known and important structures can be defined as a Fraïssé limit, e.g. the (rational) Urysohn metric space, the random graph and the ordered rationals are all Fraïssé limits of their respective classes. Moreover, as mentioned, the class of all finite rational diversities is a Fraïssé class. Hence, the class has a limit, which we will call the rational Urysohn diversity and denote by U Q . It is not difficult to check that the completion of U Q is (isomorphic to) U itself, and thus we obtain the mentioned new construction of U. Moreover, using results of Kechris and Rosendal from [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF], it is easy to show that the conjugacy action of Aut(U Q ) on itself admits a dense orbit. Furthermore, we will see that Aut(U Q ) embeds densely into Aut(U), from which it follows that the conjugacy action of Aut(U) on itself also admits a dense orbit. Having a dense conjugacy class is also known as the topological Rokhlin property because the well-known Rokhlin Lemma of ergodic theory implies that the automorphism group of a standard measure space (X, µ) has this property. Having a dense conjugacy class is equivalent to the following generic ergodicity property: every conjugacy invariant Baire measurable subset of the group is either meagre or comeagre (cf. [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF]). Recall that a subset of a topological space is meagre if it is a countable union of nowhere dense sets, i.e. sets whose closure has empty interior. A comeagre set is then simply a set whose complement is meagre. A set A is Baire measurable if there is an open set U such that the symmetric difference between A and U is meagre. Today there are many groups that are known to admit a dense conjugacy class. Besides Aut(X, µ), the unitary group of a separable infinite dimensional Hilbert space, and the homeomorphism groups of the Hilbert cube [0, 1] N and the Cantor space 2 N all share this property, just to mention a few (cf. [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF]).

We conclude the thesis by showing that Aut(U Q ) has so-called ample generics. Ample generics is a powerful property with many strong implications. It was first introduced by Hodges, Hodkinson, Lascar and Shelah in [START_REF] Hodges | The small index property for ω-stable ω-categorical structures and for the random graph[END_REF] in order to study the small index property for automorphism groups of certain countable structures. This property says that any subgroup of index < 2 ℵ 0 must be open. In [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF], Kechris and Rosendal studied ample generics further and showed that it implies several strong properties. One of these is the automatic continuity property. A classical result of Pettis says that any Baire measurable homomorphism from a Polish group G to a separable group must be continuous (cf. [START_REF] Pettis | On continuity and openness of homomorphisms in topological groups[END_REF]). If G has ample generics, this is simply true for any group homomorphism. Kechris and Rosendal also show that a Polish group with ample generics has a unique Polish group topology and that it cannot be the countable union of non-open subgroups.

The thesis is organised as follows: Chapter 1 covers the basics of Polish group theory and classical descriptive set theory that we will need. In Chapter 2 we introduce metric model theory and the concepts and results we will need later in the thesis. We will also show a few smaller lemmas and propositions that will come in handy from time to time. Chapter 3 is concerned with infinitary continuous logic. Chapter 4 studies Polish groups as automorphism groups of metric structures and the Roelcke completion as a set of types. In Chapter 5 we study the Urysohn metric space as a metric structure. The last chapter, Chapter 6, studies both the rational and complete Urysohn diversities as well as their automorphism groups.

Chapter 1

Polish Group Theory

In this first, short chapter of the thesis we have included some preliminary facts and results from descriptive set theory and Polish group theory that we will need throughout. There are no new results here and very few proofs. We invite the reader to consult e.g. Kechris' excellent book [START_REF] Kechris | Classical descriptive set theory[END_REF] for a thorough exposition of classical descriptive set theory and for instance Becker and Kechris' book [START_REF] Becker | The descriptive set theory of Polish group actions[END_REF] for more on Polish groups. Finally, Dierolf and Roelcke's [START_REF] Roelcke | Uniform structures on topological groups and their quotients[END_REF] is a good reference for uniform spaces and group completions.

Polish spaces

The perhaps most basic object of descriptive set theory is a Polish space. These spaces are so named because they were first studied by a large number of wellknown Polish mathematicians. Today they provide a general framework for the study of the main topics of this thesis, i.e. descriptive set theory, metric model theory and Polish group theory.

In this short section we only introduce the absolute essentials that we will need. There are many good references for Polish spaces and we invite the reader to consult e.g. [START_REF] Kechris | Classical descriptive set theory[END_REF] for a thorough exposition.

The definition of a Polish space reads as follows:

Definition 1.1.1. A topological space X is separable if there is a countable dense subset. X is completely metrisable if there is a complete metric inducing the topology. A Polish space is a separable completely metrisable topological space.

Polish spaces are abundant in mathematics. We have listed two well-known examples below.

Example 1.1.2. The reals R with their usual topology is a Polish space. So is the open unit interval (0, 1) since it is homeomorphic to R (even though the usual metric is not complete).

Example 1.1.3 (The Cantor Space). Consider the set C = 2 ω consisting of all sequences in 2 = {0, 1}. We endow {0, 1} with the discrete topology and C with the product topology. It has as basis the sets

U s = {x ∈ C : s ⊆ x},
where s ∈ 2 <ω is a finite sequence. The space C is called the Cantor space. It is a Polish space with a complete compatible metric defined by d(x, y) = 2 -n-1 for x = y, where n is the least number such that x(n) = y(n). The set

{x ∈ C : ∃N ∀n ≥ N x(n) = 0}
is a countable dense subset.

Remark 1.1.4. In the example above one could just as easily have used any countable ordinal instead of 2. The space N = ω ω is called the Baire space.

The following proposition is easily verified. (ii) Countable products of Polish spaces are Polish. In particular, countable powers of a Polish space are again Polish.

Recall that a G δ set is a countable intersection of open sets. We have the following important characterisation of those subspaces of a Polish space that are themselves Polish (cf. [START_REF] Kechris | Classical descriptive set theory[END_REF]Theorem 3.11]).

Theorem 1.1.6. A subspace of a Polish space is Polish in the subspace topology if and only if it is G δ .

Baire category

We will from time to time encounter notions from Baire category. Therefore, we provide a short introduction to this area. The reader is again referred to [START_REF] Kechris | Classical descriptive set theory[END_REF] for a more comprehensible introduction to Baire category.

Definition 1.2.1. Let X be a topological space. A subset A ⊆ X is nowhere dense if its closure A has empty interior, i.e. A • = ∅.
A is meagre if it is a countable union of nowhere dense sets. The complement of a meagre set is called comeagre.

Remark 1.2.2. Observe the following:

• A set A is nowhere dense if and only if the complement of its closure A c is dense.

• A is meagre if and only if it is contained in a countable union of closed nowhere dense sets. The meagre sets of a topological space can be used to define an important σ-algebra consisting of the sets with the Baire property. Definition 1.2.3. A subset A ⊆ X of a topological space X has the Baire property if there is an open set U such that the symmetric difference A U is meagre.

It is not hard to see that the sets with the Baire property form a σ-algebra. Moreover, it is the smallest σ-algebra containing the open sets and the meagre sets (cf. [START_REF] Kechris | Classical descriptive set theory[END_REF]Proposition 8.22]).

Since the sets with the Baire property form a σ-algebra, we can define maps that are measurable with respect to it. These are the Baire measurable maps.

Definition 1.2.4. A map f : X → Y between topological spaces is Baire measurable if the preimage of any open set in Y has the Baire property in X.
It is of course evident that continuous maps are Baire measurable. In fact, most natural maps are. A general rule of thumb says that if you can write down an explicit definition of the map, then it is (probably) Baire measurable.

The crown jewel of Baire category is the Baire Category Theorem proved by R. Baire himself in 1899.

Theorem 1.2.5 (Baire Category Theorem, [START_REF] Baire | Sur les fonctions de variables réelles[END_REF]). Let X be a completely metrisable space. Then any countable intersection of dense open subsets of X is dense.

Proof. Let d be a complete metric on X and let (D n ) n∈N be a sequence of dense open subsets of X. Let U be a non-empty open set. We must show that D n ∩ U is non-empty. Since D 0 is dense and open, we can find a ball, B 0 , of some radius, r 0 , such that B 0 ⊆ U ∩ D 0 . Since B 0 is open and D 1 is dense and open, we can find a ball, B 1 , of radius r 1 ≤ 1 2 r 0 and such that B 1 ⊆ B 0 ∩ D 1 . Continuing in this way, we get a sequence of balls B n of radius r n , where r n ≤ 1 n+1 r 0 and such that B n ⊆ B n-1 ∩ D n . If we let x n denote any point of B n , then the sequence (x n ) n∈N is a Cauchy sequence. Thus, by completeness, (x n ) converges to some x. For any n ∈ N we must have that x ∈ B n since B n is closed, and thus x ∈ B n . Furthermore, we have that B n ⊆ B 1 ⊆ U and for each n ∈ N we have that B n ⊆ D n . Hence, B n ⊆ D n as well, so that x ∈ D n ∩ U . Remark 1.2.6. The following are equivalent in a topological space X:

(i) A countable intersection of dense open sets in X is dense.

(ii) Any non-empty open set is not meagre. (iii) Every comeagre set in X is dense.

A space satisfying these properties is called a Baire space.

The Baire Category Theorem is one of the most important and useful theorems of descriptive set theory, with a multitude of applications in many areas of mathematics. We will be using this theorem ourselves in Chapter 3.

Polish groups

One of the main objects of study in the thesis is Polish groups. In general, a topological group is a group equipped with a topology such that the group operations are continuous. Polish groups are then defined as follows: Definition 1.3.1. A Polish group is a topological group whose topology is separable and completely metrisable.

Polish groups appear throughout mathematics and they have been studied for around a century. Below is a list of some examples of Polish groups.

Examples. The following groups are Polish:

(i) Any countable group equipped with the discrete topology.

(ii) (R, +) with the usual topology.

(iii) If G n is a Polish group for each n ∈ N, then G n with the product topology is Polish.

(iv ) The group of invertible n × n-matrices GL(n, R) viewed as a subspace of R n 2 .

(v ) The permutation group of N, S ∞ , equipped with the pointwise convergence topology. This group is particularly important in model theory, since its closed subgroups are exactly the automorphism groups of countable structures (cf. [START_REF] Becker | The descriptive set theory of Polish group actions[END_REF] for the details).

(vi) If H is a separable infinite dimensional Hilbert space, then the unitary group U (H) of all bounded linear operators T : d) is a separable complete metric space, then the isometry group of X, Iso(X), is a Polish group with the pointwise convergence topology. A compatible complete metric is given by

H → H such that T T * = T * T = I is a Polish group. (vii) If (X,
d(f, g) = ∞ n=0 2 -n-1 d 1 (f (x n ), g(x n )) + d 1 (f -1 (x n ), g -1 (x n )) ,
where (x n ) is a countable dense sequence in X and where

d 1 (x, y) = d(x, y) 1 + d(x, y) .
(viii) More generally, the automorphism group Aut(M) of a separable metric structure M is Polish since it is a closed subgroup of the isometry group of M.

We will get back to metric structures in the next chapter. Furthermore, we will see in Chapter 4 that any Polish group may be viewed as the automorphism group of a separable metric structure. Thus, the study of Polish groups can be considered as contained in the study of metric structures and their automorphism groups. General metrisable topological groups were completely characterised by Birkhoff and Kakutani. Moreover, and this will be important to us, they showed that when the group is metrisable, it is always possible to find a leftinvariant compatible metric, i.e. a metric d inducing the topology such that d(gh, gf ) = d(h, f ) for all g, h and f . Theorem 1.3.2 (Birkhoff-Kakutani). A topological group G is metrisable if and only if it is Hausdorff and has a countable neighbourhood basis at the identity. Moreover, if G is metrisable, the metric can be chosen to be leftinvariant.

Remark 1.3.3. If G is metrisable with left-invariant metric d L , then the metric d R (g, h) = d L (g -1 , h -1 ) defines a right-invariant compatible metric.
In particular, it follows that any Polish group admits a compatible left-invariant metric. However, and this is very important, the left-invariant metric need not be complete. For instance, S ∞ does not admit such a metric (cf. [START_REF] Becker | The descriptive set theory of Polish group actions[END_REF]). This opens up the study of completions of Polish groups with respect to for instance left-invariant metrics. We will get back to that below.

A useful theorem due to Pettis [START_REF] Pettis | On continuity and openness of homomorphisms in topological groups[END_REF] tells us that Baire measurable homomorphisms between Polish groups are always continuous. Theorem 1.3.4 (Pettis [START_REF] Pettis | On continuity and openness of homomorphisms in topological groups[END_REF], cf. also [START_REF] Kechris | Classical descriptive set theory[END_REF]Theorems 9.9 and 9.10]). Let G be a topological group. If A ⊆ G has the Baire property and is non-meagre, then A -1 A := {gh : g ∈ A -1 , h ∈ A} contains an open neighbourhood of the identity. It follows that if ϕ : G → H is a homomorphism from a Polish group G to a separable group H and ϕ is Baire measurable, then ϕ is continuous.

Therefore, most natural homomorphism that one will encounter when working with Polish groups will be continuous. However, using the axiom of choice, it is possible to construct homomorphisms between some Polish groups that are not continuous. On the other hand, there are Polish groups G, such that any homomorphism ϕ : G → H into a separable group H is continuous. This property is known as the automatic continuity property and we will get back to it in Chapter 6, where we provide a new example of a Polish group with this property.

Another important notion, that we will come across a few times in this thesis, is the notion of a group action. These are defined as follows:

Definition 1.3.5. Let G be a group and let X be a set. An action of G on X is a map G × X → X denoted (g, x) → g • x, such that for all x ∈ X, 1 G • x = x and (gh) • x = g • (h • x).
We say that G acts on X and denote it by G X.

If G is a topological group and X is a topological space, we say that an action G X is continuous if the action is continuous as a map G×X → X.

For instance, if we are given a structure M, then the automorphism group Aut(M) acts on M simply by g • a = g(a). We will therefore use these two notations interchangeably. We will also often omit the '•' and simply write ga for g ∈ Aut(M) and a ∈ M.

Uniform spaces

In this section, we provide a short introduction to uniform spaces and uniformities on Polish groups. We will need these concepts in Chapter 4.

A uniform space should be thought of, in terms of generality, as a structure sitting between topological spaces and metric spaces. The idea is to define the minimal structure needed in order to define uniform notions such as uniform continuity, Cauchy sequences and completeness. Thus a uniform space should be a generalisation of a metric space, since all these notions can be defined using a metric. On the other hand, uniform continuity implies continuity, so a uniform space must in particular be a topological space.

There are several equivalent ways to define a uniform space. We have chosen the definition using entourages, which was the definition used when André Weil first introduced the concept in the 1930's. Definition 1.4.1. Let X be a set. A collection U of subsets of X × X is a uniformity if it satisfies the following conditions:

(i) If U ∈ U, then ∆X = {(x, x) : x ∈ X} ⊆ U . (ii) If U ∈ U and U ⊆ V , then V ∈ U. (iii) If U, V ∈ U, then U ∩ V ∈ U. (iv ) If U ∈ U, then there is V ∈ U such that V • V ⊆ U , where V • V := {(x, y) ∈ X × X : ∃z ∈ X(x, z), (z, y) ∈ V }. (v ) If U ∈ U, then U -1 ∈ U, where U -1 := {(x, y) : (y, x) ∈ U }.
Elements of U are referred to as entourages.

Given two uniformities U and V on a set X, we say that U is coarser than V and V is finer than U if U ⊆ V. If neither is contained in the other, it is natural to ask whether there is a finest, respectively coarsest, uniformity contained in, respectively containing, both U and V. This is indeed the case. The coarsest uniformity containing both is called the join of U and V and is denoted by U ∨ V. The finest uniformity contained in both is called the meet of U and V and is denoted U ∧ V.

A base for a uniformity U is a subset B ⊆ U such that for all U ∈ U there is B ∈ B with B ⊆ U .

If we are given a uniform space (X, U), we can define a topology on X as follows: a set A ⊆ X is open if for every x ∈ A, there is U ∈ U such that U [x] := {y : (x, y) ∈ U } ⊆ A. We denote this topology by τ U and say that it is induced by the uniformity U.

We can also define the notion of uniform continuity.

If f : X → Y is a map between uniform spaces (X, U) and (Y, V), then f is uniformly continuous if for any V ∈ V, there is U ∈ U, such that for all x, x ∈ X, if (x, x ) ∈ U , then (f (x), f (x )) ∈ V .
It is also possible to define Cauchy sequences and completions for uniform spaces using the notion of a Cauchy filter. If F is a filter on the uniform space (X, U), we say that F is a Cauchy filter if for any entourage U ∈ U, there is F ∈ F such that F × F ⊆ U . We say that (X, U) is complete if any Cauchy filter on X converges to some x ∈ X in the induced topology τ U . Recall that F converges to x if for all neighbourhoods of x, there is an element of the filter contained in the neighbourhood. Finally, any uniform space (X, U) has a completion, i.e. a complete uniform space in which X is dense. We refer the reader to [START_REF] Roelcke | Uniform structures on topological groups and their quotients[END_REF] for a proof of this fact and for a more thorough introduction to uniform spaces.

Uniformities on Polish groups

On a topological group there are four canonical uniformities that all induce the group topology. These are the left, right, two-sided and the Roelcke uniformity. In this thesis, we will mainly be interested in the Roelcke uniformity (cf. Chapter 4). However, the left uniformity will also be important to us, as we will use it in several constructions. Of course we could just as easily have used the right uniformity in these constructions, but it seems to be the left one that most authors prefer.

The left uniformity of a Polish group G is generated by the entourages

{(g, h) : g ∈ hV },
where V varies over neighbourhoods of the identity. It is not hard to check that the compatibility of the group operations with the topology implies that these sets do in fact generate a uniformity that induces the group topology.

The left completion of G, denoted G L , is the completion of G with respect to the left uniformity.

Similarly, the right uniformity is generated by sets of the form

{(g, h) : g ∈ V h},
where V again varies over identity neighbourhoods. The right completion of G is the completion with respect to this uniformity and is denoted G R . The two-sided uniformity is now defined as the join of the left and the right uniformity. Therefore, it is generated by intersections of entourages from these two uniformities. However, this uniformity is always complete for Polish groups, and it will therefore not play any further role for us here.

As mentioned, we are mainly interested in the Roelcke uniformity. It is defined as the meet of the left and right uniformities and is generated by entourages of the form

{(g, h) : g ∈ V hV },
where V varies over identity neighbourhoods. The Roelcke completion of G is the completion of G with respect to this uniformity and is denoted G L∧R .

For a Polish group G, the Birkhoff-Kakutani Theorem tells us that it is always possible to find a left-invariant metric d L compatible with the topology. Moreover, it is easy to see that any such left invariant metric induces the left uniformity on G by considering entourages of the form {(g, h) : d L (g, h) < r} for r > 0. Given such a left-invariant metric d L , we can define a metric d L∧R inducing the Roelcke uniformity on G as follows:

d L∧R (g, h) = inf f ∈G max{d L (g, f ), d L (f -1 , h -1 )}.
Thus, the Roelcke completion G L∧R is the completion of G with respect to this metric.

The Roelcke uniformity has received quite a lot of attention in the literature (see e.g. [START_REF] Ben | Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups[END_REF][START_REF] Roelcke | Uniform structures on topological groups and their quotients[END_REF][START_REF] Rosendal | Coarse geometry of topological groups[END_REF][START_REF] Uspenskij | The Roelcke compactification of unitary groups[END_REF][START_REF] Uspenskij | The Roelcke compactification of groups of homeomorphisms[END_REF][START_REF] Uspenskij | Compactifications of topological groups[END_REF][START_REF] Uspenskij | On subgroups of minimal topological groups[END_REF][START_REF] Zielinski | Locally roelcke precompact polish groups[END_REF]). One reason for this is that the Roelcke precompact groups (defined below) have particularly nice properties, as we mentioned in the introduction. In Chapter 4 we will study the Roelcke completion from the viewpoint of metric model theory. It turns out that there is a close connection between the Roelcke completion and type spaces of continuous logic. We will use this connection to study the class of the socalled locally Roelcke precompact groups. These groups include the Roelcke precompact ones but also many groups that are not Roelcke precompact, such as, for instance, the isometry group of the Urysohn metric space. Roelcke precompactness and local Roelcke precompactness are defined as follows:

Definition 1.4.2. Let G be a Polish group. A subset A ⊆ G is said to be Roelcke precompact if for all neighbourhoods V ⊆ G of the identity, there is a finite set F ⊆ G such that A ⊆ V F V .
If G is a Roelcke precompact subset of itself, we say that G is Roelcke precompact.

If there is an open Roelcke precompact set A ⊆ G, we say that G is locally Roelcke precompact.

It is not difficult to see that a set

Chapter 2

Model Theory for Metric Structures

This chapter surveys the main theoretical framework of the thesis: metric model theory also known as continuous logic. Continuous logic (in some form or another) was already defined in the 1960's (cf. [START_REF] Chang | Continuous model theory[END_REF]). Back then, the authors allowed any compact Hausdorff space as the set of truth values, which perhaps was too general for a successful theory. The modern version of continuous logic focuses on the special case where only compact intervals of reals are allowed as "truth values". This has the key advantage that two natural quantifiers similar to the classical ones may be defined, namely the supremum and the infimum. This version of continuous logic has been developed over the last 20 years or so with the works of a large group of authors (cf. e.g. [START_REF] Ben | Fraïssé limits of metric structures[END_REF][START_REF] Ben | Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups[END_REF][START_REF] Ben | On d-finiteness in continuous structures[END_REF][START_REF] Ben | Continuous first order logic and local stability[END_REF][START_REF] Ben | Topometric spaces and perturbations of metric structures[END_REF][START_REF] Ben Yaacov | Model theory with applications to algebra and analysis[END_REF][START_REF] Ben Yaacov | Metric Scott analysis[END_REF][START_REF] Farah | Model theory for C * -algebras[END_REF] and the list could go on for a while). The driving force behind the development has been to define a model theory that is better suited than the classical discrete model theory for describing structures appearing naturally in other parts of mathematics, such as e.g. Banach spaces, metrisable groups, measure algebras etc. The classical model theory does not capture very well the somehow continuous nature of these structures. Continuous model theory, on the other hand, has been developed exactly to do just that. Today continuous logic has reached a point where more and more applications to other parts of mathematics have appeared. Furthermore, not only does continuous logic pose a better suited theory for these sort of continuous structures, but it is also strikingly similar to classical model theory. Most of the main results from the classical theory have a continuous counterpart. These include e.g. the Compactness Theorem, the Löwenheim-Skolem Theorem and the Omitting Types Theorem, just to mention a few (cf. [START_REF] Ben Yaacov | Model theory with applications to algebra and analysis[END_REF]). This, together with the many fruitful applications, emphasises that the continuous logic we have today is not only a powerful tool, but also the "good" way to generalise classical model theory to a continuous version.

The main idea behind metric model theory is to use complete metric spaces as the foundation on which structures can be defined. In order for the Compactness Theorem to hold, we have to require the metric to be bounded which 31 can be a little annoying. However, usually it is not hard to work around this requirement as we will see below. We will think of the metric as the generalisation of equality in classical logic, and think of something being "close to the truth" if the distance is close to 0. Thus, 0 plays the role of truth. This might cause a bit of confusion at first, but when working with the theory it will be clear right away why this is the natural choice. On top of the metric space, just as in classical logic, we allow predicates, functions and constants with one crucial requirement: everything must be uniformly continuous. Thus, we think of these objects as being maps from the metric space into either the space itself or into the reals. The first-order language is constructed recursively just as in the classical case, using continuous map as connectives and infimum and supremum as quantifiers. This makes sense, since we think of predicates as maps into e.g. [0, 1]. It might seem peculiar at first, but if one considers the classical connectives as maps into the set of truth values {0, 1}, it becomes clear that the obvious generalisation to the continuous setting is to use continuous maps.

For further details on the basics of continuous logic we refer the reader to the survey [START_REF] Ben Yaacov | Model theory with applications to algebra and analysis[END_REF], where everything is explained in full detail.

Languages and structures

In this first proper section of the chapter, we will develop the formalism and theoretical framework of metric model theory that we will need in the chapters to come. We begin by defining what a metric structure is, before moving on to define the formal language of metric model theory.

Signatures

The first thing we need is a tool that will help us control the uniform continuity of the formulas in our language. Definition 2.1.1. A modulus of uniform continuity is a function

δ : (0, ∞) → (0, ∞).
For metric spaces (X, d X ) and (Y, d Y ), we say that a map f : X → Y respects the modulus δ if for all x, x ∈ X and ε > 0, we have

d X (x, x ) ≤ δ(ε) =⇒ d Y (f (x), f (x )) ≤ ε.
In this case, f is said to be uniformly continuous with respect to δ.

We will often simply write 'δ is a modulus' instead of 'δ is a modulus of uniform continuity'.

With the moduli of uniform continuity established, we proceed to define the alphabets for our languages. These are called metric signatures and they will usually be denoted by an L with various subscripts. As for all alphabets (natural and formal), L is merely a collection of symbols without meaning. It is only later that the symbols will be given meanings, when they are interpreted in a structure. Definition 2.1.2. A metric signature is a countable collection

L = {(P i , n i , δ i , I i ) : i ∈ I} ∪ {(F j , m j , ρ j ) : j ∈ J }
where for each i ∈ I, j ∈ J

• n i , m j ∈ N,

• P i is a predicate symbol of arity n i , • I i is a compact interval of reals, called the bound of P i , • F j is a function symbol of arity m j ,

• δ i and ρ j are moduli of uniform continuity of arities n i and m j , respectively.

Moreover, the signature contains a special binary predicate d (intended for the metric) equipped with the modulus δ d (ε) = ε/2.

Predicates will also be referred to as relations and 0-ary function symbols will be referred to as constant symbols.

Remark 2.1.3. The countability assumption on the signatures is in general not necessary. However, this assumption ensures that the logic topology on the type spaces defined below is metrisable, which will be convenient for some of our arguments. Moreover, we will only consider countable signatures in this thesis, so this is no real restriction for us.

Metric structures

With the signatures defined, we are now ready to explain what a metric structure is exactly. They are defined as follows:

Definition 2.1.4. Let L = {(P i , n i , δ i , I i ) : i ∈ I} ∪ {(F j , m j , ρ j ) : j ∈ J } be a metric signature. A metric L-structure M consists of the following data:

• A bounded complete metric space (M, d M ). M will be referred to as the universe of M.

• A map F M j : M m j → M respecting ρ j for each j ∈ J .

• A function P M i : M n i → I i respecting δ i for each i ∈ I.

For a symbol S ∈ L the map S M is the interpretation of S in M. The special symbol d is always interpreted as the metric d M .

Metric L-structures will also be referred to as L-models or simply structures or models. To ease the notation, we will not discern between the structure M and the underlying metric space M . This will not cause any confusion and it will be perfectly clear from context whether we consider the structure or the metric space.

The boundedness condition on the metric is, as mentioned, a bit of a nuisance. However, the boundedness is important for the theory to work smoothly and to make sure that the type spaces (defined below) are compact. Fortunately, there are ways to work around this requirement. One way is to consider many-sorted structures, where we allow several bounded complete metric spaces as part of the structure together with maps between the sorts. However, many-sorted structure are a little cumbersome to work with, so when dealing with unbounded spaces we have chosen another approach explained below.

The work-around we will implement is the following: if we want to consider an unbounded space as a metric structure, we will add countably many predicates to the language that will encode the unbounded distance up to some bound. In particular, this approach works out nicely when considering the unbounded Urysohn metric space and diversity in Chapters 5 and 6 below.

Before we move on, let us provide a few examples of metric structures to indicate how versatile the concept is.

Examples. The following can be considered as metric structures.

(i) Complete metric spaces. If (M, d M ) is a bounded complete metric space, we can consider it as a structure in the empty signature. If d M is not bounded, we can add predicates to the language as described above.

(ii) Classical first-order structures. As the metric we simply use the discrete metric. Predicates and functions are clearly uniformly continuous. Thus, metric model theory generalises the classical discrete model theory.

(iii) Hilbert spaces over R. They can be considered as many-sorted structures or, equivalently, we can restrict ourselves to the unit ball. The signature is

L Hilb = {0, +, -, •, • , (m λ ) λ∈Q }.
With some care, one can also consider Hilbert spaces over C as metric structures by considering the real and imaginary parts separately.

(iv ) Measure algebras. Suppose (X, µ) is a probability space and let MALG µ denote the corresponding measure algebra, i.e. the measurable sets where A and B are identified if the symmetric difference A B has measure 0. The map d µ (A, B) = µ(A B) defines a metric on MALG µ and therefore MALG µ is a metric structure in the signature L Bool = {0, 1, ∩, ∪, µ}.

Just as in classical logic, we can augment a given signature with constants coming from some given structure. Definition 2.1.5. Let L be a signature and let M be an L-structure. Given a countable subset A ⊆ M we denote by L(A) the signature obtained by augmenting L with constant symbols c a for each element a ∈ A. M is naturally an L(A)-structure simply by setting c M a = a.

We can also talk about embeddings and isomorphisms of structures.

Definition 2.1.6. Let M and N be two L-structures. An embedding of M into N is a map ι : M → N that commutes with the interpretations of all predicate and function symbols of L. That is, if F ∈ L is an n-ary function symbol and P a k-ary predicate symbol, we have ι F M (a 0 , . . . , a n-1 ) = F N ι(a 0 ), . . . , ι(a n-1 )

and

P M b 0 , . . . , b k-1 = P N ι(b 0 ), . . . , ι(b k-1 )
for any a i , b j ∈ M. In particular, ι must be an isometry and is therefore always injective. If there is an embedding ι : M → N , we write M ∼ -→ N and say that M embeds into N .

If M ⊆ N as sets and the inclusion map is an embedding, we say that M is a substructure of N and we write M Ă ∼ N .

An isomorphism is a surjective embedding and we write M ∼ = N when there is an isomorphism between M and N .

An isomorphism from M to itself will be called an automorphism.

Automorphisms will play a significant role for us. Let us therefore fix some notation concerning them. Definition 2.1.7. Let M be an L-structure. The automorphism group of M, denoted Aut(M), is the group of all automorphisms of M.

Since the metric is always included in the signature, automorphisms are in particular isometries. Thus, Aut(M) is a subgroup of Iso(M). Moreover, it is easy to check that Aut(M) is a closed subgroup in the pointwise convergence topology. Hence, Aut(M) is a Polish group if M is separable. In fact, as we shall see in Chapter 4, any Polish group is (isomorphic to) the automorphism group of some separable metric structure.

Syntax

We will now define the language of a given signature consisting of the expressions that we will be able to assign meaning to. Just like for natural languages, only certain combinations of letters make sense as words, and we are only allowed to combine words in certain ways depending on the grammar of the language. As in classical model theory, the language is built recursively creating terms and formulas that we can combine according to certain rules using the symbols of the signature and create new formulas. One of the key differences between continuous and classical logic is that we allow uniformly continuous functions as our connectives instead of the classical first-order connectives. This will allow us to introduce methods and terminology from functional analysis, because the collection of all formulas may be viewed as a Banach algebra of real valued continuous functions on a compact Hausdorff space.

Fix a set of variables V. Since we will only need countably many variables, we can assume that V is countable, but this is not essential. V should formally be considered as a part of any metric signature. Terms are then defined as follows:

Definition 2.1.8. Let L be a metric signature. The class of L-terms is recursively defined as follows:

• All variables of V are L-terms.

• If t 0 , . . . , t n-1 are L-terms and F ∈ L is an n-ary function symbol, then F (t 0 , . . . , t n-1 ) is an L-term.

Notice that the definition above includes constants as 0-ary function symbols. If v = (v 0 , . . . , v n-1 ) are the variables used to define some term t, we will write t(v) to indicate this and say that t depends on v.

We can now use the terms to define the simplest meaningful expressions, namely the atomic formulas. Definition 2.1.9. Let L be a signature. An atomic formula is an expression of the form P (t 0 , . . . , t n-1 ), where P is an n-ary predicate symbol from L and t 0 , . . . , t n-1 are terms. The bound for P (t 0 , . . . , t n-1 ) is the bound for P (determined by L).

The bound for an atomic formula is used in the definition below to define a bound for all formulas. Formulas are defined recursively starting from the atomic formulas using uniformly continuous functions defined on the products of the bounds. Note, however, that since these functions are defined on compact spaces, it is enough to assume them to be merely continuous. Definition 2.1.10. Let L be a signature. The class of L-formulas is defined recursively as follows:

• Every atomic formula is a formula.

• If ϕ 0 , . . . , ϕ n-1 are formulas with bounds I i and f : i I i → R is a continuous function, then f (ϕ 0 , . . . , ϕ n-1 ) is a formula with bound Im(f ). Such functions f are referred to as connectives.

• If ϕ is a formula with bound I and v is a variable, then sup v ϕ and inf v ϕ are formulas with bound I. sup and inf are called quantifiers.

The collection of all L-formulas is denoted L(L) or simply L.

The notions of free and bound variables are defined as in the classical case: an occurrence of a variable v in a formula ϕ is bound if it occurs within the scope of a quantifier, i.e. if ϕ contains a formula of the form inf v ψ or sup v ψ and the occurerrence of v in question is in ψ. If v occurs outside the scope of any quantifier it is called free. A sentence is a formula with no free variables and a collection of sentences is called a theory. As usual, a formula is quantifier free if it is built without using the quantifiers. We will use the notation ϕ(v 0 , . . . , v n-1 ) to denote that the formula ϕ has free variables among v 0 , . . . , v n-1 , and we will say that ϕ depends on v 0 , . . . , v n-1 . Note that a formula only depends on finitely many free variables.

Like in classical logic, we can define the complexity of a term or a formula. This will come in handy, since many arguments and definitions are done by induction or recursion on the complexity of formulas. Definition 2.1.11. The complexity or rank of a term t, denoted rk(t), is defined recursively as follows:

• If t is a variable, then rk(t) = 0. • If t is F (t 1 , . . . , t m ), then rk(t) = max i rk(t i ) + 1.
We can then define the complexity of a formula recursively as follows: Definition 2.1.12. The complexity or rank of a formula ϕ ∈ L(L), denoted rk(ϕ), is defined recursively as follows:

(i) If ϕ is an atomic formula, then rk(ϕ) = 0. (ii) If ϕ is f (ϕ 1 , . . . , ϕ n ) for formulas ϕ j and a connective f , then

rk(ϕ) = max i rk(ϕ i ) + 1. (iii) If ϕ is inf v ψ or sup v ψ, then rk(ϕ) = rk(ψ) + 1.

Semantics

The above establishes the syntax of our language. The semantics are provided by structures. Any term t(v 0 , . . . , v n-1 ) is naturally interpreted in a structure M as a map t M : M n → M. Likewise, any formula ϕ(v 0 , . . . , v n-1 ) is naturally interpreted in M as a map ϕ M : M n → I ϕ , i.e. to the bound of ϕ. Moreover, these maps will respect moduli of uniform continuity that are determined by the signature and therefore independent of the structure. Hence, no matter where we interpret a formula or term, it will always respect the same modulus.

The formal definitions of the interpretations of terms and formulas are given recursively as follows: Definition 2.1.13. Let L be a signature and let M be an L-structure. Let moreover t(v) be an L-term depending on the variables v = (v 0 , . . . , v n-1 ). The map t M : M n → M is defined recursively on the complexity of t as follows:

• If t is the variable v i , then t M (ā) := a i for any ā ∈ M n .

• If t is the term F (t 1 (v), . . . , t m (v)) for a function symbol F , then

t M (ā) := F M (t M 1 (ā), . . . , t M m (ā))
for any ā ∈ M n .

The interpretations of formulas are then defined as follows:

Definition 2.1.14. Let L be a signature and let M be an L-structure. Let moreover ϕ(v) be a formula with free variables among v = (v 0 , . . . , v n-1 ). The map ϕ M : M n → I ϕ is defined by recursion on the complexity of ϕ as follows:

• If ϕ(v) is the atomic formula P (t 1 (v), . . . , t m (v)) for a predicate P and terms t i , then

ϕ M (ā) := P M (t M 1 (ā), . . . , t M m (ā))
for all ā ∈ M n .

• If ϕ(v) is the formula f (ϕ 1 (v), . . . , ϕ m (v)) for a connective f and formulas ϕ i , then ϕ M (ā) := f (ϕ M 1 (ā), . . . , ϕ M m (ā)).

• If ϕ(v) is the formula sup x ψ(v, x) or inf x ψ(v, x), then ϕ M (ā) := sup b∈M ψ M (ā, b) or ϕ M (ā) := inf b∈M ψ M (ā, b),
respectively.

As we have mentioned, the interpretation of any term or formula is uniformly continuous. Moreover, the interpretation will respect a syntactically defined modulus of uniform continuity, i.e. a modulus that only depends on the signature L and not the structures. The proof of this is a simple induction that we leave for the reader. Proposition 2.1.15 ([9, Theorem 3.5]). Let L be a signature. For any L-term t(v) and any L-formula ϕ(v) there are moduli of uniform continuity δ t and δ ϕ , such that for any L-structure M, the map t M will respect δ t and the map ϕ M will respect δ ϕ .

Of course, strictly speaking, we should specify which metric we put on finite powers of some structure M before we can talk about moduli of uniform continuity. The usual convention is to choose the maximum distance, so that is what we will do, but of course any metric inducing the product uniform structure would work as well. We will also need to fix metrics on countably infinite powers of structures. In particular, we will need metrics on M ω and M ω⊕ω (i.e. pairs of infinite sequences) for reasons that will become clear in Chapter 4. As we shall see there, the type space S ω⊕ω (Th(M)) (defined below) contains the Roelcke completion of Aut(M). Since this type space uses the metric on M ω⊕ω , we need to be specific about how d M ω⊕ω is defined. Again, we can choose any metric inducing the product uniformity, but to make things explicit and work smoothly we have to make some choice. The metrics are defined below. Definition 2.1.16. Let M be an L-structure with metric d M . The metric d M n on M n for n ∈ N is defined to be the maximum

d M n (ā, b) = max{d M (a i , b i ) : i < n}.
On M ω , we define the metric d M ω to be the sum

d M ω (ā, b) = i∈N 2 -i-1 d M (a i , b i ). (2.1)
Finally, on M ω⊕ω = {(ā, b) : ā, b ∈ M ω }, we define the metric d M ω⊕ω to be the following:

d M ω⊕ω ((ā, ā ), ( b, b )) = max{d M ω (ā, b), d M ω (ā , b )}. (2.2)
We will for the most part omit the n, ω and ω ⊕ ω subscripts to make the notation more readable.

Observe that the metric d M n can be defined using the L-formula ϕ(x, ȳ) = max{d(x i , y i ) : i < n}, where max is considered as a connective defined on I n d . However, d M ω and d M ω⊕ω cannot directly be defined using a formula, since they depend on infinitely many variables. Fortunately, they can be approximated uniformly by formulas, and hence we will consider them as formulas as well. We will explain this in more details in a subsequent subsection.

Basic model theoretical concepts

In this subsection, we define several basic concepts of metric model theory that we will need throughout the thesis. These concepts are metric generalisations of concepts from classical first-order logic.

The first concept we define is satisfaction of formulas in a structure.

Definition 2.2.1. Let L be a signature and let M be an L-structure. We say that M satisfies the formula ϕ(v) with the tuple ā ∈ M |v| if ϕ M (ā) = 0. This is denoted M ϕ(ā).

In particular, if we are given a sentence σ, then M satisfies σ if σ M = 0. We will denote this by M σ.

The notions of satisfiable and complete theories are then defined as follows:

Definition 2.2.2. We say that a theory T is satisfiable if there is a structure M satisfying all sentences of T . In this case, we say that M is a model of T and write M T .

If M is a structure, then the theory of M is the collection of all sentences that are satisfied by M. The theory of M is denoted by Th(M).

A theory T is said to be complete if there is a structure M such that T = Th(M).

We can also define the concept of elementarity as follows: Definition 2.2.3. Let L be a signature and let M and N be two L-structures. We say that M and N are elementarily equivalent, denoted M ≡ N , if Th(M) = Th(N ).

If M Ă ∼ N , i.e. if M is a substructure of N , and for all formulas ϕ(x) and all tuples ā of elements of M, we have ϕ M (ā) = ϕ N (ā), we say that M is an elementary substructure of N and that N is an elementary extension of M. We denote this by M N .

An elementary embedding of M into N is a map Φ : M → N such that for all formulas ϕ(x) and all tuples ā of elements of M, we have

ϕ M (ā) = ϕ N (Φ(ā)).
If there is such a map M → N , we say that M embeds elementarily into N and write M N .

Note that an isomorphism is automatically an elementary embedding (by induction on complexities), so there is no need to define an "elementary isomorphism".

Another well-known notion from the classical case that has a metric counterpart is the notion of quantifier elimination. As it is so often the case in the metric setting, we need to introduce some ε's in the appropriate places in order to generalise the classical notion. Definition 2.2.4. A theory T admits quantifier elimination if for any formula ϕ(x) and any ε > 0, there is a quantifier free formula ψ(x) such that

sup{|ϕ M (ā) -ψ M (ā)| : M T, ā ∈ M |x| } < ε.
In the next section we will see an equivalent formulation of quantifier elimination that can be easier to verify. However, it uses so-called saturated models and quantifier free types and these concepts have not yet been introduced. We invite the impatient reader to confer Proposition 2.4.5 below.

The last basic concept of this subsection, which we will need in Chapters 5 and 6, is the concept of a prime model. Definition 2.2.5. A model M of a theory T is a prime model if it is elementarily embeddable into any other model of T .

Prime models are of course interesting in their own right. However, we are mainly interested in them because they are examples of so-called atomic models. In fact, a separable model is prime if and only if it is atomic. Atomic models models will play an important role for us in Chapter 4, and they will be defined in Section 2.5.

The Banach algebras of formulas

In this section we introduce notions from functional analysis that will be helpful when dealing with formulas and, more importantly, uniform limits of formulas.

The collection of all formulas L(L) generated by a signature L carries the structure of a commutative unital algebra in a natural way. If we fix some enumeration vω = (v 0 , . . .) of the variables of V (recall we assumed V to be countable), then we may consider the formulas of L(L) as uniformly continuous maps M ω → R for any L-structure M. Since multiplication and addition are continuous maps, they are connectives, and thus we can add and multiply formulas and multiply with real scalars as well. Therefore, L(L) becomes an algebra. Moreover, given any complete theory T , we can define a natural semi-norm on L(L) as follows: Definition 2.3.1. Let L be a metric signature and let L(L) denote the formulas it generates. Let moreover T be a complete theory. Given ϕ(v ω ) ∈ L(L), we define the map

• T : L(L) → [0, ∞) by ϕ(v ω ) T = sup{|ϕ M (ā)| : M T & ā ∈ M ω }.
(2.3)

We will omit the variables from time to time and simply write ϕ T for a formula ϕ. If the theory is empty, or if it is clear from context which theory we are considering, we will just write ϕ .

Since any formula has bounded image, it follows that • T only attains finite values. Moreover, it is easy to see that this map defines a semi-norm. However, there are several formulas such that ϕ T = 0, so • T is not an actual norm. Therefore, we make the following definition: Definition 2.3.2. Two formulas ϕ, ψ ∈ L(L) are said to be logically equivalent with respect to T if ϕ -ψ T = 0.

It is easy to see that logical equivalence is in fact an equivalence relation on L(L). If we let I T denote the equivalence class of the zero-map M ω → R, then I T is an ideal in L(L). Hence, we obtain a normed commutative unital algebra by considering the quotient L(L)/I T with the norm

ϕ • T := ϕ T , (2.4)
where ϕ • denotes the class of ϕ in L(L)/I T .

Definition 2.3.3. The completion of L(L)/I T with respect to the norm • T is called the Banach algebra of L-formulas over T and is denoted F(T ).

Elements of F(T ) are called L-definable predicates or simply formulas.

We denote F(∅) simply by F.

For the most part, we only consider formulas up to logical equivalence. Since two equivalent formulas always define the same map on any structure, we will still write ϕ even if we are actually considering the class ϕ • .

We can now show that the metrics d M ω and d M ω⊕ω defined above, for a structure M, are definable predicates. Proof. The expressions

ϕ n (x, ȳ) = n i=0 2 -i-1 d(x i , y i )
for each n ∈ N are formulas and converge in • to the formula d ω we are looking for. It is clear that the interpretation of d ω in any M is exactly d M ω . Similarly, the expressions

ψ n ((x, x ), (ȳ, ȳ )) = max n i=0 2 -i-1 d(x i , y i ), n i=0 2 -i-1 d(x i , y i )
are formulas and converge in • to the formula we are looking for.

We observe that by [1, Theorem 2.3], F is isometrically isomorphic to the continuous real-valued functions on some compact Hausdorff space S. As we will see in the next section, this space and similar spaces may be viewed as the type spaces of metric model theory. In some sense, this is a better definition of types than the usual one, since it generalises naturally to the infinitary case that we will consider in Chapter 3. However, there is no need to cause too much confusion in this chapter, so we will stick to the classical definition of types in the next section.

It is easy to see that the formulas with free variables among some fixed set I ⊆ V is a subalgebra of L(L). Thus, it follows that everything we have defined so far can be done for this subalgebra. The corresponding Banach algebra will be denoted F I (T ) or F I when T = ∅. Elements of F I (T ) are called I-ary definable predicates or I-ary formulas.

Similarly, it is easy to see that the quantifier free formulas with free variables among I constitute a subalgebra of L(L). The corresponding Banach algebra is denoted QF I (T ). The subalgebra of all quantifier free formulas is denoted QF(T ). Observe that T eliminates quantifiers if and only if QF(T ) = F(T ).

Type spaces

In this section we will define one of the absolute key concepts of the thesis: types. The type of a tuple of elements from a structure tells us exactly what properties are true for the given tuple. Thus, understanding the collection of types is the same as understanding the structure and its model theory. Just as in the classical first-order setting, metric types are simply defined as the set of formulas satisfied by the given tuple. Furthermore, the set of types carries a topology, the logic topology, that turns out to be compact thanks to the Compactness Theorem -again just as in the classical setting. However, unlike the classical setting, the metric type spaces carry another topology induced by the type distance. In the classical case, this distance is always discrete, but in the general metric setting the distance turns out to provide crucial new information about the types. In fact, one of the key virtues of metric model theory is this type distance, and understanding its interplay with the logic topology will be one of our major tools. Types are defined as follows:

Definition 2.4.1. Let v be a tuple of distinct variables (finite or infinite) and let T be a complete L-theory. A v-type over T is a collection p(v) of Lformulas with free variables among v, such that there is an L-structure M T and some tuple ā ∈ M |v| for which we have

p(v) = {ϕ(v) : ϕ M (ā) = 0}.
We say that ā realises p(v) and write ā p(v). We also say that p(v) is the type of ā over T and write p(v) = tp(ā).

The collection of all v-types over T is denoted S v(T ).

The set of v-types realised by a fixed structure M T is denoted

S M v (T ), i.e. S M v (T ) = {p(v) ∈ S v(T ) : ∃ā ∈ M |v| ā p(v)}. A quantifier free v-type over T is a collection of the form {ϕ(v) ∈ QF v(T ) : ∃M T ∃ā ∈ M |v| ϕ M (ā) = 0}.
This set is denoted tp qf (ā) and we say that tp qf (ā) is the quantifier free type of ā over T . The collection of quantifier free types is denoted QS v(T ).

For the most part, we will not really care about the specific variables in a given type. Therefore, we will usually omit the variables and simply write 'p' instead of 'p(v)' and refer to p as a 'type' instead of 'v-type over T '. Likewise, we will write S n (T ) for the collection of types with free variables among some n-tuple of variables and refer to these types as 'n-types' or simply 'types'. All of this terminology is completely standard and should not cause any confusion.

Note that we allow infinitely many free variables as well. Two important cases for us using infinitely many variables are S ω (T ) and S ω⊕ω (T ), where we allow the formulas to have free variables among, respectively, one and two countably infinite tuples of free variables.

Another important observation to make is that the type of a tuple depends on the structure it comes from. That is, if ā is a tuple of elements from a structure M, which is contained in some other structure N , it is not necessarily true that the type of ā in M is the same as the type of ā in N . However, it is easy to see that if M N , i.e. if M is an elementary substructure of N , then the type of ā in M is the same as the type of ā in N . We should therefore actually write tp M (ā) to emphasise that we are considering ā as a tuple in M. However, it will usually be clear from context where a given tuple comes from, so our notation should not cause any confusion.

Next, we observe that there are natural projection maps between the various type spaces. We define them here explicitly for future reference.

π ω (p)(ϕ(x)) = p(ϕ(x, ȳ)) and π ω (p)(ϕ(ȳ)) = p(ϕ(x, ȳ)).
That is, π ω (p) is the type p restricted to formulas with free variables among the first infinite tuple x, and π ω (p) is the type p restricted to formulas with the second infinite tuple ȳ as free variables.

These maps will be referred to as the projection maps between the type spaces.

As always the super-and subscripts will be omitted whenever there is no chance of confusion.

Once the logic topology and type distance are defined (cf. Definitions 2.4.7 and 2.4.10), it will we straightforward to verify that these projections maps are continuous with respect to both topologies.

Like in classical model theory, we can also talk about types over some (countable) subset A ⊆ M T for some given structure M. This is simply defined using the signature L(A), where we have added constant symbols for each element of A, and the theory T A of the L(A)-structure M A based on M. We will denote the type of a tuple b over T A in M A by tp( b | A) to emphasise that we have augmented our language. As in the classical case, saturated structures are structures that realise all types over any subset of certain cardinalities. Since we have restricted ourselves to countable signatures, we can only define ω-saturation. However, it is of course possible to define saturation for any cardinal. Fortunately, we will only need ω-saturation. This is defined as follows:

Definition 2.4.3. A model M of a theory T is ω-saturated if for all finite subsets A ⊆ M and all n ∈ N, we have S M A n (T A ) = S n (T A ), i.e. M realises all n-types over A.
Using this notion of saturation, we can now give an equivalent formulation of quantifier elimination, as promised earlier. The equivalent condition is the so-called back-and-forth property defined as follows:

c ∈ M, it holds that if tp qf (ā) = tp qf ( b), then there is c ∈ N such that tp qf (ā, c) = tp qf ( b, c ).
We then have: Proposition 2.4.5 [START_REF] Ben | Continuous first order logic and local stability[END_REF]Theorem 4.16]). The following are equivalent for a theory T :

(i) T admits quantifier elimination.

(ii) T has the back-and-forth property.

In Chapter 5 and 6 we will use this proposition to show that the theories of the Urysohn metric space and diversity eliminate quantifiers.

Any type p(v) defines a map from the Banach algebra F v(T ) of v-ary formulas, defined in the previous section, to the reals, by letting p(ϕ) be the value ϕ M (ā) for a realisation of p in a structure M T . In fact, this map is a multiplicative linear functional on F v(T ).

Proposition 2.4.6. Let p(v) ∈ S v(T ) be realised by ā in some model M of T . Then the map p : F v(T ) → R defined by p(ϕ(v)) := ϕ M (ā) is a multiplicative linear functional on F v(T ).
Moreover, there is a one-to-one correspondence between multiplicative linear functionals on F v(T ) and types in S v(T ). We could therefore just as well define types in this way. Indeed, this turns out to be the good way to generalise types to infinitary metric model theory, as we will discover in Chapter 3. However, we have chosen to stick to the classical definition in this chapter to not confuse the reader unnecessarily. It is, however, useful in metric model theory to think of types as functionals instead of collections of formulas.

The logic topology

In classical first-order logic, the set S v(T ) is endowed with a topology called the logic topology. As mentioned earlier, in metric model theory we have not only one, but two important topologies on S v(T ). The first topology is the natural generalisation of the topology from the classical setting. It is therefore also called the logic topology. Definition 2.4.7. The logic topology τ v on S v(T ) is the topology generated by the sets

[ϕ < r] = {p ∈ S v(T ) : p(ϕ) < r}
for a formula ϕ(v) and real number r > 0.

We will also denote this topology simply by τ v or τ . It is not hard to see that τ is Hausdorff. Furthermore, if we view a type as a multiplicative linear functional, we see that τ is the topology of pointwise convergence on formulas. Moreover, by the Compactness Theorem for metric structures (cf. [9, Corollary 5.12]), it follows that τ is compact. In fact, the compactness of (S I (T ), τ ) for any set of variables I is just a restatement of this theorem.

Proposition 2.4.8. Let L be a signature and let T be a complete L-theory.

For any set of variables I ⊆ V, the type space (S I (T ), τ ) is a compact Hausdorff space.

Proof. Apply [9, Corollary 5.12].

Another easy fact is that the projection maps between the various type spaces defined above are continuous with respect to the logic topologies. We state it here as a lemma for future reference. The proof is left for the reader.

Lemma 2.4.9. All the projections π n m , π ω m , π ω and π ω from Definition 2.4.2 above are τ -continuous.

The type distance

The other topology on the type spaces is induced by the type distance which is defined below. This distance is a new phenomenon in metric model theory compared to the classical discrete setting. In the classical case, the type distance is always discrete because the structures are viewed as discrete metric spaces. However, in the general metric setting the type distance provides interesting new information on the structures we are dealing with. Definition 2.4.10. Let I be an ordered tuple of variables from V and let T be a complete theory. We define the type distance ∂ I on S I (T ) by

∂ I (p, q) = inf{d M I (ā, b) : M T & ā, b ∈ M I & ā p & b q}.
Of course, the definition above only makes sense once we have chosen some metric on powers of a given structure M. As long as I is countable, this can always be done. The important cases for us will be when I is either finite, a countable tuple or a pair of countable tuples. In all these cases, we have already specified the metrics on the corresponding powers of M in Definition 2.1.16 above. That ∂ is in fact a metric can be shown (once again) using the Compactness Theorem.

Proposition 2.4.11 ([9, Section 8]). ∂ is a metric.

Proof. It is easy to see that ∂ is a pseudo-metric. If p and q are two types, we can use the Compactness Theorem to construct a model realising the type distance ∂(p, q). In particular, if ∂(p, q) = 0, we must have p = q.

Another easy fact is that ∂ induces a finer topology than τ . We will denote this topology by τ ∂ . Moreover, ∂ is lower semi-continuous with respect to τ . Recall that this means that the sets {(p, q) : ∂(p, q) ≤ r} are τ -closed for any r > 0. Finally, ∂ is complete, which essentially follows from the Compactness Theorem (or the compactness of τ to be precise). We collect these easy facts in the following proposition for future reference.

Proposition 2.4.12 ([9, Propositions 8.7 and 8.8], [START_REF] Ben | Continuous first order logic and local stability[END_REF]Section 4.3] and [6, Lemma 1.9]). The type distance is a complete lower semi-continuous metric inducing a finer topology than τ .

In fact, because of these properties, the study of type spaces has motivated the study of so-called topometric spaces in [START_REF] Ben | Lipschitz functions on topometric spaces[END_REF] and [START_REF] Ben | Topometric spaces and perturbations of metric structures[END_REF]. These spaces are defined as follows:

Definition 2.4.13. A topometric space is a triplet (X, τ X , d X )
, where X is a set, τ X is a topology on X and d X is a lower semi-continuous metric inducing a finer topology than τ X .

In Chapter 3 we will need some of the general results concerning topometric spaces, but for now we content ourselves with the definition. It is, however, important to mention here that we will follow the usual convention when dealing with topometric spaces, i.e. all words and concepts coming from topology (open, closed, compact etc) will refer to the topology and all words and concepts from metric space theory (balls, distances etc) will refer to the metric.

We end the section with an easy observation, namely that the projection maps mentioned earlier are uniformly continuous with respect to ∂. The proof is again left for the reader. Lemma 2.4.14. All the projection maps mentioned earlier are ∂-uniformly continuous.

Atomic & homogeneous structures

Two important concepts of classical model theory, that generalise nicely to the metric setting, are atomic and homogeneous structures. In this section, we study these concepts and show that an atomic structure knows the type distance, i.e. that the infimum defining the type distance between two types realised by the structure may be taken over realisations in the structure itself instead of over all models of the given theory. The concept of knowledge of ∂ will be important to us in Chapter 4, when we study the Roelcke completion of a Polish group as a set of types.

Homogeneity of metric structures is defined as follows:

Definition 2.5.1. A metric structure M is approximately homogeneous or simply homogeneous if for any n ∈ N and any two n-tuples ā, b ∈ M n of the same type, we have

Aut(M) • ā = Aut(M) • b.
M is said to be exact homogeneous if for any n ∈ N and ā, b ∈ M n of the same type, we have b ∈ Aut(M) • ā.

If the above holds for all tuples of the same quantifier free type, we say that M is approximately ultrahomogeneous, respectively exact ultrahomogenous.

An atomic model only realises those types that it absolutely must realise. These types are known as the principal or isolated types. Since we have too many topologies around and "isolated" could be misunderstood, we prefer the notion "principal". Principality and atomicity are defined as follows:

Definition 2.5.2. A type p ∈ S I (T ) is principal if the following holds: ∀r > 0 ∃ϕ ∃s > 0 p ∈ [ϕ < s] ⊆ B(p, r).
In other words, p belongs to the τ -interior of any ∂-ball around p.

A model M of T is atomic if all types realised by M are principal.

As mentioned, and just as in the classical setting, principal types are realised everywhere.

Theorem 2.5.3 (Omitting Types Theorem, [START_REF] Ben Yaacov | Model theory with applications to algebra and analysis[END_REF]Theorem 12.6]). Let T be a complete theory in a countable signature and let p ∈ S n (T ). Then the following are equivalent: (i) p is principal.

(ii) p is realised in every model of T .

In the next section we will show a version of this theorem for infinitary continuous logic.

As a corollary to the Omitting Types Theorem, we can show, as promised earlier, that prime models are atomic. Proposition 2.5.4. Let T be a complete theory. Then any prime model M of T is atomic.

Proof. Suppose M is a prime model of T that is not atomic. Then there is a type p realised in M which is non-principal. Therefore, by the Omitting Types Theorem, there is a model N of T omitting p. But since M is prime, it embeds elementarily into N . Therefore, any realisation of p in M is mapped to a realisation of p in N under this embedding. Thus, M must be atomic.

In fact, the other direction is also true: if M is a separable atomic model, then M is prime. However, we will not use this implication. We refer the reader to [14, Proposition 1.18] for a proof.

Another key property of separable atomic models is that they are always homogeneous.

Proposition 2.5.5. Suppose M is a separable atomic metric structure. Then M is approximately homogeneous.

Proof. This follows from a standard back-and-forth argument. See for instance [43, Propositions 6.8 and 6.9].

Another useful lemma that will come in handy later on is the following. It says that in order to check principality of a type, it is enough to check that any ball around it has non-empty τ -interior. Lemma 2.5.6 ([9, Proposition 12.5]). Let T be a complete theory and let n ∈ N. Then p ∈ S n (T ) is principal if and only if for all r > 0, the ball B ∂ (p, r) has non-empty τ -interior.

In order to check whether or not a type is principal, it is moreover enough to check the inclusion of the τ -open neighbourhood in the ∂-ball for a τ -dense subset. More precisely we have the following useful lemma: Lemma 2.5.7. Let T be a complete theory and let p ∈ S I (T ). Suppose that D ⊆ S I (T ) is a τ -dense subset. Then p is principal if and only if for all r > 0 there is a formula ϕ and an s > 0 such that p(ϕ) < s and

[ϕ < s] ∩ D ⊆ B ∂ (p, r).
Proof. If p is principal, it is clear that the condition above is satisfied, so we must show the other direction. Let therefore r > 0 be given and find ϕ and s > 0 such that p(ϕ) < s and

[ϕ < s] ∩ D ⊆ B ∂ (p, r/2).
We claim that [ϕ < s] ⊆ B(p, r). To see this, let q ∈ [ϕ < s]. By density of D, we find a sequence (q n ) of types in [ϕ < s] ∩ D converging in τ to q (recall that our signatures are countable, so τ is metrisable). Therefore, ∂(p, q n ) < r/2 so (p, q n ) is a sequence converging to (p, q) in τ inside the set

{(p 1 , p 2 ) : ∂(p 1 , p 2 ) ≤ r/2},
which is τ -closed by lower semi-continuity of ∂. Hence, ∂(p, q) ≤ r/2 < r, which was what we wanted.

As it turns out, the set of types realised in a model of T is always τ -dense. Thus, to show principality of some type, we can restrict ourselves to the types realised in a model. This will prove very convenient for us in many cases. Proof. Let [ϕ < r] be a non-empty basic τ -open set. Take some q such that q(ϕ) < r. Then the sentence inf x max{0, ϕ(x) -q(ϕ)} must be in T , so it is satisfied by M. Hence, there is some ā ∈ M such that ϕ M (ā) < r and therefore tp(ā) ∈ [ϕ < r] as we wanted.

Knowledge of ∂

The following property will be important for us when we study the Roelcke completion as a set of types in Chapter 4. It is a very natural property saying that a given structure M contains "enough" realisations of the types it realises to calculate the distance between them. In lack of a better term for it, we have chosen to say that M knows the distance. Since it is such a basic property, it might already be defined in the literature, but we have not (yet) stumbled upon it.

Definition 2.5.9. We say that a metric structure M knows the type distance ∂ I if for any p, q ∈ S M I (T ), we have

∂ I (p, q) = inf{d M I (ā, b) : ā, b ∈ M I & ā p & b q}.
The main example of a model that knows ∂ n for each n ∈ N is an atomic model.

Proposition 2.5.10. Any atomic model M of T knows ∂ n for all n ∈ N.

Proof. Let p, q ∈ S M n (T ) and let ā p and b q in M. Let moreover ε > 0 be given and set ∂(p, q) =: s. We will find b0 realising q such that d M (ā, b0 ) ≤ s+ε as required. To do that, we will construct a Cauchy sequence ( bi ) such that tp( bi ) → q in ∂ and such that d M (ā, bi ) ≤ s + ε. It follows that bi converges to a realisation of q close enough to ā.

First we find a formula ϕ 1 and r 1 such that

q ∈ [ϕ 1 < r 1 ] ⊆ B(q, ε/3).
By considering the formula ε 3r 1 ϕ 1 , we may moreover assume that r 1 is actually equal to ε 3 . To find the first element of our Cauchy sequence, we note that

p(inf ȳ max{d(x, ȳ) . -s, |ϕ 1 (ȳ)|}) < ε/3,
where .

denotes the map R 2 → R given by

s . -t = s -t if s ≥ t 0 otherwise.
The inequality essentially follows from the Compactness Theorem, since it allows us to construct a model realising the type distance between p and q. It follows that we can find b1 in M n such that d M (ā, b1 ) < s + ε 3 and such that |ϕ 1 ( b1 )| < ε 3 . Therefore, ∂(q, tp( b1 )) < ε 3 by our choice of ϕ 1 . Set q 1 := tp( b1 ). To find the second element of our Cauchy sequence, we let ϕ 2 be a formula such that

q ∈ [ϕ 2 < ε/3 2 ] ⊆ B(q, ε/3 2 ).
Again, by the Compactness Theorem, it follows that

q 1 (inf ȳ max{d(x, ȳ) . -ε/3, |ϕ 2 (ȳ)|}) < ε/3 2 .
Therefore, we can find b2 ∈ M n such that

d M ( b1 , b2 ) < ε 3 + ε 3 2 and |ϕ 2 ( b2 )| < ε 3 2 . It follows that ∂(q, tp( b2 )) < ε 3 2 . Continuing in this fashion, we obtain a sequence ( bi ) of elements in M n such that d M ( bi , bi+1 ) < ε/3 i + ε/3 i+1
and such that ∂(q, tp( bi )) < ε 3 i . It follows that ( bi ) is a Cauchy sequence converging to a realisation bω of q. Moreover, d M (ā, bi ) ≤ s + ε for all i and thus d M (ā, bω ) ≤ s + ε, which was what we wanted.

By definition, if M is atomic, all types in S M n (T ) are principal. It turns out that the same holds for the types in S M ω (T ) and S M ω⊕ω (T ).

Lemma 2.5.11. Let M be an atomic metric structure with theory T . Then all types in S M ω (T ) and S M ω⊕ω (T ) are principal.

Proof. We will only consider types in S ω (T ), since the argument for types in S M ω⊕ω (T ) is completely analogous. Let p ∈ S ω (T ) be a type realised in M and let r > 0 be given. To show principality of p, it is enough, by Lemma 2.5.7, to find a τ -open set U such that p ∈ U ∩ S M ω (T ) ⊆ B(p, r). Now, for each n, let π n : S ω (T ) → S n (T ) denote the τ -continuous projection map. Let N ∈ N be such that i>N 2 -i < r/2. By atomicity of M, we may find a formula ϕ and r > 0 such that

π N (p) ∈ [ϕ < r ] ⊆ B(π N (p), r/2). Set U := π -1 N ([ϕ < r ]).
Then of course p ∈ U and for any q ∈ U realised in M, we have that ∂(π N (p), π N (q)) < r/2. Moreover, by Proposition 2.5.10, M knows ∂ n for all n. Hence, we can find realisations āN π N (p) and bN π N (q) in M such that d M (ā N , bN ) < r/2. Since q and p are realised in M, we can, by homogeneity of M, find realisations ā p and b q, such that for all

i ≤ N , d M (a i , b i ) < r/2. Since i>N 2 -i < r/2, it follows that d M (ā, b) < r
and therefore ∂(p, q) < r as well.

By the same argument as in Proposition 2.5.10, this implies that any atomic structure knows ∂ ω and ∂ ω⊕ω .

Lemma 2.5.12. Let M be an atomic metric structure. Then M knows ∂ ω and ∂ ω⊕ω .

Proof. The proof from Proposition 2.5.10 carries over by Lemma 2.5.11.

We will need these results in the chapters to come.

The exact homogeneous case

In this subsection we will focus on the exact homogeneous structures. It turns out that atomicity of an exact homogeneous structure is preserved when we add constant symbols to the language. This will follow from the well-known result below due to Effros in [START_REF] Edward | Transformation groups and C*-algebras[END_REF]. First recall that an action G X is transitive if for all x ∈ X we have G • x = X. Theorem 2.5.13 (Effros' Theorem, cf. [66, Theorem 1.1]). Suppose G X is a transitive continuous action of a Polish group G on a separable metrisable space X. Then the following are equivalent:

• X is Polish.

• For every x ∈ X and every 1 G -neighbourhood U ⊆ G the set

U • x := {y ∈ X : ∃g ∈ U y = g • x} is a neighbourhood of x.
We now have: Proposition 2.5.14. Let M be a separable atomic metric structure which is exact homogeneous and let c ∈ M n be a tuple of elements. Then M c is also exact homogeneous and atomic.

Proof. Exact homogeneity of M c is clear.

Let c be a tuple of elements of M and let p ∈ S n (T c) be a type over c realised in M c, say by ā, and let r > 0 be given. Set

U := {g ∈ G : d(ā, gā) < r/2},
where G = Aut(M). Then U is a symmetric open neighbourhood of the identity. Since M is exact homogeneous, all orbits are closed and hence Polish. Therefore, by Effros' theorem, the set

U • c is open inside G • c. Thus, we find δ > 0 such that B(c, δ) ∩ G • c ⊆ U • c. Set δ := min{δ, r/2} and find a formula ϕ(x, ȳ) such that tp(ā, c) ∈ [ϕ < δ ] ⊆ B(tp(ā, c), δ ).
We now claim that p ∈ [ϕ(x, c) < δ ] ⊆ B(p, r), showing that p is principal. To see this, it is enough, by Lemma 2.5.7, to show this inclusion for types realised in M c. Let therefore q be a type over c with q(ϕ) < δ realised in M c, say by b. Then tp( b, c) ∈ B(tp(ā, c), δ ), so since M knows the type distance and is homogeneous, there is g ∈ G such that d(g • ( b, c), (ā, c)) < δ . In particular, gc ∈ B(c, δ), so gc = hc for some h ∈ U . Thus,

h -1 g ∈ Stab(c) = Aut(M c). Moreover, h -1 g • b tp( b | c) = q and d(h -1 g • b, ā) ≤ d(h -1 g • b, h -1 ā) + d(h -1 ā, ā) < δ + r/2 ≤ r.
It follows that ∂(q, p) < r as we wanted.

We conclude that p ∈ [ϕ(x, c) < δ ] ⊆ B ∂c (p, r). Since r > 0 was arbitrary, we conclude that p is principal and therefore that M c is atomic. This proposition will be important for us in Chapter 4, when we study stabiliser subgroups of locally Roelcke precompact automorphism groups of exact homogeneous structures.

Chapter 3 Infinitary Metric Model Theory

In this chapter, we will develop a version of infinitary model theory for metric structures and prove generalisations of several results from finitary metric model theory, i.e. the model theory developed in the previous chapter. Most notably, we prove a version of the Omitting Types Theorem for infinitary metric model theory (Theorem 3.4.2 below). This result is originally due to Eagle [START_REF] Christopher | Omitting types for infinitary [0, 1]-valued logic[END_REF], but our proof here is new. The proof uses the Baire Category Theorem, like Eagle's original proof, but our proof is directly inspired by a proof of the Omitting Types Theorem from classical logic. Therefore, we hope it will be easier to follow than Eagle's proof. Moreover, the precise statement of Theorem 3.4.2, which was suggested by Itaï Ben Yaacov, is slightly more general than Eagle's version. As a corollary to Theorem 3.4.2, we obtain an infinitary version of the Ryll-Nardzewski Theorem describing the ℵ 0 -categorical structures using type spaces. Moreover, we obtain a new useful description of the type distance on infinitary type spaces.

As in the classical case, the main idea is to allow conjunctions and disjunctions over countably infinite sets of formulas, thus gaining expressive power of the language. However, just as in the classical case, the price we pay is the loss of the Compactness Theorem, which has been an essential tool for us so far. Therefore, we must be more careful and creative when we are building the theory and proving results.

Most of the concepts that we have defined so far in Chapter 2 are defined exactly in the same way for infinitary logic. However, there are a few minor differences, mainly due to technical reasons. The main difference comes in the recursive definition of formulas. First, to make some arguments work more smoothly, we want our set of connectives to be countable. Therefore, we restrict ourselves to use polynomials with rational coefficients as connectives. This does not make that much of a difference, since it does not change the definable predicates by the Weierstraß Approximation Theorem. Secondly, and more importantly, we allow the supremum and infimum over countably many formulas respecting the same modulus of continuity and bound. These 55 connectives correspond to countable conjunctions and disjunctions in the classical case and are hence called infinitary connectives. By requiring the formulas to respect the same modulus we ensure that the interpretations of the formulas are uniformly continuous, as they should be.

Infinitary languages

In this section we will explain how the infinitary language is defined. The basic definitions are the same as in Chapter 2, so we will only explain the differences and adjustments one needs to do, to set up the infinitary language properly. As in Chapter 2 there are two parts: syntax and semantics. We begin with the syntax.

Infinitary syntax

Signatures and structures are defined in the same way as for finitary metric model theory. In particular, we stress that a signature is still assumed to be countable. Terms and atomic formulas are also defined in the same way, but general formulas are defined slightly differently. We therefore make the following revision of Definition 2.1.10 describing the recursive construction of formulas. Definition 3.1.1. Let L be a signature. The class of infinitary L-formulas is denoted L ω 1 ω (L) and is defined recursively as follows:

• Atomic formulas are in L ω 1 ω (L).

• If ϕ 0 , . . . , ϕ n are formulas in L ω 1 ω (L) with bounds I i and if f : i I i → R is any polynomial with rational coefficients, then f (ϕ 0 , . . . , ϕ n ) is a formula in L ω 1 ω (L).

• If ϕ is a formula in L ω 1 ω (L), then sup v ϕ and inf v ϕ are formulas in L ω 1 ω (L) too for any variable v.

• If {ϕ n : n ∈ N} is a countable family of formulas in L ω 1 ω (L) with the same bound I and respecting the same modulus δ, then n ϕ n and n ϕ n are formulas in L ω 1 ω (L).

When the signature is insignificant we will allow ourselves to write L ω 1 ω instead of L ω 1 ω (L).

i and i are, as mentioned, referred to as the infinitary connectives and the other connectives are referred to as finitary. The finitary formulas are the formulas we get if we refrain from using the infinitary connectives. The collection of all finitary formulas is denoted L ωω (L) or simply L ωω .

Sentences and free variables are defined as in the finitary case. By ϕ(v) we denote that the formula ϕ has free variables among v. Note that we allow a formula to have infinitely many free variables. However, any formula may be approximated uniformly by formulas with finitely many free variables (cf. Proposition 3.3.1), so this does not change much compared to the finitary case.

We will also need to extend the notion of complexity of formulas to incorporate the infinitary formulas. The definition is a straightforward extension of Definition 2.1.12 above. Definition 3.1.2 (Addendum to Definition 2.1.12). Let {ϕ n : n ∈ N} be a countable collection of formulas respecting the same modulus and bound. The complexity or rank of the formulas n ϕ n and n ϕ n is defined to be sup n rk(ϕ n ) + 1. As before, the complexity of any formula ϕ is denoted rk(ϕ).

Observe that in the infinitary case, the complexity of a formula may be infinite and will, by definition, always be a successor ordinal (or 0).

As mentioned above, we will give a new proof of the Omitting Types Theorem below. In the classical infinitary setting, this theorem is stated only for countable fragments of the language. Hence, we will require the same thing here, although we could just as well allow separable fragments. The countability assumption does, however, simplify some of the arguments. Moreover, we lose no real generality, since the definable predicates do not change.

Definition 3.1.3. A fragment of L ω 1 ω (L) is a countable subset F ⊆ L ω 1 ω (L)
that contains L ωω (L) and that is closed under the following operations, (i) Subformulas, (ii) Substitution of terms for variables, i.e. if ϕ( t) ∈ F and v is a tuple of variables, then ϕ(v) ∈ F , (iii) The finitary continuous connectives of Definition 3.1.1 Moreover, any ϕ ∈ F is required to have only finitely many free variables.

The collection of those formulas of a fragment F that have free variables among some fixed set of variables I ⊆ V will be denoted F I . Remark 3.1.4. Note that since we assume the fragments to be countable, we tacitly have to assume that the signature is countable as well. Otherwise, if we were working with an uncountable signature, there would be no fragments at all. Remark 3.1.5. Recall that L ωω is assumed to be countable since we only use a countable set of connectives. Therefore we can ask a fragment to be both countable and contain L ωω .

If we want to relativise a notion to some fragment F , we will add the pre-fix "F -" to the name of the notion. Thus, for example, an F -theory is a collection of sentences from the fragment F . Likewise, we make the following definition of a complete F -theory. Definition 3.1.6. An F -theory T is complete if there is a model M such that T is the F -theory of M. That is, T is all the F -sentences that are satisfied by M. T is satisfiable if there is a structure M satisfying all sentences of T .

Similarly, the notion of categoricity can be made with respect to a fragment. Definition 3.1.7. Let F be a fragment and let T be an F -theory. We say that T is F -ℵ 0 -categorical if any two separable models of T are isomorphic.

As usual, we will try to avoid the cumbersome "F "-notation as often as possible.

Infinitary semantics

The interpretation of an L ω 1 ω -formula in a structure is defined recursively as in the finitary case. The only thing we need to define here is therefore how the infinitary connectives are interpreted. Definition 3.1.8 (Addendum to Definition 2.1.14). Let M be a structure and let {ϕ n : n ∈ N} be a countable collection of formulas with the same modulus δ and the same bound I. The interpretations of the infinitary formulas n ϕ n and n ϕ n are defined as the maps

n ϕ n M , n ϕ n M : M ω → I given by n ϕ n M (ā) = inf n ϕ M n (ā) and n ϕ n M (ā) = sup n ϕ M n (ā).
With the interpretations of the infinitary formulas established, we can now show that they always give uniformly continuous maps on the structures in which they are interpreted. Proposition 3.1.9. If {ϕ n : n ∈ N} is a collection of formulas with the same modulus δ and bound I, then there is a syntactically defined modulus δ 0 such that for any structure M, ( n ϕ n ) M will respect δ 0 and I. Similarly, there is a syntactically determined modulus and bound for n ϕ n . It follows that all L ω 1 ω -formulas can be assigned a syntactically determined modulus of uniform continuity.

Proof. It is clear that the interpretations of the infinitary formulas must respect the (compact) bound I. Moreover, the argument for the two kinds of infinitary formulas is the same, so we will only provide it for n ϕ n . Furthermore, the conclusion follows from a straightforward argument using induction on the complexity of formulas.

We claim that δ 0 (ε) := δ(ε/3) is a modulus for ( n ϕ n ) M for any structure M. To see this, suppose ε > 0 and that ā and b are sequences from M such that d(ā, b) ≤ δ 0 (ε). Then we can find some N ∈ N such that

| inf n ϕ M n (ā) -ϕ M N (ā)|, | inf n ϕ M n ( b) -ϕ M N ( b)| < ε/3.
It follows that

| inf n ϕ M (ā) -inf n ϕ M n ( b)| ≤ |ϕ M N (ā) -ϕ M N ( b)| + 2ε/3 < ε,
which was what we wanted.

Elementary substructures

We move on to show generalisations of two classical theorems of model theory. These are the downwards Löwenheim-Skolem Theorem and the Tarski-Vaught Test. In order to state them properly, we need to generalise some of the concepts of finitary metric model theory to the infinitary case. What this means more precisely is that we need to make some of the definitions relative to a fragment. First of all, embeddings, substructures, isomorphisms and automorphisms are all defined as in the finitary case (cf. Definition 2.1.6). The notion of elementarity is then defined with respect to some fragment. Suppose therefore that F is a fragment of L ω 1 ω (L). Definition 3.2.1. We say that two structures M and N are F -elementarily equivalent if they have the same F -theory, i.e. if it holds that for any sentence σ ∈ F , we have σ M = σ N . We denote this by M ≡ F N .

If M Ă ∼ N and it holds that for all formulas ϕ ∈ F we have

ϕ M (ā) = ϕ N (ā)
for all tuples ā of elements of M, we say that M is an F -elementary substructure of N and we write M F N . In this case, we will also say that N is an F -elementary extension of M.

We will mostly omit the F sub-and prescripts in the definition above, because we always tacitly assume we are given some fixed fragment.

We now have the following version of the Tarski-Vaught Test for infinitary metric model theory. The proof of it is very similar to the proof in the finitary case, and the reader might want to confer [9, Proposition 4.5]. Proposition 3.2.2 (Tarski-Vaught Test). Let L be a metric signature and let F be a fragment of L ω 1 ω (L). Suppose M and N are L-structures such that M Ă ∼ N . Then the following are equivalent:

(i) M F N
(ii) For all n ∈ N and all n-ary formulas ϕ(x) ∈ F and all ā ∈ M n we have

inf{ϕ N (ā a i =b ) : b ∈ N } = inf{ϕ N (ā a i =b ) : b ∈ M}
for each i ≤ n, where ā a i =b denotes the tuple obtained by replacing a i with b in ā.

Proof. If M N , then (ii) must hold since the equality holds for any formula of F.

Conversely, suppose (ii) holds. We proceed by induction on the complexity of formulas. If ϕ is an atomic formula then the interpretations of ϕ in the two models must coincide on M, since by assumption M Ă ∼ N .

Suppose now that for all formulas ϕ ∈ F with rk(ϕ) ≤ α we have

ϕ N (ā) = ϕ M (ā).
Let ψ be a formula of rank α + 1.

If ψ is f (ψ 0 , . . . , ψ n ) for some formulas ψ i and connective f , it follows by the induction hypothesis that ψ N (ā) = ψ M (ā).

If ψ is n ψ n , then each ψ n has rank less than or equal to α so we see that

ψ N (ā) = inf n ψ N n (ā) = inf n ψ M n (ā) = ψ M (ā).
Similarly, the interpretations must coincide if ψ is n ψ n . Finally, if ψ is inf x i ϕ(x), then by (ii) we directly get that ψ N (ā) = ψ M (ā) for any tuple of elements from M.

We conclude that M F N , which was what we wanted.

Remark 3.2.3. Note that any closed set M ⊆ N satisfying the second condition in the proposition above is automatically a substructure, because given any function symbol F and a tuple ā from M , we have

inf c∈M d N (F N (ā), c) = 0. Thus, F N (ā) ∈ M .
With the Tarski-Vaught Test at our disposal, we can now prove a version of the downwards Löwenheim-Skolem Theorem for infinitary metric model theory. We will only use the separable case, so that is all we will prove here. The proof is more or less the same as in [9, Proposition 7.3], although we have included a few more details. Theorem 3.2.4 (Downward Löwenheim-Skolem Theorem). Let L be a signature and let F be a fragment. Suppose M is an L-structure and that A ⊆ M is a separable subset of M. Then there exists a separable F -elementary substructure N M such that A ⊆ N .

Proof. Let A 0 ⊆ A be a countable dense subset. For ϕ ∈ F , ε ∈ Q ≥0 , i ∈ N and ā a finite tuple of elements of A 0 , let

C ϕ,ε,ā,i = {c ∈ M : ϕ M (ā a i =c ) ≤ ε}. Let B 0 consist of exactly one element of C ϕ,ε,ā,i for each tuple (ϕ, ε, ā, i) such that C ϕ,ε,ā,i is non-empty. Let B 1 = A 0 ∪ B 0 .
We now recursively iterate the above procedure to obtain

B n ⊇ B n-1 such that B n contains one element of each non-empty set C ϕ,ε, b,i , for finite tuples b ∈ B <ω n-1 . Let N denote the closure of B ω := n∈N B n ⊆ M. Note that since B ω is countable, N is separable.
We now claim that N satisfies the Tarski-Vaught Test. Let therefore ε > 0, ā ∈ N n and ϕ(x 0 , . . . , x n-1 ) ∈ F be given. For each i ≤ n -1 we set

r i := inf c∈M {ϕ M (ā a i =c )} + ε/2
By the uniform continuity of ϕ M , we can find a tuple b ∈ (B ω ) n such that for all c ∈ M and i ≤ n -1 we have

|ϕ M (ā a i =c ) -ϕ M ( b b i =c )| < ε/2. Find m ∈ N such that b ∈ (B m ) n . Then for any i ≤ n -1, B m+1 contains an element c i of C ϕ,r i , b,i . Therefore, ϕ M ( b b i =c i ) ≤ r i and thus it follows that inf c∈N {ϕ M (ā a i =c )} ≤ ϕ M (ā a i =c i ) ≤ ϕ M ( b b i =c i ) + ε/2 ≤ r i + ε/2.
Since this holds for any ε > 0, we conclude that N satisfies the Tarski-Vaught Test. It follows that N is in fact an L-structure and moreover A ⊆ N M, which was what we wanted.

Type spaces in infinitary logic

We will now define types and type spaces in infinitary metric model theory. As hinted to a few times, the definition uses the Banach algebras of definable predicates for infinitary metric model theory, so we begin by explaining exactly how they are defined in the infinitary setting. We should, however, first mention that the definition of types presented here is taken from [START_REF] Ben Yaacov | Metric Scott analysis[END_REF].

Banach algebras of infinitary formulas

The construction of the Banach algebras of infinitary formulas or infinitary predicates is more or less identical to what we did in the previous chapter, but we do need to fix some notation.

The semi-norm on the commutative algebra (over Q) of L ω 1 ω -formulas is defined as in Definition 2.3.1 above. Logical equivalence is then defined as in Definition 2.3.2. The associated Banach algebra is denoted F ω 1 ω or, if we consider equivalence with respect to a theory T , F ω 1 ω (T ). We will still write ϕ • if we want to emphasise that we are considering the class of the formula ϕ. Elements of F ω 1 ω (T ) are called L ω 1 ω -definable prediates or, by an abuse of language, simply formulas.

If we make this construction starting with a given fragment F , the resulting algebra is denoted

F F (T ) or, if T is empty, simply F F .
Similarly, if we restrict ourselves to formulas with free variables among some set I ⊆ V, we obtain Banach algebras denoted F ω 1 ω,I (T ) and F F I (T ). To simplify our notation, these algebras will also be denoted F I (T ) whenever it is clear whether we consider a fragment or the full algebra.

We recall that since all these algebras are commutative Banach algebras over R, it follows from [1, Theorem 2.3] that they are isometrically isomorphic to the space of continuous functions on some compact Hausdorff space. It is these spaces that we will use as our type spaces in infinitary metric model theory (cf. Definition 3.3.2 below). However, before moving on, we have the following convenient proposition, saying that the formulas with finitely many free variables are dense in L ω 1 ω (L) and therefore give us the same Banach algebras.

Proposition 3.3.1. Let L be a signature and T a complete theory. The formulas with finitely many free variables are dense inside L ω 1 ω (L) with respect to the semi-norm • T . Hence, they give rise to the same Banach algebra over R.

Proof. As we mentioned, the proof is done by transfinite induction on the complexity of formulas. We will show that any formula may be approximated in • T by formulas with finitely many free variables.

Any formula of complexity 0 must be atomic and hence it only contains finitely many free variables by definition.

Suppose now we can approximate any formula of complexity less than or equal to the ordinal α with a formula of finitely many free variables respecting the same modulus of continuity. Let ϕ be a formula with rk(ϕ) = α + 1 and let ε > 0 be given. We must find a formula ϕ 0 with finitely many free variables that respects the same modulus as ϕ and such that ϕ -ϕ 0 ≤ ε.

If ϕ is f (ϕ 1 , . . . , ϕ n ) for a polynomial f , then it simply follows from the (uniform) continuity of f and our induction hypothesis that we may find ϕ 0 with finitely many free variables such that ϕ -ϕ 0 ≤ ε. Moreover it is easy to check that ϕ 0 can be chosen to respect the same modulus as ϕ.

If ϕ is sup x i ψ(x), then by the induction hypothesis, we find ψ 0 with finitely many free variables that respects the modulus of ψ and such that ψ-ψ 0 ≤ ε. This means that

sup{|ψ M (ā) -ψ M 0 (ā n)| : M is a model, ā ∈ M ω } ≤ ε,
where ā n := (a 0 , . . . , a n-1 ). We claim that this implies

sup{| sup b∈M ψ M (ā a i =b ) -sup c∈M ψ M 0 ((ā n) a i =c )| : M, ā ∈ M ω } ≤ ε, (3.1)
where ā a i =b denotes the sequence ā with b on the i'th coordinate. To see this, let M be a model and ā ∈ M ω . We want to show that

| sup b∈M ψ M (ā a i =b ) -sup c∈M ψ M 0 ((ā n) a i =c )| ≤ ε.
We may suppose that one of the supremums is greater than the other, say

sup b∈M ψ M (ā a i =b ) ≥ sup c∈M ψ M 0 ((ā n) a i =c ).
For any ε > 0 we may find b 0 such that

sup b ψ M (ā a i =b ) ≤ ψ M (ā a i =b 0 ) + ε . Therefore, we obtain | sup b∈M ψ M (ā a i =b ) -sup c∈M ψ M 0 ((ā n) a i =c )| ≤ ψ M (ā a i =b 0 ) + ε -sup c∈M ψ M 0 ((ā n) a i =c ) ≤ ψ M (ā a i =b 0 ) + ε -ψ M 0 ((ā n) a i =b 0 ) ≤ sup b∈M ω {|ψ M ( b) -ψ M 0 ( b n)|} + ε ≤ ε + ε .
Since ε was arbitrary, we conclude the inequality holds for ε = 0 as well. Since M and ā ∈ M ω were arbitrary, we conclude that (3.1) holds. Finally we note that ϕ respects the same modulus as ψ. Hence ϕ 0 := sup x i ψ 0 is a formula with finitely many free variables that respects the same modulus as ϕ and such that ϕ -ϕ 0 ≤ ε as we wanted. If ϕ is inf x i ψ(x), we use that ϕ is logically equivalent to sup x i -ψ(x) and make the same calculations as above.

Assume next that ϕ is n ϕ n (x), where all ϕ n respect the same modulus δ and bound. Since each ϕ n has rank less than α, we may find ϕ 0 n with finitely many free variables respecting the same modulus as ϕ n and such that ϕ n -ϕ 0 n T < ε. We wish to form their infinitary conjunction to obtain the formula we are looking for. However, the number of free variables they depend on may be unbounded. Therefore, we find some N ∈ N such that for any model M T and any two sequences ā, b ∈ M ω we have

d M ω (ā, b) ≤ N i=0 2 -i-1 d M (a i , b i ) + δ(ε), (3.2)
where d M ω is the metric on M ω defined in Definition 2.1.16. Let now ψ n be the formula inf x i ,i>N ϕ 0 n (x). Then each ψ n has at most the first N + 1 variables as free variables. Moreover, they respect the same modulus δ as the ϕ n 's. Furthermore, for any M T and any ā ∈ M ω we have that

|(ϕ 0 n ) M (ā) -ψ M n (ā)| ≤ ε
because of (3.2) above. Therefore, we are allowed to form the formula n ψ n . Denote this formula by ψ. Since all the ψ n 's respect δ it follows that ϕ and ψ respect the same modulus as well. Furthermore, we claim that ϕ -ψ T ≤ 3ε, which of course is enough for what we want. Let therefore M T and let ā ∈ M ω be given. Then we can find an n 0 ∈ N such that

|ϕ M (ā) -ψ M (ā)| = | inf n ϕ M n (ā) -inf m ψ M m (ā)| ≤ |ϕ M n 0 (ā) -ψ M n 0 (ā)| + ε ≤ |ϕ M n 0 (ā) -(ϕ 0 n 0 ) M (ā)| + |(ϕ 0 n 0 ) M (ā) -ψ M n 0 (ā)| + ε ≤ ε + ε + ε,
which was what we wanted.

Finally, a similar argument works if ϕ is ϕ n .

All in all, we conclude that the formulas with finitely many free variables are dense in L ω 1 ω (L).

Infinitary types

We are now ready to define the types of our infinitary metric model theory. Again, we would like to stress that this defintion is taken from [START_REF] Ben Yaacov | Metric Scott analysis[END_REF]. We assume throughout that we are given a fragment F , so that everything below is done with respect to this fragment. Definition 3.3.2 ([11, Section 7]). Let T be a complete F -theory and let v be a tuple of distinct variables. A v-type over T (or simply a type) is a multiplicative bounded linear functional on F F ,v (T ) preserving the unit. We equip the set of types with the pointwise convergence topology it inherits as a subspace of the dual space. This topology is denoted τ v or simply τ and is called the logic topology. We will denote the resulting space by S v(T ).

Again, the specific variables will most of the time not be important. Hence, we will often use the notation S n (T ) for the type space associated to some ntuple of variables. By S ω (T ), we denote the type space associated to some infinite tuple of variables.

Remark 3.3.3. We note that if we want to augment our signature L with new constant symbols from a given subset of a structure M, we may define types with respect to this augmented language just as in the finitary case. In this light, the types defined above may be considered as types over the empty set. However, since it will not be necessary for us to consider types over given sets in this chapter, we will stick to the types defined above and not indulge ourselves in such generalisations.

Since S n (T ) is a closed subset of the unit ball of the dual space, it is a compact space by Alaoglu's Theorem (cf. [41, Theorem 1.6.5]). Furthermore, we note that τ is generated by the sets

[ϕ < r] := {p ∈ S n (T ) : p(ϕ) < r}.
This is similar to the finitary setting described in Chapter 2.

The somewhat artificial -notation in the definition of the type space is used because we want to denote the subset of the so-called realisable types by S v(T ). These types are defined in the usual way. That is, a type p ∈ S v(T ) is realisable if there is a structure M T and a tuple ā ∈ M |v| such for any ϕ ∈ F v(T ) we have p(ϕ) = ϕ M (ā). In this case, we say that ā realises the type p and write ā p. As usual, the type of a given tuple ā is denoted tp(ā). As mentioned, the subspace of all realisable types in a given type space S v(T ) will be denoted by S v(T ) or just S n (T ). We leave it as an exercise for the reader to show that there are types that cannot be realised anywhere, so that in general we have S n (T ) S n (T ). This essentially follows from the failure of the Compactness Theorem in infinitary logic. However, we do have the following proposition:

Proposition 3.3.4. The space of realisable types S v(T ) is τ -dense in S v(T ).
Proof. Let ϕ(v) be a formula and let r > 0. We must show that there is a realisable type in [ϕ < r] whenever this set is non-empty. Suppose not and that no type in [ϕ < r] is realised. We claim that then the set is empty. If not, there is some p such that p(ϕ) < r. Moreover, since no type in [ϕ < r] is realised, we must have that for any model M T and any tuple ā ∈ M |v| we have ϕ M (ā) ≥ r (otherwise such a tuple ā would realise a type in [ϕ < r]). Let ε > 0 be such that p(ϕ) < r -ε and consider the formula ψ = ϕ-(r -ε). Since p is multiplicative, p(ψ) -ψ = p(ϕ) -ϕ is not invertible in F v(T ). However, for any structure M and any ā ∈ M |v| we have

(p(ϕ) -ϕ) M (ā) = p(ϕ) -ϕ M (ā) < r -ε -r = -ε < 0.
This implies that p(ψ) -ψ is invertible -a contradiction.

We will see below that S v(T ) is, in fact, a Polish space, so in particular it is a G δ subset of S v(T ).

The Gelfand transform from functional analysis allows us to identify F v(T ) with the commutative Banach algebra over R C( S v(T )) of continuous realvalued functions on the type space equipped with the uniform norm. Recall that the Gelfand transform Γ is defined by Γ(ϕ)(p) = p(ϕ). Since we are working with real valued Banach algebras, it is not a priori the case that Γ is an isometric isomorphism. However, in our case this is not hard to verify. Proposition 3.3.5. Let v be any tuple of distinct variables (finite or infinite) and let T be a complete F -theory. Then the Gelfand transform

Γ : F v(T ) → C( S v(T ))
is an isometric isomorphism of Banach algebras.

Proof. It is clear that Γ is a homomorphism. To see that Γ preserves the norm, note that

Γ(ϕ) := sup{|p(ϕ)| : p ∈ S v(T )} = sup{|p(ϕ)| : p ∈ S v(T )},
because the realisable types are dense in S v(T ) with respect to τ . Moreover,

sup{|p(ϕ)| : p ∈ S v(T )} = sup{|ϕ M (ā)| : M T, ā ∈ M |v| } = ϕ .
Thus, Γ is an isometry and therefore injective as well. Furthermore, the image is closed, contains the constant maps and separates points, so by the Stone-Weierstraß Theorem (cf. e.g. [26, Theorem 16, IV.6.15, p. 272]), Γ is surjective as well.

In the light of this proposition, we will simply identify F v(T ) and C( S v(T )). Hence, we will write ϕ(p) instead of Γ(ϕ)(p) for a formula ϕ and type p.

Connection to the usual definition

We will now explain exactly how the definition of types presented here and the usual definition relates. In finitary logic, the usual definition of a type is, that it is a set of formulas of the form

{ϕ ∈ F v(T ) : M ϕ(ā)} = {ϕ : ϕ M (ā) = 0}
for a given model M of T and a fixed tuple ā ∈ M |v| (cf. Definition 2.4.1 from the previous chapter). It is not hard to see that a set of this form is a maximal ideal in F v(T ). From the theory of Gelfand duality, we know that maximal ideals correspond to kernels of bounded multiplicative linear functionals, and this is the connection to our definition of types. Given a set of the form above, the tuple ā gives us, of course, a multiplicative linear functional with {ϕ : ϕ M (ā) = 0} as its kernel. Conversely, given a multiplicative linear functional p, we may, by the Compactness Theorem, find some realisation of ā p. Hence, the correspondence that maps sets of the form above to the multiplicative functional it defines, is a bijective correspondence between types and maximal ideals of F v(T ).

Unfortunately, in the infinitary case we are working with, the Compactness Theorem fails. It follows that there are types that cannot be realised anywhere. However, the correspondence between maximal ideals in F v(T ) and elements of S v(T ) still holds. This follows essentially from the Gelfand-Mazur Theorem for algebras over R. The theorem states that any real normed division algebra, i.e. an algebra where every element except 0 is invertible, is isomorphic to R, C or the quarternions H (cf. for instance [START_REF] Bonsall | Complete normed algebras[END_REF]§14 Theorem 7]). Using this theorem, we have the following proposition: Proposition 3.3.6. There is a bijection between maximal ideals of F v(T ) and the type space.

Proof. It is clear that the map taking a type to its kernel is an injective map into the set of maximal ideals. We need to show this map is surjective.

Let therefore I ≤ F v(T ) be a maximal ideal. Then F v(T )/I is a normed division algebra over the reals. By the Gelfand-Mazur Theorem, it follows that this quotient is isomorphic to R, C or H. If F v(T )/I is isomorphic to either C or H, we find some ϕ ∈ F v(T ) and ψ ∈ I such that ϕ 2 -ψ = -1. Thus, given a model M and a tuple ā ∈ M |v| , we have

ψ M (ā) = (ϕ M (ā)) 2 + 1 ≥ 1 > 0.
Therefore, ψ is invertible, so I = F v(T ). This is a contradiction, since a maximal ideal must be proper. We conclude that F v(T )/I ∼ = R.

We now define a type p ∈ S v(T ) by letting p(ϕ) be the real number corresponding to ϕ + I under the isomorphism obtained above. It is clear that the kernel of p is exactly I, so p is mapped to I under our correspondence. Hence, this correspondence is surjective as well.

Following these considerations, it is therefore natural to consider types as multiplicative linear functionals instead of certain sets of formulas -also in finitary metric model theory. The key advantage of this approach is that it generalises nicely to other more general logics where the Compactness Theorem fails, such as the L ω 1 ω -logic we are working with here.

The infinitary type distance

We move on to define a distance on the spaces of types. We will need this for the formulation of the Omitting Types Theorem. Recall that in finitary continuous logic we define the distance between two types to be

∂ F IN (p, q) = inf{d M (ā, b) : M T, ā, b ∈ M n , ā p and b q}.
Unfortunately, this definition does not work in infinitary fragments, since one of the types might not be realisable or there might not be one model realising both types, due to the failure of the Compactness Theorem. Nevertheless, we can still define a type distance. Recall that in the definition below .

denotes the map R 2 → R given by [START_REF] Ben Yaacov | Metric Scott analysis[END_REF]Section 7]). Let F be a fragment and v be a tuple of variables. Let moreover T be a complete F -theory. On S v(T ) we define the type distance ∂ Fv by

s . -t = s -t if s ≥ t 0 otherwise. Definition 3.3.7 ([
∂ Fv (p, q) ≤ s ⇐⇒ ∀ϕ ∈ F v q inf ȳ max{(d(v, ȳ) . -s), |ϕ(ȳ) -p(ϕ)|} = 0.
The F v-subscript will mostly be omitted, since we will only be dealing with one fixed fragment. Note also that the formulas above are (by our usual abuse of notation) considered as elements of the algebra F v(T ), and hence it makes sense to use any continuous function and not just polynomials as connectives. Thus, the absolute value and .

make sense (alternatively they could simply be allowed as connectives without changing our countability assumption). We would also like to note here that if two types p, q ∈ S v(T ) are realised in the same model by ā, b ∈ M |v| , respectively, then ∂(p, q) ≤ d M (ā, b) just as in the finitary setting. To see this, we simply observe that for any formula ϕ, if we let s be equal to d M (ā, b) in the above definition, the infimum is obviously 0 when we evaluate in b (by plugging in ā in the place of ȳ). It should also be noted that if our given fragment F is just L ωω , then by the Compactness Theorem, ∂ is equal to the usual type distance.

The above definition is, as noted, taken from [START_REF] Ben Yaacov | Metric Scott analysis[END_REF]. However, the authors there only define ∂ on the space of realisable types. We claim that ∂ is in fact a metric on all of S v(T ). Moreover, we claim that ( S v(T ), τ, ∂) is a topometric space (cf. Definition 2.4.13).

Proposition 3.3.8. ( S v(T ), τ, ∂) is a topometric space for any tuple of variables v.

Proof. We will first check that ∂ is in fact a metric. Below we will denote inf ȳ max{d(v, ȳ) .

-r, |ϕ(ȳ) -p(ϕ)|} by ψ p,r,ϕ (v) for a type p, r ∈ R and formula ϕ. As usual, the distance symbol d is interpreted according to which power of M we are working with, as defined in Definition 2.1.16 above. We begin by noting that for any realisable type q 0 , we have q 0 (ψ p,r,ϕ ) ≥ 0, since we take the infimum over positive numbers. Therefore, by τ -density of the realisable types, we have that q(ψ p,r,ϕ ) ≥ 0 for all types q.

We will now verify that ∂ is a metric.

∂(p, q) ≥ 0 for all p, q.

Suppose not, so that ∂(p, q) ≤ r < 0 for some types p and q. Then q(ψ p,r,ϕ ) = 0 for every ϕ. Fix some ϕ and let 0 < ε < |r|. By τ -density, we can find a realisable type q 0 such that |q 0 (ψ p,r,ϕ ) -q(ψ p,r,ϕ )| = q 0 (ψ p,r,ϕ ) < ε.

Suppose ā ∈ M |v| realises q 0 . Then

q 0 (ψ p,r,ϕ ) = inf b∈M n max{d M (ā, b) . -r, |ϕ M ( b) -p(ϕ)|} < ε, so we may find some b ∈ M n such that |r| ≤ d M (ā, b) + |r| = d M (ā, b) . -r < ε < |r|,
which is a contradiction.

∂(p, q) = 0 ⇐⇒ p = q.
First we show ∂(p, p) = 0. To do this, we must show p(ψ p,0,ϕ ) = 0 for every ϕ. Thus, we let ϕ be a formula and ε > 0. By τ -density of the realisable types, we may find some p 0 ∈ S v(T ) such that both |p 0 (ϕ) -p(ϕ)| < ε/2 and |p 0 (ψ p,0,ϕ ) -p(ψ p,0,ϕ )| < ε/2. Suppose ā p 0 in M. Then

0 ≤ p 0 (ψ p,0,ϕ ) ≤ |ϕ M (ā) -p(ϕ)| < ε/2.
Therefore, p(ψ p,0,ϕ ) < p 0 (ψ p,0,ϕ ) + ε/2 < ε and since this holds for any ε > 0, we must have p(ψ p,0,ϕ ) = 0. Conversely, suppose ∂(p, q) = 0. We show that for all v-ary formulas ϕ and ε > 0 we have |p(ϕ) -q(ϕ)| < ε, which of course implies p = q.

Let therefore ϕ and ε be given. Since ϕ respects its modulus of continuity, we can find a δ > 0 such that for any model M T and ā, b

∈ M |v| we have that if d M (ā, b) < δ, then |ϕ M (ā) -ϕ M ( b)| < ε/3.
Next, we let q 0 be a realisable type such that q 0 (ψ p,0,ϕ ) < min{ε/3, δ} =: ε and |q 0

(ϕ)-q(ϕ)| < ε . Let ā q 0 in M T . Then we find b ∈ M |v| such that d M (ā, b) < ε and |ϕ M ( b) -p(ϕ)| < ε . We now have that |p(ϕ) -q(ϕ)| ≤ |p(ϕ) -ϕ M ( b)| + |ϕ M ( b) -q 0 (ϕ)| + |q 0 (ϕ) -q(ϕ)| < ε.
We conclude that p = q.

Symmetry.

Suppose towards a contradiction that symmetry fails for some p, q so that ∂(p, q) ≤ s but ∂(q, p) > s. This means that for some ϕ, p(ψ q,s,ϕ ) =: r > 0.

We may find q 0 ∈ S v(T ) such that |q 0 (ϕ) -q(ϕ)| < r/2 and q 0 (inf ȳ max{d(v, ȳ) .

-s, |ψ q,s,ϕ (ȳ) -p(ψ q,s,ϕ )|}) < r/2.

Let ā q 0 in M. Then we may find b ∈ M |v| such that d(ā, b) .

-s < r/2 and |ψ M q,s,ϕ ( b) -p(ψ q,s,ϕ )| < r/2. The latter implies

ψ M q,s,ϕ ( b) > p(ψ q,s,ϕ ) -r/2 = r/2.
On the other hand, we have

ψ M q,s,ϕ ( b) ≤ max{d M (ā, b) . -s, |ϕ M (ā) -q(ϕ)|} < r/2,
which is a contradiction.

Triangle inequality.

The argument is very similar to the other arguments. Since we must show something for each formula ϕ, we will approximate a given type on finitely many formulas arbitrarily with a realisable type. Since the triangle inequality is true for the realisable ones, it will hold for the non-realisable types as well.

The details are as follows: suppose that ∂(p 1 , p 2 ) ≤ s and ∂(p 2 , p 3 ) ≤ r. We must show that ∂(p 1 , p 3 ) ≤ s + r. Let ϕ be a v-ary formula. Then we know that

p 2 (inf ȳ max{d(v, ȳ) . -s, |ϕ(ȳ) -p 1 (ϕ)|}) = 0.
This implies that

p 3 inf ȳ max d(v, ȳ) . -r, inf z max{d(z, ȳ) . -s, |ϕ(z) -p 1 (ϕ)|} = 0 (3.3)
by using the definition of ∂(p 2 , p 3 ) ≤ r with the formula

inf z max{d(z, ȳ) . -s, |ϕ(z) -p 1 (ϕ)|},
which is mapped to 0 by p 2 . Of course, strictly speaking, the type p 2 is only defined for formulas with free variables among v, but we may naturally view it as defined on formulas with free variables among ȳ as well. Hence, we may apply p 2 to the formula above simply by setting p 2 (ψ(ȳ)) := p 2 (ψ(v)). We now claim that (3.3) implies

p 3 inf z max{d(v, z) . -(s + r), |ϕ(z) -p 1 (ϕ)|} = 0,
why it follows that ∂(p 1 , p 3 ) ≤ s + r. We will show the claim, but first, to ease the notation a bit, put

Ψ 1 (v) := inf ȳ max d(v, ȳ) . -r, inf z max{d(z, ȳ) . -s, |ϕ(z) -p 1 (ϕ)|} and Ψ 2 (v) := inf z max{d(v, z) . -(s + r), |ϕ(z) -p 1 (ϕ)|} .
To show the claim, let ε > 0 and let p 0 3 be a realisable type such that

|p 3 (Ψ 1 ) -p 0 3 (Ψ 1 )| = p 0 3 (Ψ 1 ) < ε and |p 3 (Ψ 2 ) -p 0 3 (Ψ 2 )| < ε. Suppose that ā p 0 3 in M T . Then we may find b, c ∈ M |v| such that d M (ā, b) < r + ε, d M ( b, c) < s + ε and |ϕ M (c) -p 1 (ϕ)| < ε.
Therefore, d M (ā, c) .

-(r + s) < 2ε, so p 0 3 (Ψ 2 ) < 2ε. We conclude that p 3 (Ψ 2 ) < 3ε. Since ε was arbitrary and since p 3 (Ψ 2 ) ≥ 0, we must have p 3 (Ψ 2 ) = 0. Since this holds for any v-ary formula ϕ, we conclude that ∂(p 1 , p 3 ) ≤ s + r.

All in all we conclude that ∂ is a metric on S v(T ).

We move on to show that ∂ is lower semi-continuous and that it refines the topology.

Lower semi-continuity.

By definition, we must show that {(p, q) ∈ S v(T ) 2 : ∂(p, q) ≤ r} is τ -closed for any r ∈ R. Of course, this set is empty for r < 0 and it is the diagonal for r = 0. Therefore, the set is closed in both cases.

Let r > 0. Let (p n , q n ) → (p, q) pointwise with ∂(p n , q n ) ≤ r. We must show that ∂(p, q) ≤ r as well. Thus, we must show

q(inf ȳ max{d(v, ȳ) . -r, |ϕ(ȳ) -p(ϕ)|}) = 0
for all v-ary formulas ϕ. Let therefore ϕ be a v-ary formula and put

Ψ(v) := inf ȳ max{d(v, ȳ) . -r, |ϕ(ȳ) -p(ϕ)|} and Ψ n (v) := inf ȳ max{d(v, ȳ) . -r, |ϕ(ȳ) -p n (ϕ)|}.
We must show that q(Ψ) = 0. First, we note that q(Ψ) ≥ 0 always holds because this inequality must hold for a realisable type and hence it must hold for all types by τ -density. To show equality, we will show q(Ψ) < ε for all ε > 0.

Since p n (ϕ) → p(ϕ), we have Ψ n → Ψ. Moreover, q n (Ψ) → q(Ψ) since q n → q in τ . Thus, we may find some N such that Ψ -

Ψ N < ε/2 and |q N (Ψ) -q(Ψ)| < ε/2. Then q N (Ψ) = |q N (Ψ) -q N (Ψ N )| ≤ Ψ -Ψ N < ε/2,
and hence q(Ψ) < ε as required.

M (ā, b) < δ we have |ϕ M (ā) -ϕ M ( b)| < ε/3. Let ε := min{ε/3, δ/2}. We claim that B ∂ (p, ε ) ⊆ [ϕ < r].
Thus, we let q ∈ B ∂ (p, ε ) and must show q(ϕ) < r. We may find a realisable type q 0 such that q 0 (inf ȳ max{d(v, ȳ) .

-ε , |ϕ(ȳ) -p(ϕ)|}) < ε and |q 0 (ϕ) -q(ϕ)| < ε .

Let ā q 0 in some model M T . Then we may find b ∈ M such that

max{d M (ā, b) . -ε , |ϕ M ( b) -p(ϕ)|} < ε .
We now have

|p(ϕ) -q(ϕ)| ≤ |q(ϕ) -q 0 (ϕ)| + |q 0 (ϕ) -ϕ M ( b)| + |p(ϕ) -ϕ M ( b)| < ε/3 + ε/3 + ε/3 = ε, where |p(ϕ) -ϕ M ( b)| < ε/3 follows from d M (ā, b) . -ε < ε , since this implies d M (ā, b) < δ. Thus, q(ϕ) < p(ϕ) + ε < r. We conclude that B ∂ (p, ε ) is contained in [ϕ < r]. Therefore, [ϕ < r] is ∂-open.
All in all we conclude that ∂ is a lower semi-continuous metric on S v(T ) refining τ and hence that S v(T ) is a topometric space.

The topology that the metric ∂ induces on S v(T ) will be denoted τ ∂ .

As a corollary to the proposition above, we will obtain another way to compute the distance between two types. In order to explain this properly, we will need to introduce some terminology. The first definition deals with so-called normal topometric spaces. Definition 3.3.9. We say that a topometric space (X, τ X , d X ) is normal if the following two conditions are satisfied:

(i) Any two τ X -closed subsets F, G ⊆ X with positive distance d X (F, G) > 0
may be separated by disjoint τ X -open subsets. (ii) For any τ X -closed set F and any r > 0, the closed r-ball around F , B(F, r) := {x :

d X (x, F ) ≤ r}, is τ X -closed.
If (ii) above holds, X is said to have closed metric neighbourhoods.

The following lemma follows easily from [6, Lemma 1.8]

Lemma 3.3.10 (Essentially [6, Lemma 1.8]). Let (X, τ, ∂) be a topometric space. If τ is compact, then (X, τ, ∂) is normal.
As a direct corollary, we obtain the following:

Corollary 3.3.11. S v(T ) is a normal topometric space for any v.

Proof. Since we have already shown that ( S v(T ), τ, ∂) is a topometric space, the corollary follows directly from the lemma above.

This allows us to exploit some of the results of [START_REF] Ben | Lipschitz functions on topometric spaces[END_REF] for normal topometric spaces.

In particular, we will use the following topometric version of Urysohn's lemma.

Theorem 3.3.12 (Urysohn's Lemma, [START_REF] Ben | Lipschitz functions on topometric spaces[END_REF]Theorem 1.6]). Let (X, τ, ∂) be a normal topometric space and suppose F, G ⊆ X are τ -closed with d(F, G) > 0.

Then for any r ∈]0, d(F, G)[ there is a ∂-1-Lipschitz τ -continuous function f : X → [0, r] such that f is 0 on F and r on G.

We will use this result to show that the type distance can be computed as a supremum over the so-called 1-Lip formulas. These are defined below. First, recall that the algebra of formulas F v(T ) may be identified with the real valued continuous functions on ( S v(T ), τ ). Hence, we may use the two notations ϕ(p) and p(ϕ) interchangeably. Both notations will be convenient below.

Definition 3.3.13. A formula ϕ ∈ F v(T ) is k-Lip for k > 0 if for any model M T , we have |ϕ M (ā) -ϕ M ( b)| ≤ kd M (ā, b) for all ā, b ∈ M |v| .
That is, ϕ is always interpreted as a k-Lipschitz map.

We now have the following convenient description of the type distance:

Theorem 3.3.14. The type distance ∂ Fv on S v(T ) can be computed as the supremum over 1-Lip formulas. I.e.,

∂ Fv (p, q) = sup ϕ {|p(ϕ) -q(ϕ)| : ϕ is a 1-Lip formula in F v(T )} (3.4)
Proof. Suppose first that ∂(p, q) ≤ s. We must show that the right-hand side in (3.4) above is less than s as well.

Let ϕ be a 1-Lip formula and ε > 0. We will show that |ϕ(p)-ϕ(q)| ≤ s+ε, which is sufficient to obtain the desired inequality.

Let q 0 ∈ S v(T ) be a realisable type such that

q 0 (inf ȳ max{d(v, ȳ) . -s, |ϕ(p) -ϕ(ȳ)|}) < ε/3
and such that |q 0 (ϕ) -q(ϕ)| < ε/3. Suppose ā q 0 in some model M T and let b ∈ M |v| be such that both

d M (ā, b) . -s < ε/3 and |ϕ(p) -ϕ M ( b)| < ε/3. This gives us |ϕ(p) -ϕ(q)| ≤ |ϕ(p) -ϕ M ( b)| + |ϕ M ( b) -ϕ M (ā)| + |ϕ M (ā) -ϕ(q)| < d(ā, b) + 2ε/3 < s + ε,
which was what we wanted.

The other inequality follows from Urysohn's lemma for topometric spaces stated above:

Let ∂(p, q) > ε > 0. Since ( S v(T ), τ, ∂) is a normal topometric space, Urysohn's Lemma above (with {p} and {q} as the two closed sets) gives us a ∂-1-Lipschitz function ϕ 0 : S v(T ) → R such that ϕ 0 (p) = 0 and ϕ 0 (q) = ∂(p, q) -ε.

If we consider ϕ 0 as an element of F v(T ), then ϕ 0 is also a 1-Lip formula, since given M T and ā, b ∈ M |v| , we have

|ϕ M 0 (ā) -ϕ M 0 ( b)| = |ϕ 0 (tp(ā)) -ϕ 0 (tp( b))| ≤ ∂(tp(ā), tp( b)) ≤ d M (ā, b),
where the last inequality always holds for two types that are realised in the same model M. It now follows that

sup{|ϕ(p) -ϕ(q)| : ϕ is 1-Lip} + ε ≥ |ϕ 0 (p) -ϕ 0 (q)| + ε = ∂(p, q).
Letting ε tend to 0, we conclude that

sup{|ϕ(p) -ϕ(q)| : ϕ is a 1-Lip formula} ≥ ∂(p, q)
as well, which was what we wanted.

Principality of infinitary types

The notion of principality also generalises to the infinitary setting. However, since not all types are realised, the matter is a little more delicate. As we are dealing with a space equipped with two different topologies, i.e. the logic topology τ and the metric topology τ ∂ , it will be convenient to introduce some terminology on how the topologies interact. The first definition is a very general one concerning arbitrary topological spaces.

Definition 3.3.15. Let X be a space equipped with two topologies τ 1 and τ 2 .

We say that the topologies coincide at a given point x ∈ X if for any U ∈ τ 1 containing x, there is V ∈ τ 2 such that x ∈ V ⊆ U , and vice versa.

The next definition defines the principal types for infinitary metric model theory, i.e. those types where the two topologies interact nicely. In order for the topologies around principal types to behave as in the finitary case, we will restrict ourselves to the subspace topologies on the realisable type space. Definition 3.3.16. Let T be an F -theory and let v be a tuple of variables. A type p ∈ S v(T ) is principal if the following holds:

∀r > 0 ∃ϕ ∈ F ∃s > 0 : ∅ = [ϕ < s] ∩ S v(T ) ⊆ B ∂ (p, r) ∩ S v(T ).
In other words, any ball around p has non-empty τ -interior in the subspace topologies.

Notice that it is not a priori the case that a principal type is realised, since we do not assume that the type p is contained in any of the two intersections above. However, it turns out that principal types are always realised in any model of the theory T (cf. Lemma 3.

below).

Given the definition of principality above, another natural concept arises, namely those types where the above holds not just for the subspace topologies but for the full topologies. We have chosen to call this property compatibility of ∂. Definition 3.3.17. Let T be a complete F -theory and let v be a finite tuple of variables. We say that ∂ is compatible at a type p ∈ S v(T ) if the following hold:

∀r > 0 ∃ϕ ∈ F ∃s > 0 : ∅ = [ϕ < s] ⊆ B ∂ (p, r).
Observe that there are some obvious implications between the three concepts defined above: if τ and τ ∂ coincides at a type p, then ∂ is compatible at p, and if ∂ is compatible at p, then p is principal. Moreover, in [START_REF] Ben Yaacov | Metric Scott analysis[END_REF]Lemma 7.4] the authors show that our definition of principality and their definition are equivalent for realisable types. We state the lemma below for convenience. As mentioned, we have that a principal type is always realisable. Moreover, it is realised in all models of the given theory. Hence, if τ and τ ∂ coincide or if ∂ is compatible at p, then p is realisable. Lemma 3.3.19. Let T be a complete F -theory and let p ∈ S n (T ) be a principal type. Then p is realised in all models of T .

Moreover, if q ∈ S n (T ) is any type realised in a model M T by some b ∈ M n , we may for any ε > 0 find a realisation ā

p in M such that d M (ā, b) ≤ ∂(p, q) + ε.
Proof. To simplify the notation in the proof below we will assume that all sets considered are intersected with the set of realisable types S n (T ).

Let p be a principal type in S n (T ). The first thing we will show is that for any r, s > 0 and formula ϕ such that [ϕ < s] ⊆ B(p, r), we can find a formula ϕ 1 such that p(ϕ 1 ) = 0 and [ϕ 1 < r] ⊆ B(p, r). For this, we suppose r > 0 is given. Then we may find s > 0 and a formula ϕ such that [ϕ < s] ⊆ B(p, r/3). Pick any p 0 ∈ [ϕ < s] and set

ϕ 1 (x) := inf ȳ max{d(x, ȳ) . -r/3, |ϕ(ȳ) -p 0 (ϕ)|}.
Then p(ϕ 1 ) = 0 because ∂(p 0 , p) < r/3. Set s := min{s -p 0 (ϕ), r/3}.

We now claim that [ϕ 1 < s ] ⊆ B(p, r). To see this, suppose q(ϕ 1 ) < s and that ā q in some M T . Then we can find some tuple b in M such that d M (ā, b) < s + r/3 and |ϕ M ( b) -p 0 (ϕ)| < s . It follows that ϕ M ( b) < p 0 (ϕ) + s ≤ s and therefore that ∂(tp( b), p) < r/3. Moreover,

∂(tp( b), q) ≤ d M (ā, b) < r/3 + s ≤ 2r/3.
Therefore, we get that

∂(p, q) ≤ ∂(p, tp( b)) + ∂(tp( b), q) < r
proving our claim. If we let ϕ 1 be the formula (r/s )ϕ 1 , we get that

[ϕ 1 < r] ⊆ B(p, r)
and that p(ϕ 1 ) = 0 as we wanted.

With this observation at our disposal, we can now show the lemma. Let q be a realisable type realised by some b in some M T and let ε > 0. Set s := ∂(p, q) and find some ϕ 1 such that p(ϕ 1 ) = 0 and [ϕ 1 < ε/3] ⊆ B(p, ε/3). Then q(inf ȳ max{d(x, ȳ) .

-s, |ϕ 1 (ȳ)|}) = 0, and therefore we can find some tuple ā1 in M such that d M ( b, ā1 ) ≤ s + ε/3 and |ϕ M 1 (ā 1 )| < ε/3. Hence, ∂(tp(ā 1 ), p) < ε/3. Next, we find some ϕ 2 with p(ϕ 2 ) = 0 and [ϕ 2 < ε/3 2 ] ⊆ B(p, ε/3 2 ). Since ∂(tp(ā 1 ), p) < ε/3, it follows that

inf c max{d M (ā 1 , c) . -ε/3, |ϕ 2 (c)|} = 0.
Therefore, we can find some ā2 in M such that

d M (ā 1 , ā2 ) < ε/3 + ε/3 2 and |ϕ M 2 (ā 2 )| < ε/3 2 . Hence, ∂(tp(ā 2 ), p) < ε/3 2 as well.
Continuing in this fashion, we obtain a sequence (ā n ) such that

d M (ā n-1 , ān ) < ε 3 n-1 + ε 3 n and ∂(tp(ā n ), p) < ε
3 n . The latter implies, of course, that tp(ā n ) converges to p with respect to ∂. Moreover, the former implies that (ā n ) is a Cauchy sequence because

d M (ā n , ān+k ) ≤ ε k-1 i=0 1 3 n+i + 1 3 n+i+1 = 2ε 3 k -1 3 k+n ≤ 2ε 1 3 n → 0 when n → ∞.
Thus, by completeness of M, we must have that (ā n ) converges to some ā. Moreover, this ā must realise p by continuity of the language. Thus, p is realised in M, and since M was arbitrary, p is realised in any model of T . Finally, for the 'moreover'-part, we have that

d M ( b, ān ) ≤ s + ε 3 + n-1 i=1 ε 3 i + ε 3 i+1 = s + ε 1 - 2 3 n ≤ s + ε.
It follows that d(ā, b) ≤ s + ε as well, which was what we wanted.

Omitting types

In this section we will give a new proof of the Omitting Types Theorem for infinitary metric model theory. This theorem was originally proved by Eagle in [START_REF] Christopher | Omitting types for infinitary [0, 1]-valued logic[END_REF]. However, the precise formulation of the theorem below is slightly more general than that of Eagle's. This formulation for finitary continuous logic is originally due to Itaï Ben Yaacov.

The first thing we need to do, is to define exactly what we mean by "omitting". As usual, F denotes a fragment. Definition 3.4.1. Let T be a complete F -theory and suppose M T . Let v be a finite tuple of variables and let p ∈ S v(T ) be a type. We say that M omits p if there is no realisation of p in M.

We now have: Theorem 3.4.2 (Omitting Types, cf. [27, Theorem A]). Let T be a complete F -theory and suppose that for each n ∈ N, X n ⊆ S n (T ) is a τ -meagre and ∂-open set. Then there is a separable model M T omitting all types in X n .

Before proceeding to the proof, we have a few remarks.

Remark 3.4.3. Recall that we have assumed our signatures and fragments to be countable. The countability of our signatures is essential for the theorem, as it may fail for uncountable signatures. Remark 3.4.4. Note that if X ⊆ S n (T ) is τ -meagre and ∂-open, it follows that no p ∈ X can be principal. Hence, this theorem shows, in particular, that if p is non-principal, then there is a model omitting it.

The proof of the theorem depends on two lemmas. To state the first of them, we need to define a convenient subspace of S ω (T ). This is the space of the types whose realisations are actually models of T . We say that such a type enumerates a model. Below, we will denote the closure of the set of all elements of a sequence ā by cl(ā). Definition 3.4.5. We say that a realisable type p ∈ S ω (T ) enumerates a model of T if for all realisations ā p in M T , we have that cl(ā) M. We denote the space of the types that enumerate a model of T by S en ω (T ). Given p ∈ S en ω (T ) and ā p in M T , we let M ā denote the corresponding elementary substructure with universe cl(ā).

It turns out that for p to enumerate a model, it is enough that there is some realisation ā p in a model M T such that M ā T and such that ā realises p in M ā. This is the content of the lemma below. Lemma 3.4.6. p ∈ S ω (T ) enumerates a model of T if and only if there is some ā p in a model M of T such that M ā T and ā realises p in M ā.

Proof. One direction is obvious: if p ∈ S ω (T ) enumerates a model, then there is of course a realisation ā such that M ā T and ā realises p inside M ā.

Suppose conversely that ā p in a model M T such that M ā T and ā realises p inside M ā. Let b p be another realisation of p inside some model N of T . Then the map Φ : a i → b i extends to an isometry M ā → N whose image is cl( b). Moreover, Φ is an elementary embedding, since for any formula ϕ(x) in our fragment, we have

ϕ Mā (ā I ) = p(ϕ(x)) = ϕ N ( bI ),
where āI denotes the tuple (a i : i ∈ I). By uniform continuity of formulas, it follows that the above equalities hold for any tuple of elements c ∈ M ā. By the Tarski-Vaught Test, it follows that cl( b) is in fact the universe of a model Mb of T and, moreover, that Mb N , which was what we wanted.

The first main lemma that we will use to prove Theorem 3.4.2 reads: Lemma 3.4.7. (S en ω (T ), τ ) is a Polish space.

We postpone the proof of Lemma 3.4.7 until after the proof of Theorem 3.4.2. Instead, we move on to state the second lemma that we will need. In order to do that, we need to introduce some projection maps S en ω (T ) → S n (T ). We will also need to be specific about which variables we are using to construct these spaces, so let us suppose that we use the infinite tuple x = (x 0 , . . .) to construct the space S ω (T ), and that we use the first n variables of x to construct S n (T ).

For each n ∈ N and each s ∈ N n we define a map π s : S en ω (T ) → S n (T ) by π s (p)(ϕ(x 0 , . . . , x n-1 )) = p(ϕ(x s 0 , . . . , x s n-1 )).

The second lemma now says the following: 8.45]). Hence, by the Baire Category Theorem, we find some p ∈ S en ω (T ) \ X. Let ā be a sequence realising p in some model M T . Then we claim M ā is a separable model omitting all types in n X n . To see this, we first note that no tuple of elements from ā can realise a type in any X n , since π s (p) / ∈ X n for all s ∈ N n and n ∈ N. Moreover, if for some tuple b ∈ M n ā we had that tp( b) ∈ X n , then we could find some ∂-ball around tp( b) contained in X n , say B ∂ (tp( b), ε) ⊆ X n . Since the elements of ā are dense in M ā by definition, we could find some s ∈ N n such that ās = (a s 0 , . . . , a s n-1 ) would be less than ε from b. Thus, ∂(tp(ā s ), tp( b)) < ε as well, which of course would imply that π s (p) ∈ X n , and this is not possible. Hence, we conclude that no tuple of elements from M ā realises a type in any X n .

It remains of course to prove the two lemmas.

Proof of lemma 3.4.7. We will prove that S en ω (T ) is a G δ subset of S ω (T ). For each formula ϕ(x) ∈ F x and ε > 0 define

U ϕ,ε,i = {p ∈ S ω (T ) : inf j∈N p(ϕ(x i=x j )) < p(inf x i ϕ(x)) + ε},
where x i=x j denotes the tuple x but with x j on the i'th coordinate.

U ϕ,ε,i is open since it is equal to j [ϕ(x i=x j ) -inf x i ϕ(x) < ε].
Moreover, these sets are dense. To show this, we must show that the intersection U ϕ,ε,i ∩[ψ < r] is non-empty for any formula ψ. Recall that all formulas in our fragment are assumed to have finitely many free variables. Thus, suppose ψ has free variables among x 0 , . . . , x m-1 and that ϕ has free variables among x 0 , . . . , x n-1 . We can without loss of generality assume m ≥ n, since we can always consider ψ as having free variables among arbitrarily large (finite) sets of variables. Take some type p 0 ∈ [ψ < r]. Since we have shown that the realisable types are τ -dense, we may assume p 0 is realised, say by ā ∈ M ω . Let s := p 0 (inf

x i ϕ(x)) = inf c∈M ϕ M (ā i=c ).
Then for some b ∈ M we must have ϕ M (ā i=b ) < s + ε. Consider the type p = tp(ā m=b ). Our choice of b implies that

p(ϕ(x i=xm )) = ϕ M (ā i=b ) < s + ε.
Therefore, we have

inf j p(ϕ(x i=x j )) ≤ p(ϕ(x i=xm )) < p 0 (inf x i ϕ(x)) + ε = p(inf x i ϕ(x)) + ε.
In other words, p ∈ U ϕ,ε,i . Moreover, as we made sure not to change the first m coordinates, we still have p(ψ) < r. We conclude that U ϕ,ε,i is dense.

Next, for each ε > 0 and each formula Φ ∈ F x of the form ϕ k , we let V Φ,ε be the set

V Φ,ε = {p ∈ S ω (T ) : inf k p(ϕ k ) < p(Φ) + ε}.
It is clear that V Φ,ε is an open set for each Φ and ε. Moreover, V Φ,ε contains all realisable types and is therefore dense.

We will prove that S en ω (T ) is the intersection of all the U ϕ,ε,i 's and all the V Φ,ε 's. The idea is, roughly, that the sets V Φ,ε will make sure any type in the intersection will be realisable, and the sets U ϕ,ε,i will make sure that the type enumerates a model. The latter follows from the Tarski-Vaught Test (cf. Proposition 3.2.2). What we must show is therefore that

S en ω (T ) = n,i∈N ϕ∈Fx U ϕ, 1 n ,i ∩ n∈N Φ∈Fx V Φ, 1 n . (3.5)
By the Tarski-Vaught Test (cf. Proposition 3.2.2), S en ω (T ) is included in the intersection on the right-hand side. Therefore, we only need to show the other inclusion.

Let p be a type from the set on the right-hand side of (3.5). Now, since the Compactness Theorem holds for finitary logic (cf. [9, Theorem 5.8]), we may find a model M T ∩ L ωω where p restricted to the finitary fragment L ωω is realised by some ā ∈ M ω , i.e. for any finitary formula ϕ, we have p(ϕ) = ϕ M (ā). We claim that M 0 := cl(ā) is a model of T and that ā realises p in both this model and in M. From Lemma 3.4.6 it then follows that p enumerates a model of T .

We begin by showing that ā does in fact realise all of p in M. To show this, we proceed by transfinite induction on the complexity of formulas.

If ϕ ∈ F x is atomic, then ϕ is from the finitary part, so by our choice of ā, p(ϕ) = ϕ M (ā).

Suppose now that p(ψ) = ψ M (ā) for every ψ ∈ F x of rank less than the ordinal α and let ϕ ∈ F x have rank α + 1.

If ϕ is f (ψ 1 , . . . , ψ n ) for a connective f , then since p is a multiplicative linear functional, we have

p(ϕ) = f (p(ψ 1 ), . . . , p(ψ n )).
Since ψ i has rank less than α, and since F x is closed under subformulas, we have p

(ψ) = ψ M i (ā). Thus, p(ϕ) = ϕ M (ā) as well. If ϕ is inf x i ψ(x), then since p ∈ n U ψ, 1
n ,i , we have p(inf

x i ψ(x)) = inf j∈N p(ψ(x i=x j )).
By our induction hypothesis, we have p(ψ(x i=x j )) = ψ M (ā i=a j ), and so we obtain p(inf

x i ψ(x)) = inf j∈N ψ Mā (ā i=a j ) = inf b∈M ψ M (ā i=b )
as required. Finally, we assume that ϕ is of the form ϕ k , where the complexity of each ϕ k is less than or equal to α. Since p ∈ n V ϕ, 1 n , we have

p( ϕ k ) = inf k p(ϕ k ) = inf k ϕ M k (ā) = ϕ M (ā).
We conclude that p is realised by ā in M.

We proceed to show that M 0 satisfies condition (ii) of the Tarski-Vaught Test, so that it is the universe of a structure. Let therefore ϕ(x 0 , . . . , x m-1 ) be a formula from our fragment F and let b = (b 0 , . . . , b m-2 ) be a tuple of elements from M 0 . It is enough to show that

inf{ϕ M ( b, c) : c ∈ M} = inf{ϕ M ( b, c) : c ∈ M 0 }.
Clearly, the left hand side is less than the right hand side. Therefore we proceed to show the other inequality. For this, we let ε > 0 be given. For each element b i of the tuple b we may find, by the uniform continuity of formulas, a n i such that the following holds:

inf c∈M 0 ϕ M ( b, c) ≤ inf c∈M 0 ϕ M ((a n i ), c) + ε ≤ inf c∈M ϕ M ((a n i ), c) + 2ε ≤ inf c∈M ϕ M ( b, c) + 3ε.
Here we used that p ∈ U ϕ,ε,| b|+1 and that ā p in M in the second inequality. Since ε > 0 was arbitrary, we conclude that the equality above is satisfied. It follows that M 0 is in fact the universe of a model M ā of T . Moreover, by the Tarski-Vaught Test, M ā M.

By Lemma 3.4.6, we conclude that p ∈ S en ω (T ), which was what we wanted.

Finally, we will prove the second lemma.

Proof of lemma 3.4.8. Let s ∈ N n be a finite injective sequence. We begin by showing continuity of π s . Let ϕ(x 0 , . . . , x n-1 ) ∈ F n (T ) and r > 0. We want

to show π -1 s ([ϕ < r]) is open. But this is clear as π -1 s ([ϕ(x 0 , . . . , x n-1 ) < r]) = {p : p(ϕ(x s 0 , . . . , x s n-1 )) < r} = [ϕ(x s 0 , . . . , x s n-1 ) < r].
Next we show surjectivity. Let p ∈ S n (T ) and let ā p in M T . By the downwards Löwenheim-Skolem Theorem (cf. Theorem 3.2.4), we find a separable substructure M 0 M containing all elements of ā. If we let p be the type realised by a dense sequence of M 0 with the elements of ā on the coordinates given by s, we see that p enumerates a model and that π s (p) = p.

Finally, we show that the map is open. By Proposition 3.3.1, it is enough to show that π s ([ϕ < r]) is open for each formula ϕ(x 0 , . . . , x m-1 ) with finitely many variables and r > 0. We may assume m > max{s i : i < n} by simply considering ϕ as a formula with free variables among the variables x 0 , . . . , x k for k sufficiently large. We claim that

π s ([ϕ < r]) = {p ∈ S n (T ) : p(ψ) < r}, (3.6)
where ψ is defined by ψ(x s 0 , . . . , x s n-1 ) := inf{ϕ(x 0 , . . . , x m-1 ) : {x i : i / ∈ s}}.

To see why (3.6) holds, let π s (p) ∈ π s ([ϕ < r]) and let xk denote the first k variables of x for each k ∈ N. Then we have that

π s (p)(ψ(x n )) = p(ψ(x s 0 , . . . , x s n-1 )) = p(inf{ϕ(x m ) : {x i : i / ∈ s}}) ≤ p(ϕ(x 0 , . . . , x m-1 )) < r.
Thus, π s (p) is in the set on the right-hand side of (3.6). Conversely, if p 0 is a type such that p 0 (ψ) < r, we take a realisation ā p 0 . Then we find elements b witnessing the infimum inf ϕ < r. Finally, we use the Löwenheim-Skolem Theorem to find a separable model containing the elements of ā and b, which gives us a type p enumerating a model such that π s (p) = p 0 and such that p(ϕ) < r. We conclude that the map π s is in fact open as we wanted.

The infinitary Ryll-Nardzewski Theorem

As a corollary to Theorem 3.4.2, we obtain a version of the Ryll-Nardzewski Theorem for infinitary metric model theory (cf. [9, Theorem 12.10] for a continuous finitary version of this theorem). Theorem 3.5.1. Let F be a fragment and let T be a complete F -theory. Then the following are equivalent:

(i) T is F -ℵ 0 -categorical. (ii) All p ∈ S n (T ) are principal for all n. (iii) τ S n (T ) = τ ∂ S n (T ) for all n.

Proof. (i) =⇒ (ii).

We will show the contrapositive. Suppose p ∈ S n (T ) is not principal. Then there is r > 0 such that the ∂-ball B(p, r) around p has empty τ -interior. Therefore, B(p, r) ∩ S n (T ) is a ∂-open and τ -meagre subset of the realisable types. Hence, by the Omitting Types Theorem, there is a separable model M T omitting B(p, r) ∩ S n (T ) (of course it will omit the whole ball B(p, r), but we do not need that here). However, p is realisable, so there is some separable model N T realising p. It follows that M and N cannot be isomorphic since they don't realise the same types. Therefore, T cannot be ℵ 0 -categorical.

(ii) ⇐⇒ (iii).

First we note that if the two subspace topologies agree, then every realisable p is of course principal, so (iii) implies (ii).

We proceed to show the other implication. From Lemma 3.3.18 above we know that τ S n (T ) coincides with τ ∂ S n (T ) at every realisable p ∈ S n (T ).

Thus, what we must show is that the two subspace topologies actually are the same. Let therefore p and r > 0 be given and suppose that q ∈ B(p, r)∩S n (T ). Then we find some s > 0 such that B(q, s) ⊆ B(p, r). By the above, the two topologies coincide at q, so we find some ϕ such that

q ∈ [ϕ < s] ∩ S n (T ) ⊆ B(q, s) ∩ S n (T ) ⊆ B(p, r) ∩ S n (T ).
We conclude that B(p, r) ∩ S n (T ) is τ S n (T )-open. Since τ δ refines τ we conclude that the two subspace topologies are the same.

(ii) =⇒ (i).

Suppose all types are principal. We will show that T is ℵ 0 -categorical. Thus, let M and N be separable models of T . We will construct an isomorphism between the two models by a back and forth argument more or less identical to the one in the proof of Fact 1.5 in [START_REF] Ben | On d-finiteness in continuous structures[END_REF] Let {a i : i ∈ N} =: M 0 ⊆ M and {b i : i ∈ N} =: N 0 ⊆ N be dense countable subsets. We will recursively define elementary maps, f i : A i → N and g i : B i → M, defined on finite subsets, A i and B i , of M and N respectively, such that the following holds:

-A i+1 = {a 0 , . . . , a i } ∪ A i ∪ g i (B i ). -B i+1 = {b 0 , . . . , b i } ∪ B i ∪ f i+1 (A i+1 ). -For all a ∈ A i we have d(a, g i f i (a)) ≤ 2 -i . -For all b ∈ B i we have d(b, f i+1 g i (b)) ≤ 2 -i .
Let us first note why this will be enough. For any a ∈ A i , we have that

d(f i+1 (a), f i (a)) ≤ d(f i+1 (a), f i+1 g i f i (a)) + d(f i+1 g i f i (a), f i (a)) ≤ d(a, g i f i (a)) + 2 -i ≤ 2 -i+1 .
Thus, the maps f i converge to a map f :

A i → N . Moreover, f must be elementary, since if we let c = (a i 0 , . . . , a i n-1 ), we have

ϕ N (f (c)) = lim i ϕ N (f i (c)) = ϕ M (c),
where we have used the continuity of the language. Since f is in particular an isometry, we may extend it to a map f : Ā = M → N . Again by the continuity of the language, f is an elementary embedding of M into N .

Similarly, we find an elementary embedding ḡ : N → M using the g i 's. Moreover we see that ḡ must be the inverse of f , since for any a ∈ M 0 , we have that d(a, ḡ f (a)) ≤ d(a, g j+1 f j (a)) + d(g j+1 f j (a), ḡf j (a)) + d(ḡf j (a), ḡ f (a)), and we see that each of these values tend to 0 as j → ∞. Thus, ḡ f is the identity on M 0 and therefore also on M. Similarly, we see that f ḡ is the identity on N as well.

It remains of course to construct the sequences (f i ) and (g i ). We begin by letting f 0 = g 0 = ∅ and A 0 = B 0 = ∅. Then, since T is complete and M, N T , we have M ≡ N so f 0 and g 0 are elementary.

Suppose now that f i and A i are given. Then B i is also defined by the requirement above. Let A i = (a i 0 , . . . , a i n ) and B i = (f i (a i 0 ), . . . , f i (a i n ), . . .) be enumerations as tuples of these two sets. Let p = tp(B i ) and let ϕ(x 0 , . . . , x m ) be a formula such that [ϕ < 2 -i ] ⊆ B(p, 2 -i ) (in the subspace topologies) and such that p(ϕ) = 0. This can be done since all types are principal, so the subspace topologies agree by (ii) ⇔ (iii).

Now, ψ = inf x n+1 ,...,xm |ϕ(x)| is mapped to 0 by p. Moreover, by elementarity, tp(A i )(ψ) = 0 as well. Hence, we may find a tuple A of elements of M such that tp(A i , A ) ∈ B(p, 2 -i ). Then by Lemma 3.3.19, we can find a realisation B of p in M such that d((A i , A ), B ) ≤ 2 -i . We can now define g i by mapping B i to this tuple B . Since tp(B i ) = tp(B ), this defines an elementary map. Moreover, since d((A i , A ), B ) ≤ 2 -i , the third condition above is also satisfied.

We construct f i+1 from g i in an analogous way. By our comments above, we conclude M and N are isomorphic.

Observe that in the argument of (ii) =⇒ (i) we only used that all types realised by M and N are principal. Thus, we have actually shown the following proposition: Proposition 3.5.2. Any two separable atomic models of the complete theory T are isomorphic.

As a corollary to the theorem above, we obtain that for ℵ 0 -categorical T , the realisable type space is complete with respect to ∂. Corollary 3.5.3. Let T be a complete theory. Then the space of principal ntypes is complete with respect to the type distance for each n ∈ N. In particular, if T is an ℵ 0 -categorical complete theory, then S n (T ) is closed with respect to the type distance ∂ for each n ∈ N.

Proof. Let (p k ) be a sequence of principal types converging to some p. We may assume that ∂(p k , p k+1 ) < 2 -k by simply passing to a subsequence.

Take ā1 p 1 in some M T . Then, by Lemma 3.3.19, we find ā2 p 2 in M such that d M (ā 1 , ā2 ) ≤ 2 -1 . Since p 3 is principal, we can use the lemma again to obtain a realisation ā3

p 3 with d M (ā 2 , ā3 ) ≤ 2 -2 .
Continuing in this fashion, we find realisations āk p k in M such that (ā k ) k∈ω is Cauchy. Therefore, this sequence converges to some ā in M n . Moreover, this ā must realise p, since for any formula ϕ(x), we have

ϕ M (ā k ) → ϕ M (ā)
by uniform continuity of the language. At the same time

ϕ M (ā k ) = p k (ϕ) → p(ϕ), since convergence in ∂ implies pointwise convergence.
The 'in particular'-part of the corollary follows by Theorem 3.5.1 above, since if T is ℵ 0 -categorical, it follows that all realisable types are principal.

We do not know if the above result holds for more general theories. Thus, we finish the chapter with the following open problem: Question 3.5.4. Is the space of realisable types closed with respect to the type distance for any complete theory T ? Chapter 4

Polish Groups & Metric Model Theory

There is a very close connection between Polish groups and automorphism groups of separable metric structures. In fact, these are just two sides of the same coin: a group is Polish if and only if it is (isomorphic to) the automorphism group of a separable metric structure. We have already mentioned in Chapter 1 that isometry groups of Polish spaces equipped with the pointwise convergence topology are Polish. It follows that automorphism groups of complete separable metric structures are Polish as well, since they are closed subgroups of the isometry group of the metric space the structure is based on. In the first section of this chapter, we explain the other direction of this equivalence by constructing the so-called canonical metric structure M G associated to a Polish group G.

We are going to use the canonical metric structure to study the Roelcke completion of a Polish group as a certain set of types. Recall from Chapter 1 that this completion is the completion of the Roelcke uniformity, which is the meet of the left and right uniformities. If d L is a left-invariant compatible metric on the Polish group G, then the metric given by

d ∧ (g, h) = inf f max{d L (g, f ), d L (f -1 , h -1 )}
induces the Roelcke uniformity. Hence, the Roelcke completion is the completion of G with respect to d ∧ . Note moreover that this works for any leftinvariant metric.

The connection between the Roelcke completion and type spaces of continuous logic was studied in [START_REF] Ben | Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups[END_REF]. In that paper, the authors show that there is an intimate connection between ℵ 0 -categorical metric structures and Roelcke precompact groups, i.e. the groups with compact Roelcke completion. Using 87 the Ryll-Nardzewski Theorem of continuous logic, they obtain the following theorem: Theorem 4.0.1 ([13]). Let G = Aut(M) be the automorphism group of a separable metric structure and suppose that the space of orbit closures M G is compact. Then the following are equivalent:

(i) G is Roelcke precompact. (ii) Th(M) is ℵ 0 -categorical.
Of course, the assumption that G is the automorphism group of a separable metric structure will turn out to be no real restriction, since the canonical metric structure will allow us to view any Polish group in this way. Thus, this theorem is really a characterisation of when a Polish group in general is Roelcke precompact.

Using this result, it is easy to show that many well-known metric structures have Roelcke precompact automorphism groups. For instance, S ∞ is Roelcke precompact because it is the automorphism group of the countably infinite set (which is ℵ 0 -categorical of course). Similarly, the automorphism group of the dense linear order (Q, <) is Roelcke precompact because its theory is ℵ 0categorical. Likewise, the unitary group of a separable Hilbert space [START_REF] Uspenskij | The Roelcke compactification of unitary groups[END_REF], the automorphism group of a standard probability space [START_REF] Glasner | The group Aut(µ) is Roelcke precompact[END_REF], the isometry group of the bounded Urysohn space [START_REF] Uspenskij | On subgroups of minimal topological groups[END_REF] and the automorphism group of the random graph are all Roelcke precompact, which may be seen using this theorem.

The key observation made in [START_REF] Ben | Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups[END_REF] that connects type spaces and Roelcke completions, is, that for an ℵ 0 -categorical structure it is possible to embed the Roelcke completion of the group isometrically into the type space S ω⊕ω (T ).

In the present chapter, we will show that under weaker assumptions than ℵ 0categoricity it is still possible to make this embedding (cf. Proposition 4.2.2 below). In particular, one can view the Roelcke completion of any Polish group G as a subspace of the type space S ω⊕ω (Th(M G )), where M G is the canonical metric structure associated to G. We will use this embedding to study a property of the Roelcke completion known as local Roelcke precompactness introduced in Chapter 1.

Local Roelcke precompactness is a natural generalisation of Roelcke precompactness. Instead of having a compact Roelcke completion, locally Roelcke precompact (lRpc) groups have a precompact identity neighbourhood, i.e. a neighbourhood whose closure in the Roelcke completion is compact. Due to a recent theorem of Zielinski from [START_REF] Zielinski | Locally roelcke precompact polish groups[END_REF], the lRpc groups are, as expected, the groups with a locally compact Roelcke completion. Using this, and inspired by the results on the Roelcke precompact groups mentioned above, we will obtain a characterisation of the lRpc Polish groups in terms of type spaces of continuous logic (cf. Theorem 4.2.5). Of course, since we do not have compactness of the Roelcke completion, our analysis of the connection to metric model theory must and will be much more delicate and careful than in the Roelcke precompact case.

We will also study what happens in the exact homogeneous case. Recall that a metric structure is exact homogeneous if whenever two tuples have the same type, we can find an automorphism of the structure mapping one tuple to the other. In general, homogeneity for metric structures usually only means approximate homogeneity, meaning that we can move two tuples of the same type arbitrarily close to each other with automorphisms. However, many wellknown metric structures are actually exact homogeneous, and hence it is not an unreasonable assumption to impose on the structure even in the general metric setting. For instance, the two most important metric structures for us, the Urysohn metric space and diversity, are both exact homogeneous.

As mentioned in the introduction, the automorphism group of a discrete structure can be viewed as a closed subgroup of the permutation group S ∞ . These groups are exactly the so-called non-archimedean groups, i.e. the Polish groups having a basis at the identity consisting of clopen subgroups. It follows that the automorphism group of a discrete structure is locally Roelcke precompact if and only if it contains a Roelcke precompact open subgroup. However, some caution should be taken here, since in general it is not true that a subgroup which is Roelcke precompact as a subset is also Roelcke precompact as a group. On the other hand, if the subgroup is open, it follows directly from the definition that the subgroup is Roelcke precompact as a group if and only if it is Roelcke precompact as a subset.

As usual, things are more complicated when we pass to the general metric setting. However, if the structure is exact homogeneous, we do get one direction of the above equivalence for locally Roelcke precompact groups: if the automorphism group of an exact homogeneous structure is locally Roelcke precompact, then there is a finite tuple of elements from the structure such that the stabiliser of this tuple is Roelcke precompact. Furthermore, any stabiliser subgroup is a locally Roelcke precompact group. We will investigate the exact homogeneous case in the last section of the chapter.

The canonical metric structure

In this section, we will go through the construction of the canonical metric structure associated to a Polish group. Moreover, we will observe that this structure is in fact atomic, which we believe to be a new result.

The construction goes as follows: given a Polish group G, denote its left completion by G L with left-invariant metric d L . G acts naturally on G L by

g • x = g • lim n g n := lim n gg n
for g ∈ G, x ∈ G L and a sequence (g n ) in G converging to x with respect to d L . This is well-defined by left-invariance of d L . Moreover, this action is continuous, since d L is compatible with the group topology. Note that this implies that G acts continuously on any countable power of G L simply by the diagonal action. We want to use G L as the universe of a metric structure M G such that G is the automorphism group of M G . In order to do that, for each n ∈ N and n-tuple ā of elements from some dense countable subset of G L , we let P ā be an n-ary predicate. Since the metric d L may be assumed to be bounded by 1, we let [0, 1] be the bound for each P ā. Moreover, the modulus of continuity will just be the identity so that the predicates become 1-Lipschitz maps when interpreted. We interpret P ā in M G by setting

P M G ā ( b) := d L (G • ā, G • b), where d L (G • ā, G • b) denotes the infimum inf{d L (ā , b ) : ā ∈ G • ā, b ∈ G • b}.
Then it is easy to verify that each P M G ā respects its modulus and bound, and thus

M G = ( G L , {P M G ā }
) is a metric structure. We now have the following proposition, originally due to Melleray (cf. [START_REF] Melleray | A note on Hjorth's oscillation theorem[END_REF]Theorem 6]), saying that M G is ultrahomogeneous and has G as its automorphism group. Moreover, M G is atomic which is, as mentioned, a new result. Therefore we only include the proof of this fact. Proof. As mentioned, we will only show atomicity, since that is the only new result. Let therefore p ∈ S n (T ) be a type realised in M G and let r > 0 be given. Suppose first that there is a predicate P ā for a tuple ā realising p. We then claim that

[P ā < r] ∩ S M G n (T ) ⊆ B(p, r),
which by Lemma 2.5.7 is enough to show principality of p. Suppose therefore q ∈ [P ā < r] is a type realised by M G , say by b. Then d L (G • ā, G • b) < r, so we find g, h ∈ G such that d L (gā, h b) < r as well. Since gā p and h b q, it follows that ∂(p, q) ≤ d L (gā, h b) < r as well. Hence, q ∈ B(p, r) as we wanted. Suppose now that p is realised by an arbitrary tuple ā of elements of G L . By density, we find a predicate P ā0 for a tuple ā0 such that d L (ā, ā0 ) < r. It follows that tp(ā 0 ) = q ∈ B(p, r), and hence that there is some r > 0 such that B(q, r ) ⊆ B(p, r). Since q is principal by the above, we find a formula ϕ and an s > 0 such that [ϕ < s] ⊆ B(q, r ). Hence, B(p, r) has non-empty τ -interior and therefore, by Lemma 2.5.6, we get that p is principal. We conclude that M G is atomic.

The structure M G will prove to be extremely useful to us. In particular, the atomicity of M G is crucial for the proof of Theorem 4.2.5 below, because it implies knowledge of the type distance.

The Roelcke completion & model theory

In this section, we will study the Roelcke completion G L∧R of a Polish group G as a set of types. By what we have done in the previous section, we can always assume that G is given as the automorphism group of a separable atomic metric structure, namely the structure M G . This construction encourages us to study the connection between G L∧R and metric model theory further. The connection has, as mentioned, already been studied in [START_REF] Ben | Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups[END_REF], where the authors show that for ℵ 0 -categorical structures, one can embed G L∧R into S ω⊕ω (Th(M)), where M is a structure with G = Aut(M). Below we show that for a homogeneous model which knows ∂ ω⊕ω it is still possible to make this embedding. Since M G is atomic, it does indeed know the type distance, and thus this embedding allows us to give a characterisation of local Roelcke precompactness in Theorem 4.2.5. We begin with a simple but useful observation connecting Roelcke precompact subsets and type spaces. Therefore, we have

∂(tp( b, uf u b), tp( b, f b)) ≤ d M ((u -1 b, f u b), ( b, f b)) = max{d M (u -1 b, b), d M (f u b, f b)} < ε
as we claimed. Thus, {tp( b, g b) : g ∈ A} is totally bounded.

Conversely, suppose {tp( b, g b) : g ∈ A} is totally bounded for any finite tuple b and let U be a neighbourhood of the identity. Without loss of generality, let us assume that U is of the form Ub ,r := {g : d(g b, b) < r} for some finite tuple b and r > 0. Let {p i } be a finite set of types such that the balls of radius r centered at p i cover {tp( b, g b) : g ∈ A}. We can assume without loss of generality that p i = tp( b, f i b) for some f i ∈ A. We now claim that A ⊆ U {f i }U . To see this, suppose g ∈ A and find some f i such that ∂(tp( b, g b), tp( b, f i b)) < r. By homogeneity and knowledge of ∂, we find some h ∈ Aut(M) such that

d(h • ( b, f i b), ( b, g b)) < r. It follows that h ∈ U . Moreover, f -1 i h -1 g ∈ U as well. Thus, g ∈ hf i U ⊆ U {f i }U as claimed.
Next, we will explain how to embed the Roelcke completion of a Polish group isometrically into the type space S ω⊕ω (T ). To do this, it will be convenient for us to fix, once and for all, a metric inducing the Roelcke uniformity. Since we are only dealing with automorphism groups of separable metric structures, suppose that G = Aut(M) is such a group and fix a dense sequence ξ ∈ M ω , i.e. a sequence whose elements form a dense subset of M. We can then define a left-invariant metric d L on G by

d L (g, h) = d M ω (g • ξ, h • ξ) = i∈ω 2 -i-1 d M (gξ i , hξ i ),
where G acts on M ω via the diagonal action. It is easy to check that d L is in fact left-invariant. Thus, we obtain a metric d L∧R inducing the Roelcke completion by setting

d L∧R (g, h) = inf f ∈G max{d L (g, f ), d L (f -1 , h -1 )}.
We now have: Proof. We have by definition that

d L∧R (g, h) = inf f ∈G max{d L (g, f ), d L (f -1 , h -1 )}, where d L (g, h) := d M ω (g • ξ, h • ξ).
On the other hand, we have

∂ ω⊕ω (ι(g), ι(h)) = ∂(tp(ξ, gξ), tp(ξ, hξ)) = inf f ∈G d M ω⊕ω (f -1 (ξ, gξ), (h -1 ξ, ξ)) = inf f ∈G max{d M ω (f -1 ξ, h -1 ξ), d M ω (gξ, f ξ)} = d L∧R (g, h),
where we used homogeneity and knowledge of ∂ ω⊕ω in the second equality.

We will therefore from now on identify G L∧R with the ∂-closure of the image of the map ι from the lemma above whenever we are working in a homogeneous model that knows ∂. However, from time to time it will be important for us to be able to distinguish the Roelcke completion and the closure of the image of ι, since we may work in models that do not a priori know the type distance. Therefore, we will denote the ∂-closure of the image of ι by S(G), but the reader is encouraged to think of S(G) simply as G L∧R . Since the construction of S(G) depends on the dense sequence ξ, we will from now on tacitly assume that we have fixed such a sequence in any separable structure under consideration. Since S(G) is a subset of a type space, it inherits the logic topology τ in addition to the τ ∂ -topology. By S(G), we will denote the τ -closure of S(G). Note that S(G) is compact, being a closed subset of the compact space S ω⊕ω (T ). We wish to exploit the compactness of (S(G), τ ) to show local compactness of (S(G), ∂), and hence that G is locally Roelcke precompact. Thus, if we can make sure that (S(G), τ ) is homeomorphic to (S(G), ∂), it follows that G is locally Roelcke precompact if and only if S(G) is τ -open in S(G). Fortunately for us, it turns out that (S(G), ∂) and (S(G), τ ) are homeomorphic whenever G is the automorphism group of a separable atomic structure. Moreover, as we have seen above, we are in the fortunate situation that any Polish group is the automorphism group of such a structure. Thus, we obtain our characterisation of locally Roelcke precompact groups in Theorem 4.2.5 below.

However, before we can prove Theorem 4.2.5, we will need to do some work. The first thing we need is the following description of S(G). Recall from Chapter 2 that π ω and π ω denote the projection maps from S ω⊕ω (T ) onto, respectively, the first and second ω variables. Proof. Suppose first that p ∈ S(G) and that M realises p. Then, by continuity of the projections, we have that π ω (p) = π ω (p) = tp(ξ), where ξ ∈ M ω is our fixed dense sequence. Furthermore, we find a realisation (ā, b) of p in M. We must show that we can approximate p in ∂ with types of the form tp(gξ, hξ) for g, h ∈ G. Let therefore ε > 0 be given, and find n ∈ N such that

i≥n 2 -i < ε/2. Since tp(ā n) = tp( b n) = tp(ξ n),
we get by homogeneity of M two automorphisms g and h such that

d M (gξ n, a n), d M (hξ n, b n) < ε/2.
It follows that ∂(tp(gξ, hξ), p) < ε as we wanted. Thus, p ∈ S(G).

Conversely, if p ∈ S(G), we must show that we can find a realisation in M. For this, we need M to know ∂ ω⊕ω . First, we find a sequence (p n ) of types of the form tp(ξ, g n ξ) converging in ∂ to p. By passing to a subsequence, we may suppose that ∂(p n , p n+1 ) < 2 -n . Since M knows ∂, we can find realisations (ā 1 , b1 ) p 1 and (ā 2 , b2 ) p 2 such that

d M ((ā 1 , b1 ), (ā 2 , b2 )) < 2 -1 .
Similarly, we can find (ā 2 , b

2 ) p 2 and (ā 3 , b 3 ) p 3 such that

d M ((ā 2 , b 2 ), (ā 3 , b 3 )) < 2 -2 .
By homogeneity, we can find some g ∈ G such that

d M ((ā 2 , b2 ), g(ā 3 , b 3 )) < 2 -2
as well. In this way, we construct a Cauchy sequence ((ā n , bn )) n , where (ā n , bn ) realises p n . By completeness, this sequence must converge to some realisation (ā, b) of p, which was what we wanted.

For separable atomic structures, we have the following convenient description of S(G). Proof. Denote the set on the right-hand side of (4.1) above by A. By continuity of the projections, A is τ -closed. Moreover, G (or rather ι(G)) is contained in A. Hence, it is enough to show that G is τ -dense in A. Let therefore p ∈ A and suppose p ∈ [ϕ < r] for a formula ϕ(x, ȳ) with finitely many free variables x and ȳ and r > 0. We can assume that p(ϕ) = 0. Let δ be the modulus of continuity for ϕ. By atomicity, we find a formula ψ(ȳ) such that ψ M (ξ |ȳ|) < r/2 and

[ψ < r/2] ⊆ B(tp(ξ |ȳ|), δ(r/2)). Then p(inf z max{ϕ(x, z), ψ(z)}) < r/2. Since p ∈ A, we have that p(inf z max{ϕ(x, z), ψ(z)}) = inf c∈M max{ϕ M (ξ |x|, c), ψ M (c)}.
Therefore, we can find some tuple b of elements from M such that

max{ϕ M (ξ |x|, b), ψ M ( b)} < r/2.
It follows that ∂(tp(ξ |ȳ|), tp( b)) < δ(r/2). By atomicity, M is homogeneous and knows ∂. Hence, we find g ∈ Aut(M) such that d M ( b, g(ξ |ȳ|)) < δ(r/2). It follows that

|ϕ M (ξ |x|, g(ξ |ȳ|)) -ϕ M (ξ |x|, b)| < r/2, which implies that ϕ M (ξ |x|, g(ξ ȳ)) < r/2 + ϕ M (ξ |x|, b) < r.
Hence, tp(ξ, gξ) ∈ [ϕ < r]. We conclude that G is τ -dense in A.

We can now state and prove our main result of this chapter.

Theorem 4.2.5. Let G = Aut(M) be the automorphism group of a separable atomic metric structure M. Then G L∧R is homeomorphic to (S(G), τ ). Moreover, the following are equivalent for any Polish group G:

(i) G is locally Roelcke precompact.

(ii) There is a separable atomic metric structure M with G = Aut(M) and such that (S(G), τ ) is open in (S(G), τ ).

Proof. Let G = Aut(M) be given. Since M is atomic, we have that it knows ∂ ω⊕ω (cf. Lemma 2.5.12). Therefore, from Proposition 4.2.2, it follows that we can identify the Roelcke completion of G with (S(G), ∂ ω⊕ω ). By Proposition 4.2.3, we know that S(G) ⊆ S M ω⊕ω (T ). Furthermore, it follows from Lemma 2.5.11 that all types of S(G) are principal, so τ ∂ and τ agree on S(G). Hence, G L∧R is homeomorphic to (S(G), τ ).

For the 'moreover'-part, suppose that G is any locally Roelcke precompact Polish group. Let M G be the canonical metric structure associated to G. Then M G is a separable atomic metric structure such that G = Aut(M G ). By Zielinski's theorem (cf. Theorem 1.4.3), it follows that G L∧R , and hence also (S(G), ∂), are both locally compact. Since

(S(G), ∂) is homeomorphic to (S(G), τ ), it follows that (S(G), τ ) is locally compact. Since (S(G), τ ) is compact, it follows that S(G) is τ -open in S(G).
For the other direction, assume we have a separable atomic metric structure

M such that G = Aut(M) with (S(G), τ ) open in (S(G), τ ). Then (S(G), τ )
is locally compact and homeomorphic to (S(G), ∂) which may be identified with G L∧R . Thus, (S(G), ∂) is locally compact, and therefore by Zielinski's theorem, G is locally Roelcke precompact.

The exact homogeneous case

The automorphism group of a countable homogeneous (classical) first-order structure is locally Roelcke precompact if and only if it contains a Roelcke precompact open subgroup. This is because, as mentioned above, automorphism groups of such structures are exactly the closed subgroups of the permutation group of N, S ∞ . In the general setting of metric model theory, one of the these implications still holds for exact homogeneous structures as the theorem below explains. In particular, this applies to the Urysohn metric space and the Urysohn diversity, since both of these structures are exact homogeneous. We will study these structures further in Chapters 5 and 6 below, In particular, if G is locally Roelcke precompact, then G c is locally Roelcke precompact for all c. Moreover, there is a tuple ā such that G ā is Roelcke precompact.

Proof. The proof uses the correspondence with type spaces developed above.

Let ξ be a dense sequence in M and let c ∈ M n be any finite tuple. Then cξ, i.e. c concatenated with ξ, is also a dense sequence. Since M is atomic, it knows ∂ ω⊕ω . Therefore, by Proposition 4.2.2, we can view the Roelcke completion of G as the ∂-closure of the set {tp(cξ, g(cξ)) : g ∈ G}.

As before, we denote this closure by S(G). Since we will need to know exactly which variables we are using for our types, we will consider S(G) as a subset of the space S x⊕ȳ (T ), i.e. the types using the two infinite tuples of free variables x = (x 0 , . . .) and ȳ = (y 0 , . . .). We can now consider the closure of G c in the Roelcke completion of G as the ∂-closure of the set

{tp(cξ, cg(ξ)) : g ∈ G c}.
Denote this closure by S(G c). By the same argument as in Proposition 4.2.3 and using exact homogeneity, it follows that

S(G c) = {p ∈ S(G) : ∃ā, b ∈ M ω (cā, cb ) p}.
Let now M c be the structure M with the elements of c as constant symbols and with automorphism group G c and theory T c. By Proposition 2.5.14, the structure M c is atomic and exact homogeneous. Moreover, ξ is a dense sequence in M c, so we can identify the Roelcke completion of G c with the ∂-closure of the following set of types over c:

{tp(ξ, gξ | c) : g ∈ G c}.
Denote this closure by S c(G c). To make our notation easier, we will view this set as a subset of the type space S x ⊕ȳ (T c), where we use the infinite tuples of variables x = (x n , . . .) and ȳ = (y n , . . .).

The claim is now that (S c(G c), ∂) is homeomorphic to (S(G c), ∂). To show this, we first note that any type p ∈ S c(G c) is realised in M c by Proposition 4. First of all, we should argue why this map is well-defined. Suppose therefore (ā, b) and (ā , b ) realise the type p ∈ S c(G c) over c. We must show that tp(cā, cb ) = tp(cā , cb ). Let therefore ϕ(x, ȳ) be a formula with free variables among x and ȳ. By ϕ(cx , cȳ ), we denote the formula with parameters from c, where each occurrence of x i and y i in ϕ(x, ȳ) is replaced with c i for i < n. Then

tp(cā, cb )(ϕ(x, ȳ)) = ϕ M (cā, cb ) = ϕ Mc (c Mc ā, cMc b) = p(ϕ(cx , cȳ )) = ϕ Mc (c Mc ā , cMc b ) = ϕ M (cā , cb ) = tp(cā , cb )(ϕ(x, ȳ)).
It follows that λ is well-defined. We now claim that λ is a ∂-∂-homeomorphism with image equal to S(G c). First we argue why the image of λ must be S(G c). Let therefore p ∈ S c(G c), and suppose p = tp(ā, b | c). Then λ(p) = tp(cā, cb ), and we must therefore argue why tp(cā) = tp(c b) = tp(cξ) so that λ(p) ∈ S(G) (cf. Propositions 4.2.3 and 4.2.4). Let ϕ(x) be a formula. Replace each occurrence of x i in ϕ by c i for i < n. Denote the resulting formula by ϕ(cx ). We know that tp(ā

| c) = tp(ξ | c), since p ∈ S c(G c). Hence, tp(cā)(ϕ(x)) = tp(ā | c)(ϕ(cx )) = tp(ξ | c)(ϕ(cx )) = tp(cξ)(ϕ(x)).
A similar calculation shows that tp(c b) = tp(cξ). Thus, λ(p) ∈ S(G c).

To show the other inclusion, suppose q ∈ S(G c). Then there are sequences ā and b in M such that q = tp(cā, cb ) with tp(cā) = tp(c b) = tp(cξ). If we let p be the type tp(ā, b | c), it is easy to check that p ∈ S c(G c), and clearly λ(p) = q. We conclude that Im(λ) = S(G c).

Next we show that λ is ∂-∂-continuous. In fact, λ is a ∂-∂-1-Lipschitz map. To see this, let p, q ∈ S c(G c) be given and let ε > 0. By Proposition 4.2.3, it follows that both types are realised in M c. By Proposition 2.5.14, M c is atomic. Therefore, by Lemma 2.5.12, we can find two realisations, (ā p , bp ) p and (ā q , bq ) q, such that d((ā p , bp ), (ā q , bq )) ≤ ∂(p, q) + ε. By definition of λ, we have that (cā p , cb p ) λ(p) and (cā q , cb q ) λ(q). It follows that

∂(λ(p), λ(q)) ≤ d M ((cā p , cb p ), (cā q , cb q )) ≤ d Mc ((ā p , bp ), (ā q , bq )) ≤ ∂(p, q) + ε.
Since ε > 0 was arbitrary, we conclude that ∂(λ(p), λ(q)) ≤ ∂(p, q) as we claimed. In particular, λ is ∂-∂-continuous.

We proceed to show that λ is injective with continuous inverse. For this, let γ : S(G c) → S x ⊕ȳ (T c) be the map given by

γ(tp(cā, cb )) = tp(ā, b | c).
First of all, we must argue why this is well-defined. Let therefore (cā, cb ) and (cā , cb ) both realise p ∈ S(G c), and let ϕ(x , ȳ ) be a formula with parameters from c. Let ϕ be the formula ϕ but where all occurrences of the constant symbol c i is replaced by the free variable x i , for i < n. Then ϕ (x, ȳ ) is an formula with free variables among (x, ȳ ). In particular, the free variables are among (x, ȳ), and thus we can apply p to ϕ and obtain:

tp(ā, b | c)(ϕ(x , ȳ )) = ϕ Mc (ā, b) = ϕ M (cā, b) = p(ϕ (x, ȳ )) = ϕ M (cā , b ) = ϕ Mc (ā , b ) = tp(ā , b | c)(ϕ(x , ȳ )).
We conclude that tp(ā, b | c) = tp(ā , b | c), rendering γ well-defined. Furthermore, clearly γ(λ(p)) = p and λ(γ(p)) = p, so λ is injective. All we are missing now is therefore that γ is ∂-∂-continuous.

To show this, we show continuity at each p ∈ S(G c). Let therefore ε > 0 and p = tp(cā, cb ) be given. We will produce a δ > 0 such that if ∂(p, q) < δ, then ∂(γ(p), γ(q)) < ε.

First we find

N ∈ N such that ∞ i=N 2 -i-1 diam(M) < ε/3. Let U ⊆ G be the basic open set U = {g ∈ G : d M (g(ā N , bN ), (ā N , bN )) < ε/3},
where āN and bN denote the tuples consisting of the N first elements of ā and b, respectively. By exact homogeneity and Effros' Theorem (Theorem 2.5.13 above), we get a δ > 0 such that

B((c, c), δ ) ∩ G • (c, c) ⊆ U • (c, c).
Set δ := min{ε/3, δ }/2 n , where we recall that n = |c| is the length of c. Suppose now that ∂(p, q) < δ. We will show ∂(γ(p), γ(q)) < ε. By atomicity of M, we find a realisation (c 0 ā , c1 b ) q such that d M ((cā, cb ), (c 0 ā , c1 b )) < δ.

Since q ∈ S(G c), it follows (by τ -continuity of projections between type spaces) that tp(c 0 , c1 ) = tp(c, c). By exact homogeneity, we find h ∈ G such that h • (c, c) = (c 0 , c1 ). In particular, this means c0 = c1 , so we let c denote this tuple. Moreover, we have that

d M ((c, c), (c , c )) = d M (c, h • c) ≤ 2 n n-1 i=0 2 -i-1 d M (c i , c i ) ≤ 2 n d M ((cā, cb ), (c ā , c b )) < 2 n δ ≤ δ . It follows that h • (c, c) ∈ B((c, c), δ ) ∩ G • (c, c). Therefore, there is h ∈ U such that h • (c, c) = h • (c, c). We now have that d M (h (ā, b), (ā, b)) ≤ N -1 i=0 2 -i-1 d M (h (ā N , bN ), (ā N , bN )) + ε/3 ≤ 2ε/3. Therefore, we have that h -1 (c ā , c b ) = (c h -1 ā , c h -1 b ) q and that ∂(γ(p), γ(q)) ≤ d Mc ((ā, b), (h -1 ā , h -1 b )) = d Mc (h (ā, b), (ā , b )) ≤ d Mc (h (ā, b), (ā, b)) + d Mc ((ā, b), (ā , b )) < 2ε/3 + ε/3 = ε,
where we have used, that even though h is not (necessarily) an M c automorphism, it is still an M c-isometry. We conclude that γ is ∂-∂-continuous, and so λ is a homeomorphism between S c(G c) and S(G c).

We can now show the 'in particular'-and 'moreover'-parts of the theorem: suppose that G is locally Roelcke precompact. Then by Zielinski's theorem, S(G) is ∂-locally compact. It follows that for any finite tuple c, the ∂-closed set S(G c) ⊆ S(G) is also ∂-locally compact. Since S(G c) is homeomorphic to the Roelcke completion of G c by the above, it follows, again by Zielinski's result, that G c is locally Roelcke precompact.

Moreover, by definition of locally Roelcke precompactness, G contains a Roelcke precompact identity neighbourhood. Such a neighbourhood must contain a set of the form {g : d M (g(ā), ā) < r} for some finite tuple of elements ā. It follows that the stabiliser of ā, G ā, is a Roelcke precompact subset of G. By the above, the closure of G ā inside S(G) is homeomorphic to the Roelcke completion of G ā (as a group). Therefore, G ā is a Roelcke precompact group as well.

Chapter 5

The Urysohn Metric Space

In the 1920's, in the posthumously published paper [START_REF] Urysohn | Sur un espace métrique universel[END_REF], P. S. Urysohn constructed a remarkable metric space that we today have named after him: the Urysohn metric space U. It is remarkable because it not only contains an isometric copy of any separable metric space (i.e. it is universal ), but it is the unique such space that is also ultrahomogeneous, i.e. any isometry between finite subsets extends to an isometry of the whole space. Around the same time, Banach and Mazur constructed another much more famous universal Polish space, C([0, 1]) cf. [START_REF] Banach | Théorie des opérations linéaires[END_REF], and perhaps that is why U did not catch a lot of attention at the time. In fact, it was almost forgotten for 60 years (!) before Katětov in the 1980's gave a new construction of U in [START_REF] Katětov | On universal metric spaces[END_REF], where he used the so-called Katětov functions to build a version of U around any separable metric space. These functions correspond to one-point extensions of the metric space. Katětov showed that U is the unique complete separable metric space with the so-called extension property for metric spaces (cf. Definition 5.1.3 below), meaning that any one-point extension of a finite metric space is realised in U.

When it comes to (metric) model theory, it is not just the universality of U that is important. It is this property together with ultrahomogeneity that makes U such an interesting space from the viewpoint of model theory. However, U itself does not fit well into the setup of metric model theory because it is unbounded. That is probably why most authors content themselves with studying the Urysohn sphere U 1 from a model theoretical point of view. Recall that U 1 is the unique ultrahomogeneous universal separable complete metric space with distances bounded by 1. Viewed as a metric structure, this space is ℵ 0 -categorical, and this simplifies the model theoretical study of U 1 tremendously. In the case of the unbounded space U, things are not quite as easy. The goal for this chapter is therefore to develop the metric model theory of U with its unbounded metric. Our main result is a sort of almost-categoricity theorem for the theory of U, T U . We show that any model of T U is a disjoint union of structures, each of which is similar to U itself (cf. Theorem 5.3.3). In 101 particular, it follows that any separable model is a countable disjoint union of copies of U. Hence, there are only countably many separable models of T U . Using this theorem, we obtain a number of corollaries concerning the model theory of U and its automorphism group: T U eliminates quantifiers, U is a prime model of T U and Iso(U) is locally Roelcke precompact. This latter result was already known, but our proof here is new, since it uses Theorem 4.2.5 from the previous chapter.

We begin with a short section containing the preliminaries we need concerning the Urysohn space. Then we explain how to view even unbounded metric spaces as metric structures before finally proving the mentioned categoricitylike theorem for T U together with all its corollaries.

Preliminaries on U

By now, there is a vast body of literature on U, and we will not attempt to cover it at all. Instead, we will only include what we will be using later on. As a matter of fact, we are not going to need many results concerning the Urysohn space, so this section will be quite short. The interested reader is referred to the excellent exposition of the theory of the Urysohn space in [START_REF] Melleray | Some geometric and dynamical properties of the Urysohn space[END_REF].

Basically, all we need are the two most fundamental results concerning U, together with the building blocks of Katětov's construction. These building blocks are known as Katětov maps and the extension property. Katětov maps correspond, as mentioned, to one-point extensions of a given metric space, and the extension property says that any such extension is already present in the space. This property turns out to characterise the Urysohn space, since any two separable complete metric spaces with the extension property are isometric and hence isometric to U. Katětov maps are defined as follows:

Definition 5.1.1. Let (X, d) be a metric space. A map f : X → R is a Katětov map if for all a, b ∈ X, it holds that |f (a) -f (b)| ≤ d(a, b) ≤ f (a) + f (b).
The set of all Katětov maps on X is denoted by E(X).

If we are given a finite tuple ā ∈ X n and a tuple r ∈ R n , we say that r is a Katětov assignment (with respect to d) if the map a i → r i is a Katětov map.

Remark 5.1.2. It is worth mentioning here that one may view the elements of E(X) as the quantifier free types over X, if we consider X as a metric structure or as a subset of some larger metric structure.

Observe that if (X, d) is a metric space, y is some new element (possibly already in X) and f ∈ E(X), then the map d , defined by d (x, y) = f (x) and d (x, x ) = d(x, x ) for x, x ∈ X, defines a metric on X ∪ {y} extending d. Thus, each f ∈ E(X) corresponds to a one-point extension of X.

The extension property is now defined as follows:

Definition 5.1.3. Let (X, d) be a metric space. Then X has the approximate extension property if the following holds for all n ∈ N:

∀ā = (a i ) ∈ X n ∀f ∈ E(ā) ∀ε > 0 ∃b ∈ X ∀a i |d(a i , b) -f (a i )| < ε.
If the b above can be chosen such that d(a i , b) = f (a i ) for each i, then (X, d) is said to have the extension property.

It turns out that it is possible to axiomatise the approximate extension property in first-order continuous logic, as we will see below.

We finish the section by stating the two fundamental results concerning U that we will need. ). Any complete separable metric space with the approximate extension property is isometric to the Urysohn space.

The theory of metric spaces

In this section, we explain exactly how we can view unbounded metric spaces as metric structures. In general, there are several ways to get around this obstacle. In our case, one easy and simple way around it is to add countably many predicates to our language that are meant to "encode" the unbounded distance truncated at some finite value. Of course, this does complicate things slightly. For instance, the theory of U, T U , will no longer be ℵ 0 -categorical as it is the case for the Urysohn sphere. This follows (one is tempted to say "as usual") by the Compactness Theorem, since using that, we can construct models consisting of several copies of U with "infinite distance" between the copies. Fortunately, it turns out that these are the only models of T U (cf. Theorem 5.3.3 below). This will be the content of the next section, but first things first. We begin by defining the signature we will use and the theory of metric spaces in this signature.

For each natural number n ≥ 1, we let P n be a binary predicate symbol. The bound for P n is [0, n] and the modulus of uniform continuity for each P n is the map r → min{r, 1}/2. Denote the resulting signature by L U . To keep our notation a little simpler, we will identify the metric symbol d with P 1 .

The L U -theory of metric spaces is denoted by T met and is defined as follows:

where we used tri N , stab n and stab m . Openness of the classes follows as each predicate P M n is continuous and the class of any a ∈ M is given by

[a] E M = n {b ∈ M : P M n (a, b) < n}.
Thus, [a] E M is open and therefore also closed, since the complement is the union of all other classes.

The map ρ M is now defined as follows:

Definition 5.2.4. Let M T met . We define a map ρ M : M × M → [0, ∞] by ρ M (a, b) = sup n P M n (a, b) if aE M b ∞ otherwise. (5.1)
It is easy to check that since M T met , ρ M restricted to any E M -class is actually a metric. Moreover, each class is complete with respect to ρ M , since it is complete with respect to d M . Thus, any model of T met is a disjoint union of ρ M complete metric spaces. In the next section, we will show that if M T U , then each E M -class has the extension property with respect to the metric ρ M . It follows that if M T U is separable, then M is a disjoint union of at most countably many Urysohn spaces. From this it will follow that T U eliminates quantifiers, that U is a prime model and that Aut(U) = Iso(U) is locally Roelcke precompact.

Model theory of U

In this section we will develop the model theory of U using the constructions from the previous section. The first thing we need to do is to explain exactly how we view U as an L U -structure.

For each n ≥ 1, we let P U n be defined by

P U n (a, b) = min{ρ(a, b), n},
where ρ denotes the usual unbounded distance on U. As the metric on our structure we just truncate at 1, i.e.:

d U (a, b) = P U 1 (a, b) = min{ρ(a, b), 1}.
It is not hard to check that each P U n does indeed respect the prescribed modulus. We will denote the metric structure obtained in this way by (U, {P U n }) or simply U when it is clear from context that we consider the Urysohn space as an L U -structure. As mentioned, the theory of U is denoted by T U .

The next thing we want to do is to axiomatise the approximate extension property. As explained in the first section of the chapter, the extension property characterises U. Therefore, once we have axiomatised this property, it will follow that any model M of T U will have the extension property. Moreover, each E M class will also have this property, and so it will follow that M is a disjoint union of structures each having the extension property. The sentences that describe the approximate extension property are somewhat overwhelming at first glance, so we will make a few remarks below the definition to explain what they are saying exactly. Definition 5.3.1. For each n ≥ 2, r ∈ R n and m ∈ N and a finite tuple of variables x = (x 1 , . . . , x n ) define a sentence σ r,m by σ r,m := sup

x min

         min i,j {r i + r j . -P m (x i , x j )}, min i =j {P m (x i , x j ) . -|r i -r j |}, min i,j {m . -P m (x i , x j )}, min i {m . -r i }, inf y max i |P m (x i , y) -r i |          .
We denote the collection of all these sentences by T ext .

In fact, we will show below that T ext together with T met axiomatise T U , meaning that M T U if and only if M T ext ∪ T met . However, before moving on, a few remarks on what the sentences defined above are saying are in order.

The four minimums will, if they all fail to be 0, express that for all distinct i, j the following hold:

(i) r i + r j > P m (x i , x j ), (ii) P m (x i , x j ) > |r i -r j |, (iii) P m (x i , x j ) < m, (iv ) r i < m.
The last two items above just say that m is big enough to describe the distances between a given finite tuple of elements. The first two items say that r describes a Katětov assignment with respect to the metric P m . Thus, if items (i) through (iv) above hold, the last part of the sentence σ r,m will make sure that we can extend a finite tuple by an element with the distance prescribed by r to the elements of the tuple. We hope that these remarks will help the reader understand the sentences of T ext better.

To show that T ext together with T met axiomatise T U , we must of course verify that U satisfies T ext . Lemma 5.3.2. The Urysohn space U satisfies T ext , i.e. for any σ ∈ T ext we have σ U = 0.

Proof. Let ā = (a 1 , . . . , a n ) ∈ U n and let r ∈ R n , m ∈ N. We may assume that all the four minimums of σ U r,m fail to be 0. Thus, items (i) through (iv) above hold for ā, i.e. we have

(i) r i + r j > P U m (a i , a j ), (ii) P U m (a i , a j ) > |r i -r j |, (iii) P U m (a i , a j ) < m, (iv ) r i < m.
Therefore, since U has the extension property, we can for any ε > 0 find some

b ∈ U such that max i |P m (a i , b) -r i | = max i |ρ(a i , b) -r i | < ε.
In other words, the infimum in σ U r,m is 0. Hence, σ U r,m = 0 for all r and m.

Next we show that all models of T ext ∪T met are disjoint unions of complete metric spaces with the extension property. In particular, a separable model of T ext ∪ T met is a disjoint union of Urysohn spaces.

M . In particular, if M is separable, M is a countable disjoint union of L U -structures each of which is isomorphic to (U, {P U n }).
Proof. Let M T be given. To show that each E M -class has the extension property, we will show that each class satisfies the approximate extension property. It follows by Theorem 5.1.4 above that each class actually has the extension property, since each class is complete.

Let C be an E M -class and let ā = (a 1 , . . . , a n ) be a tuple of elements from C. Suppose that we are given a Katětov map f on (ā, ρ M ). Then we let r = (f (a i )) i and m ∈ N be large enough so that P M m (a i , a j ) < m for all i, j and so that m is a lot larger than max r i (2 max r i < m will suffice).

Let now ε > 0 be given. Without loss of generality we may assume ε < 1. Since f is Katětov, we have that

|r i -r j | ≤ P M m (a i , a j ) ≤ r i + r j .
By a slight perturbation of r, we can obtain an r such that for all distinct i, j we have |r i -r j | < P M m (a i , a j ) < r i + r j , and so that for any b ∈ M, we have that if

max i |P M m (a i , b) -r i | < ε/2, then max i |P M m (a i , b) -r i | < ε.
Since the inequalities are strict, items (i) through (iv) hold for ā, i.e.

(i) r i + r j > P M m (a i , a j ), (ii) P M m (a i , a j ) > |r i -r j |, (iii) P M m (a i , a j ) < m, (iv ) r i < m.
Hence, the infimum of σ M r ,m must be 0, so we find b in M such that

max i |P M m (a i , b) -r i | < ε/2.
By the choice of r , it follows that max i |P M m (a i , b) -r i | < ε. Finally, since ε < 1 and since m is a lot larger than max r i , it follows that b ∈ C. In other words, we see that C has the approximate extension property. Since each class is complete, it follows from Theorem 5.1.4 that all classes have the actual extension property.

For the 'in particular'-part of the proposition, assume that M is separable. Then, since each E M -class is open, it follows that there can be at most countably many classes. Moreover, any E M -class C equipped with the metric ρ M is separable, since any d M -dense subset of C is ρ M -dense as well. It follows from Theorem 5.1.5 that (C, ρ M ) is isometric to (U, ρ), which implies that (C, {P M n }) is isomorphic to (U, {P U n }). Thus, M is a countable disjoint union of Urysohn spaces.

As a corollary, we can prove that T met ∪T ext admits quantifier elimination. Proof. We show that T has the back-and-forth property. Thus, we let M and N be ω-saturated models of T and suppose two tuples ā ∈ M k and b ∈ N k have the same quantifier free type. We must show that for any c ∈ M, we can find some c ∈ N such that the quantifier free type of (ā, c) is the same as that of ( b, c ).

Suppose first that c has infinite distance to all elements of ā, i.e. that for all n ∈ N and i < k we have P M n (a i , c) = n. Let (N , b) denote the model with the elements of b named as constant symbols and let Tb denote the theory of this model. Then, by the Compactness Theorem and Theorem 5.3.3, there is some type p ∈ S 1 (Tb) such that p(P n (x, b i )) = n for all i ≤ k and n ∈ N. The existence of such a type follows, since the collection of these formulas is finitely satisfiable because we can use the extension property to find elements arbitrarily far from each other in the metric ρ N from Definition 5.2.4 above. The ω-saturation of N implies that there is some c realising this type p. It follows that ( b, c ) has the same quantifier free type as (ā, c), since they agree on each P n (x i , x j ).

The other case is similar: if c has finite distance to some a i of ā, i.e. if there is some n such that P M n (a i , c) < n, then a i E M c. Since the E N -class of b i has the extension property for ρ N and since the quantifier free types of ā and b are the same, it follows that we may find the c in N that we need. We conclude that T eliminates quantifiers by Proposition 2.4.5 above.

As another corollary, we get that U is a prime model of T ext ∪ T met .

We conclude the chapter with another corollary of our analysis above saying that the stabilisers of any a ∈ U, i.e. the group of those g ∈ Aut(U) such that g(a) = a, is a Roelcke precompact group. To show this, we show that all types in the τ -closure S(G a ) are realised in U. It follows by Proposition 4.2.3 that S(G a ) = S(G a ). Let therefore p ∈ S(G a ) be given. Then find some separable model M T U realising p with ( b1 , b2 ). By Theorem 5.3.3, M is a disjoint union of structures isomorphic to U. By Proposition 4.2.4, we have that tp( b1 ) = tp( b2 ) = tp(aξ). It follows that for all i, j ∈ N, there is some n ∈ N such that

P M n (b i 1 , b j 2 ) ≤ P n (b i 1 , a) + P n (b j 2 , a) < n.
Therefore, b1 and b2 belong to the same copy of U, and so p is realised in U.

Chapter 6

The Urysohn Diversity

Diversities were introduced by Bryant and Tupper in [START_REF] Bryant | Hyperconvexity and tight-span theory for diversities[END_REF] and further developed in [START_REF] Bryant | Diversities and the geometry of hypergraphs[END_REF] in order to generalise applications of metric space theory to combinatorial optimisation and graph theory to the hypergraph setting. The idea is very simple: instead of only assigning real numbers to pairs of elements, a diversity assigns a real number to every finite subset of the space. This turns out to generalise metric spaces quite nicely, and in [START_REF] Bryant | Hyperconvexity and tight-span theory for diversities[END_REF] and [START_REF] Bryant | Diversities and the geometry of hypergraphs[END_REF] the authors prove diversity versions of a number of results concerning or using metric spaces. The term diversity comes from a special example of a diversity that appears in phylogenetics and ecological diversities demonstrating the broad variety of applications of diversities and of mathematics in general of course. The interest in diversities from a model theoretical point of view began with the paper [START_REF] Bryant | A universal separable diversity[END_REF], where the authors construct a diversity analogue of the Urysohn metric space by adapting Katětov's construction to the diversity setting. They call the resulting structure U the Urysohn diversity. The existence of such a universal object among diversities gives rise to a plethora of questions concerning it, both from the side of model theory, but also from the side of Polish group theory, since the automorphism group of U is virtually unstudied.

In this chapter, we will develop the model theory of the Urysohn diversity along the lines of what we did for the Urysohn metric space. This study was briefly started in [START_REF] Bryant | Fraïssé limits for relational metric structures[END_REF], where the authors of [START_REF] Bryant | A universal separable diversity[END_REF] construct U as a metric Fraïssé limit in the sense of Ben-Yaacov in [START_REF] Ben | Fraïssé limits of metric structures[END_REF]. Hence, they have to explain how to view U as a metric structure. In this chapter, we will further develop the model theory of U. We will show that U is a prime model and that its theory eliminates quantifiers. This will allow us to apply Theorem 4.2.5 and show that the automorphism group of U is locally Roelcke precompact. These results follow from an almost-categoricity result similar to Theorem 5.3.3, saying that any separable model of the theory of U is a disjoint union of isomorphic copies of U.

The rest of the results of this chapter are to appear in the paper [START_REF] Hallbäck | Automorphism groups of universal diversities[END_REF]. The first of these results is that Aut(U) is a universal Polish group. We will show 111 this by using the Katětov construction of U done in [START_REF] Bryant | A universal separable diversity[END_REF] and adapt Uspenskij's proof of this fact for the isometry group of the Urysohn metric space. We have therefore, for the convenience of the reader, included the main ingredients from the construction done in [START_REF] Bryant | A universal separable diversity[END_REF].

We will also construct the rational Urysohn diversity U Q by showing that the class of finite rational diversities is a Fraïssé class (in the classical sense). Here, rational simply means that the diversity map only attains rational values. It follows that this class has a Fraïssé limit, which establishes the existence of U Q . Moreover, we will show that the completion of U Q is (isomorphic to) the Urysohn diversity, thus providing another proof of the existence of U. For the convenience of the reader, we have included a brief introduction to Fraïssé theory.

With the existence of U Q established, it is easy to show that Aut(U Q ) has a dense conjugacy class by applying a theorem of Kechris and Rosendal from [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF]. Furthermore, we will show that Aut(U Q ) embeds densely into Aut(U), from which it follows immediately that Aut(U) also has a dense conjugacy class.

In the last section of the chapter (and the thesis), we will show that Aut(U Q ) has ample generics. Ample generics is a property with many strong implications such as the automatic continuity property, the small index property, having a unique Polish group topology and the fact that the group cannot be the union of countably many non-open subgroups. All of these notions will be explained below. Moreover, it actually implies that the group has a dense conjugacy class. Nevertheless, the proof mentioned above is so easy that we have chosen to include it in any case. Ample generics of Aut(U Q ) follows from an extension theorem for diversities inspired by a result of Solecki in [START_REF] Solecki | Extending partial isometries[END_REF] and another theorem of Kechris and Rosendal from [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF].

Introduction to diversties

In this section, we have collected a few preliminaires on diversities. For further study, we refer the reader to [START_REF] Bryant | A universal separable diversity[END_REF][START_REF] Bryant | Fraïssé limits for relational metric structures[END_REF][START_REF] Bryant | Hyperconvexity and tight-span theory for diversities[END_REF][START_REF] Bryant | Diversities and the geometry of hypergraphs[END_REF].

Let us begin with the defintion of a diversity. Definition 6.1.1. A diversity is a set X equipped with a map δ, the diversity map, defined on the finite subsets of X to R such that for all finite A, B, C ⊆ X we have (D1) δ(A) ≥ 0 and δ(A) = 0 if and only if |A| ≤ 1.

(D2) If B = ∅, then δ(A ∪ B) + δ(B ∪ C) ≥ δ(A ∪ C).
As an abuse of language, we will follow [START_REF] Bryant | A universal separable diversity[END_REF] and [START_REF] Bryant | Fraïssé limits for relational metric structures[END_REF] and from time to time refer to the diversity map as a diversity as well. Hopefully this confusion of names will not cause confusion for the reader.

The following observation is useful and is easy to verify: Lemma 6.1.2. (D2) holds for δ if and only if the following two conditions hold:

(D2') Monotonicity, i.e. δ(A) ≤ δ(B) if A ⊆ B. (D2") Connected sublinearity, i.e. δ(A ∪ B) ≤ δ(A) + δ(B) if A ∩ B = ∅.
Another general observation to make is that any diversity is automatically a metric space, since the map d(a, b) = δ({a, b}) defines a metric. We refer to this metric as the induced metric. A diversity is complete, respectively separable, if the induced metric is complete, respectively separable. A bijective map f : X → Y between two diversities that preserves the values of the diversity map will be called an isoversity. If Y = X, we will call f an autoversity or simply an automorphism of X if the structure f preserves is clear from context. The group of all autoversities of a diversity X is denoted by Aut(X). It is, as usual, equipped with the pointwise convergence topology with which it becomes a topological group.

A very important observation to make is that for each n ∈ N, the diversity map induces a uniformly continuous map δ n on X n given by δ n (x 0 , . . . , x n-1 ) = δ({x 0 , . . . , x n-1 }).

We will use this fact several times, so we state it here for future reference and the convenience of the reader. The proof can be found both in [START_REF] Bryant | A universal separable diversity[END_REF] and in [START_REF] Bryant | Fraïssé limits for relational metric structures[END_REF]. Note that this implies that Aut(X) is a closed subgroup of Iso(X). It follows that Aut(X) is a Polish group whenever X is a complete and separable diversity. Lemma 6. 1.3 ([19,Lemma 21]). Let (X, δ) be a diversity and for n ∈ N let δ n denote the map on X n that δ induces. Then δ n is 1-Lipschitz in each argument. It follows that for all x, ȳ ∈ X n we have

|δ n (x) -δ n (ȳ)| ≤ δ(x i , y i ).
In particular, δ n is uniformly continuous.

In fact, to ease our notation, we will hardly discern between the maps δ n and the diversity map itself. Hence, we will from time to time write δ(ā) for an ordered tuple ā = (a 0 , . . . , a n ) instead of writing δ({a 0 , . . . , a n }). It will always be perfectly clear from the context what is meant and therefore this causes no confusion. Given a metric space, there are several ways we can use the metric to define a diversity map. Below we have included some such examples. We have also included a few other examples and the interested reader is referred to the two papers [START_REF] Bryant | Hyperconvexity and tight-span theory for diversities[END_REF] and [START_REF] Bryant | Diversities and the geometry of hypergraphs[END_REF] for a substantial list of further examples. (ii) Steiner diversity. Let (M, d) be a metric space. Given finite A ⊆ M , we let δ S (A) be the minimal length of an undirected tree connecting A.

Here, the length of a tree is the sum of all d(a, b) such that {a, b} is an edge in the tree. (M, δ S ) is called the Steiner diversity on M because such a minimal tree is known as a Steiner tree.

(iii) Hypergraph Steiner diversity. More generally, if (X, E, w) is a connected hypergraph equipped with a weight function w : E → R ≥0 , we let δ H (A) be the minimal weight of a connected subhypergraph E ⊆ E covering A.

(X, δ H ) is called the hypergraph Steiner diversity. This diversity served as inspiration for a crucial step in Theorem 6.5.6 below.

(iv ) Smallest enclosing ball diversity. Let (M, d) be a metric space. For finite A ⊆ M , we let δ(A) be the minimal diameter of a ball containing A.

(v ) Truncated diversity. Let (X, δ) be a diversity and k ∈ N. The k-truncated diversity on X is the diversity (X, δ k ), where δ k (A) is the maximum δ(B) for B ⊆ A with |B| ≤ k. Note that δ 2 is just the diameter diversity.

The diameter and Steiner diversities are important examples, since it turns out that they give lower and upper bounds for the values of any diversity. Theorem 6. 1.4 ([21,Theorem 4]). Let (X, δ) be a diversity and let δ diam and δ S denote the diameter and the Steiner diversities, respectively, that the induced metric gives rise to. Then for any finite A ⊆ X we have

δ diam (A) ≤ δ(A) ≤ δ S (A).
Intuitively speaking, the diameter gives us a lower bound because it only takes into account two elements of any finite set, i.e. the two of maximal distance. Similarly, the Steiner diversity takes all elements into account since the tree must connect all the elements of a given finite subset. Therefore it gives us an upper bound. For a detailed proof, we refer the reader to [START_REF] Bryant | Diversities and the geometry of hypergraphs[END_REF].

It is natural to ask whether or not there are intermediate diversity maps. In [START_REF] Bryant | A universal separable diversity[END_REF]Proposition 23] the authors show that an easy consequence of the universality of U is that it is neither a diameter diversity nor a Steiner diversity. Hence, U is an example of an intermediate diversity. Another good class of examples is linear combinations of a diameter and a Steiner diversity using positive coefficients.

As mentioned, the construction of the Urysohn diversity in [START_REF] Bryant | A universal separable diversity[END_REF] follows Katětov's construction of the ditto metric space. Since we shall use this construction to prove that Aut(U) is a universal Polish group, we need to introduce the diversity analogue of Katětov functions. These are the so-called admissible maps. First, we let [X] <ω denote the finite subsets of X. Admissible maps are then defined as follows: Definition 6.1.5. Let (X, δ X ) be a diversity. A map f : [X] <ω → R is admissible if the following hold:

(i) f (∅) = 0, (ii) f (A) ≥ δ X (A) for every A, (iii) f (A ∪ C) + δ X (B ∪ C) ≥ f (A ∪ B) for all A, B, C with C = ∅, (iv ) f (A) + f (B) ≥ f (A ∪ B).
The set of all admissible maps on (X, δ X ) is denoted E(X). Remark 6.1.6. Note that if we view the diversity X as a metric structure, the admissible maps are just the quantifier free 1-types over X. This is similar to the Katětov maps on a metric space (cf. Remark 5.1.2).

The reason why these maps are called admissible is because they define diversity extensions as the lemma below tells us. Lemma 6.1.7 ([18, Lemma 2]). Let (X, δ) be a diversity and f : [X] <ω → R be a map. Then f ∈ E(X) if and only if for some y, the map δ : [X ∪ {y}] <ω → R given by δ(A) = δ(A), δ(A ∪ {y}) = f (A)

for A ⊆ X finite, defines a diversity map on X ∪ {y}.

Similarly to the metric setting, we can define a diversity map on the set of admissible maps. This is defined as follows: Definition 6.1.8 ([18, Page 5]). Let (X, δ) be a diversity. On [E(X)] <ω we define a map δ by

δ({f 1 , . . . , f n }) = max j≤k sup f j i =j A i - i =j f i (A i ) : A i ⊆ X finite whenever n ≥ 2 and δ(f ) = δ(∅) = 0. Observe that δ(f 1 , f 2 ) = sup B finite |f 1 (B) -f 2 (B)|.
Moreover, as the notation suggests, δ is a diversity map on E(X).

Theorem 6.1.9 ([18, Theorem 3]). Let (X, δ) be a diversity. Then (E(X), δ) is a diversity and (X, δ) embeds into (E(X), δ) via the map x → κ x , where κ x (A) = δ(A ∪ {x}).

Unfortunately, just like in the metric setting, E(X) need not be separable even if X is. Therefore, we need to restrict ourselves to a subspace of E(X) to maintain separability. This is the subspace of the finitely supported admissible maps. These are defined as follows: Definition 6.1.10. Let (X, δ) be a diversity and let S ⊆ X be any subset. If f ∈ E(S), then we define the extension of f to X by

f X S (A) = inf f (B) + b∈B δ(A b ∪ {b}) : B ⊆ S finite, b∈B A b = A ,
where A ⊆ X is finite. We say that S is the support of f X S . The set of all finitely supported admissible maps on X is denoted E(X, ω), i.e. E(X, ω) is the set of all those h ∈ E(X) such that for some finite S ⊆ X and some f ∈ E(S), we have h = f X S .

Of course, one needs to check that the extension map f X S is in fact admissible. We refer the reader to [START_REF] Bryant | A universal separable diversity[END_REF]Lemma 6] for the details. It is easy to check that κ x is supported on {x} for any x ∈ X, and hence that X embeds into E(X, ω). Therefore, (E(X, ω), δ) is a diversity extension of X. Moreover, E(X, ω) is separable. Theorem 6. 1.11 ([18,Theorem 9]). Let (X, δ) be a separable diversity. Then (E(X, ω), δ) is a separable diversity as well.

One can then iterate this construction and obtain a Katětov tower on a given separable diversity X consisting of a sequence of separable diversities (X n , δ n ), where X n = E(X n-1 , ω) embeds into X n+1 . The union X ω of all of these diversities turns out to have an extension property similar to the extension property for metric spaces. Furthermore, it turns out that this property characterises the Urysohn diversity -just like for the metric space. The extension property for diversities is defined as follows: Definition 6.1.12. A diversity (X, δ X ) has the approximate extension property if for any finite subset F ⊆ X, any admissible map f defined on F and any ε > 0, there is x ∈ X such that |f (A) -δ X (A ∪ {x})| ≤ ε for A ⊆ F .

If the above holds for ε = 0, (X, δ X ) has the extension property.

Just like in the metric setting, it turns out that complete diversities with the approximate extension property actually have the extension property. In fact, we have: Proposition 6. 1.13 ([18,Lemmas 16 and 17]). Suppose (X, δ X ) is a separable diversity with the approximate extension property. Then its completion has the extension property.

Furthermore, as mentioned, this property characterises the Urysohn diversity, meaning that any two Polish diversities, i.e. with complete and separable induced metrics, that have the extension property are isomorphic. This is one of the main results of [START_REF] Bryant | A universal separable diversity[END_REF]. Theorem 6.1.14 ([18, Theorems 14 and 22]). Any two Polish diversities both having the extension property are isomorphic. In particular, any Polish diversity with the extension property is isomorphic to the Urysohn diversity.

Of course, we invite the reader to study [START_REF] Bryant | A universal separable diversity[END_REF] for the details. These were all the preliminaries we will need on diversities.

Model theory of the Urysohn diversity

As we have mentioned, Theorem 4.2.5 applies to the Urysohn diversity. In fact, as we shall see in this section, the model theory of the Urysohn diversity is very similar to the one of the Urysohn metric space. However, we do need to make some adjustment to the arguments and set things up properly, since we are dealing with a different structure.

In order for us to study the model theory of the Urysohn diversity, we need to explain how to describe the unbounded diversity. In [START_REF] Bryant | Fraïssé limits for relational metric structures[END_REF], the authors work in a continuous logic where unbounded predicates are allowed (cf. [START_REF] Ben | Continuous first order logic for unbounded metric structures[END_REF] for more details on this version of continuous logic). However, our set-up is slightly more restrictive and does not allow this. On the other hand, just like for the Urysohn metric space, we can amend this simply by adding more predicates to the language. Another easily fixed problem, which is addressed in [START_REF] Bryant | Fraïssé limits for relational metric structures[END_REF] as well, is that a diversity map does not depend on a fixed number of variables. Therefore, for every m ≥ 2 and n ≥ 1, we will introduce an m-ary predicate to describe the diversity map restricted to subsets of size m and truncated at n.

With the above considerations in mind, we let D n,m be an m-ary predicate symbol with bound [0, n] and modulus of continuity ∆ n,m (ε) = min{ε, 1}/m. Let L D denote the first order metric signature containing these predicates. We now turn the Urysohn diversity into an L D -structure by interpreting each predicate as the map D U n,m : U m → R given by

D U n,m (ā) = min{δ U (ā), n}.
As the metric we choose

d U (a, b) = D 1,2 (a, b) = min{δ U (a, b), 1}.
The first thing we should do is to verify that these maps actually respect the prescribed moduli. This follows from a straightforward application of Lemma 6.1.3, but for the convenience of the reader, we provide the details in the lemma below. Proof. We must show that if It is clear that if both δ U (ā), δ U ( b) ≥ n, the difference above is 0, so the inequality is true. Next, we note that since

d U m (ā, b) = max i d U (a i , b i ) < ∆ n,m (ε), then |D U n,m (ā) -D U n,m (
d U m (ā, b) < ∆ n,m (ε) < 1, we have d U (a i , b i ) = δ U (a i , b i ).
Therefore, by applying Lemma 6.1.3, we have that

|D U n,m (ā) -D U n,m ( b)| ≤ |δ U (ā) -δ U ( b)| ≤ i δ U (a i , b i ) < m∆ n,m (ε) ≤ ε.
We conclude that D U n,m respects ∆ n,m .

Thus, U equipped with this structure is a metric L D -structure. Let T U denote the theory of (U, {D U n,m }). In [START_REF] Bryant | Fraïssé limits for relational metric structures[END_REF]Section 5], there is a list of useful properties that the predicates D n,m must have in all models of T U . We have included it here for the convenience of the reader. Suppose therefore that M T U . Then we have the following: (vi) The predicates stabilise, i.e. if D M n,m (ā) < n, then for all k ≥ n we have

(i) D M n,1 (a) = 0 for all a ∈ M. (ii) D M n,m is permutation invariant. (iii) D M n,m+2 (a 0 , . . . , a m-1 , a m , a m ) = D M n,m+1 (a 0 , . . . , a m ) for a i ∈ M. (iv ) D M n,m+1 (ā, b) ≥ D M n,m (
D M k,m (ā) = D M n,m (ā)
In light of item (iii), we will write D M n,m (ā) also for tuples of length less than m, since we can just repeat the last element until we have a tuple of length m. This will simplify our notation a bit.

As in the previous chapter concerning the Urysohn metric space, we will exploit Theorem 6.1.14 to show that any model of the theory of U is a disjoint union of models having the extension property. In particular, a separable model of T U will be a disjoint union of copies of U. In order for us to state and prove this, we need to introduce some notation and make some definitions that are similar to the ones we made for the Urysohn metric space.

D M N,2 (a, c) ≤ D M n,2 (a, b) + D M m,2 (b, c) < n + m = N.
Thus, E M is transitive as well.

Openness of the classes follows from continuity of D M n,2 , since given any a ∈ M, the class of a is given by

[a] E M = n {b : D M n,2 (a, b) < n}.
Thus, each class is also closed, since the complement is the union of all other classes.

Using the predicates, we can equip any model of T U with a "diversity" map if we allow the value infinity. Recall that [M] <ω denotes the set of all finite subsets of M. Definition 6.2.4. Let M be a model of T U . Define a map

δ M : [M] <ω → [0, ∞] by δ M ({a 0 , . . . , a m-1 }) := sup n D M n,m ({a 0 , . . . , a m-1 }) if ∀i < m a 0 E M a i ∞ otherwise.
Note that if we restrict δ M to any E M -class, we obtain an actual diversity map because M T U . Moreover, it is not hard to see that δ M induces a complete metric on each E M -class. Thus, M is a disjoint union of complete δ M -diversities. Furthermore, we obtain the following theorem: Theorem 6.2.5. Let M be a model of T U and let E M and δ M be defined as above. Then each E M -class has the extension property with respect to δ M . In particular, if M is separable, M is a disjoint union of L D -structures isomorphic to U.

Proof. Let M T U be given. We will show that each E M -class has the approximate extension property with respect to δ M . It follows from Proposition 6.1.13 that each class has the actual extension property.

To do this, we will axiomatise the approximate extension property for diversities in continuous logic. For n ≥ 1, m ≥ 2 and r ∈ R P(m) (where P(m) denotes the power set of m) such that r(∅) = 0, we define a sentence σ r n,m by

sup x min                  min I⊆m {r(I) . -D n,m (x I )}, min I,J⊆m {r(I) + r(J) . -r(I ∪ J) : I, J = ∅} min I,J,K⊆m r(I ∪ J) + D n,m (x I∪K ) . -r(J ∪ K) : I = ∅ & |I ∪ K| ≥ 2 n -D n,m (x), min I⊆m {n . -r(I)}, min i∈m,J⊆m {r({i} ∪ J) -r(J)} inf y max I⊆m |D n,m+1 (x I , y) -r(I)|                 
Here, for a set I ⊆ m, xI denotes the tuple (x i : i ∈ I). The formulas within the first minimum of the sentence above will be referred to as the subformulas of the sentence. We now claim, first of all, that since U has the extension property, it will satisfy every σ r n,m . To see this, let n ≥ 1, m ≥ 2 and r ∈ R P(m) with r(∅) = 0 be given. We must show that U σ r n,m = 0, i.e. we must show that for every tuple ā ∈ U m , at least one of the subformulas of σ r n,m is 0. Of course, we can suppose that all subformulas except for the infimum are non-zero. Then what we must show is that the infimum is 0. Define a map f on P(ā) by f (ā I ) := r(I). We now claim that f is admissible for the usual diversity map δ U on U. To see this, we verify that conditions (i)-(iv) of Definition 6.1.5 hold. First, we note that δ U (ā

I ) = D U n,m (ā I ) because n > D U n,m (ā). Furthermore, we have that (i) f (∅) = r(∅) = 0. (ii) f (ā I ) = r(I) > δ U (ā I ) since the first subformula is non-zero.
(iii) For all I, J, K ⊆ m such that I = ∅ and |I ∪ K| ≥ 2 we have

f (ā I∪J ) + δ U (ā I∪K ) > r(J ∪ K) = f (ā J∪K ),
since the third subformula is non-zero. Furthermore, if |I ∪ K| = 1 and K = I, then clearly

f (ā I∪J ) + δ U (ā I∪K ) = f (ā I∪J ) = f (ā J∪K ).
If |I| = 1 and K = ∅, then by the sixth subformula, we get

f (ā I∪J ) + δ U (ā I∪K ) = f (ā I∪J ) ≥ f (ā J ) = f (ā J∪K ).
We conclude that (iii) of Definition 6.1.5 holds.

(iv ) For all non-empty I, J ⊆ m, we have

f (ā I ) + f (ā J ) = r(I) + r(J) > r(I ∪ J) = f (ā I∪J ),
since the second subformula is non-zero. Moreover, if one of I or J is empty, then trivially we have f (ā I ) + f (ā J ) ≥ f (ā I∪J ) as well. Thus, (iv) of Definition 6.1.5 holds.

It follows that f is admissible. Therefore, since U has the extension property, we find b ∈ U such that for all I ⊆ m, we have δ U (ā I ∪ {b}) = f (ā I ). Since min I {n .

-r(I)} > 0 by assumption, it follows that f (ā I ) < n, and hence

inf c∈U max I |D U n,m+1 (ā I , c) -r(I)| = 0, so U σ r n,m = 0. Therefore, if M
T U , then in particular M σ r n,m for all n, m and r ∈ R P(m) with r(∅) = 0. We claim that this implies that each E M -class has the approximate extension property with respect to δ M . Let therefore (a 0 , . . . , a m-1 ) =: ā be a tuple of E M -equivalent elements of M and let ε > 0 be given. Let moreover f be an admissible map on P(ā). Choose n large enough so that for every I ⊆ m, we have D M n,m (ā I ) < n and f (ā

I ) < n. Then δ M (ā I ) = D M n,m (ā I ) for all I ⊆ m. It follows that (i) f (∅) = 0, (ii) f (ā I ) ≥ D M n,m (ā I ) for all I ⊆ m, (iii) f (ā I∪J ) + D M n,m (ā I∪K ) ≥ f (ā J∪K ) for all I, J, K ⊆ m with I = ∅, (iv ) f (ā I ) + f (ā J ) ≥ f (ā I∪J ) for all I, J ⊆ m.
Let now r ∈ R P(m) be the image of f , i.e. r(I) = f (ā I ). By a slight perturbation of r, we can obtain an r ∈ R P(m) such that for all I ⊆ m, if

|D M n,m+1 (ā I , b) -r (I)| < ε/2, then |D M n,m+1 (ā I , b) -r(I)| < ε, and moreover such that • r (∅) = 0, • r (I) > D M n,m (ā I ) for all I ⊆ m, • r (I ∪ J) + D M n,m (ā I∪K ) > r (J ∪ K) for all I, J, K ⊆ m such that I = ∅ and |I ∪ K| ≥ 2,
• r ({i} ∪ J) > r (J) for all J ⊆ m and i / ∈ J, • r (I) + r (J) > r (I ∪ J) for I, J ⊆ m non-empty, It now follows that if M is a separable model of T U , it must be a countable disjoint union of clopen E M -classes having the extension property for δ M . Moreover, it is not hard to see that each class is δ M -separable and δ Mcomplete. Therefore, it follows that each class equipped with δ M as its diversity map is isomorphic as a diversity to U. Moreover, it is easy to check that this implies that each class is L D -isomorphic to (U, {D U n,m }) as claimed.

Remark 6.2.6. It can be shown that the collection of all the sentences σ r n,m together with the axioms for diversity maps is an axiomatisation of the theory of T U . This is completely similar to the case of the Urysohn metric space. We leave the details to be worked out by the reader.

As a corollary to the theorem above, we get that the theory of U eliminates quantifiers and that U is a prime model. Corollary 6.2.7. T U eliminates quantifiers.

Proof. We show that T U has the back-and-forth property (cf. Definition 2.4.4). Let therefore M and N be two ω-saturated models of T U and let ā and b be two tuples with the same quantifier free type from M and N , respectively, and let c ∈ M.

Like for the Urysohn metric space, there are two cases: either c is not E M related to any a i in ā or there is some a i in the E M -class of c. In the first case, using ω-saturation of N , we find a c ∈ N not E N -related to any b i . Hence, (ā, c) and ( b, c ) have the same quantifier free type. In the other case, we use Theorem 6.2.5 above and find, by the extension property, some c in the E N -class of b i such that ( b, c ) has the same quantifier free type as (ā, c).

It now follows easily that U is a prime model. Corollary 6.2.8. U is a prime model. In particular, U is atomic.

Proof. The proof is similar to the one for the Urysohn metric space: by Theorem 6.2.5 we find an isomorphic copy of U inside any model of T U , and the inclusion is elementary by quantifier elimination.

We will also say that H embeds (as a topological group) into G whenever H is isomorphic to a subgroup of G. Note that this is the same as saying that there is a map Φ : H → G such that Φ is a continuous algebraic injective homomorphism with continuous inverse.

The strategy to show that Aut(U) is universal is the following: any Polish group G can be embedded into the automorphism group of a separable diversity (X, δ X ). Denote the diversity Katětov tower on X by X ω . Then Aut(X) embeds into Aut(X ω ), which in turn embeds into Aut(U) because the completion of X ω is isomorphic to U. Moreover, these embeddings are all continuous (by Pettis' theorem) with continuous inverses. Below we elaborate each of these steps. First we need a lemma: Lemma 6.3.2. Let (X, δ X ) be a separable diversity and let X 1 := E(X, ω) denote the diversity of admissible maps on X with finite support. Then Aut(X) embeds as a topological group into Aut(X 1 ).

Proof. Let Φ : Aut(X) → Aut(X 1 ) be the map defined by Φ

(g)(f X S ) = f X g(S) , where f (g(A)) = f (A) for A ⊆ S. It is straightforward to check that Φ(g) is a bijection of X 1 extending g. Moreover, we note that Φ(g)(f X S )(A) = f X S (g -1 A) (6.1)
for any finite A ⊆ X. Using this, it is straightforward to verify that Φ(g) is an automorphism of X 1 and that Φ is injective. Furthermore, continuity of Φ follows either from Pettis' theorem (cf. [START_REF] Pettis | On continuity and openness of homomorphisms in topological groups[END_REF]) or simply by a direct argument using (6.1). Finally, continuity of the inverse of Φ can be seen as follows: Suppose Φ(g n ) → Φ(g) and let x ∈ X be given. We must show that g n (x) → g(x). For this, let κ x ∈ X 1 denote the image of x under the embedding of

X into X 1 . Then Φ(g n )(κ x ) → Φ(g)(κ x ) which means that sup B finite |Φ(g n )(κ x )(B) -Φ(g)(κ x )(B)| → 0.
In particular, this is true for B = {gx}. Therefore, we have

|Φ(g n )(κ x )({gx}) -Φ(g)(κ x )({gx})| = |δ({g -1 n gx, x}) -δ({g -1 gx, x})| = δ({gx, g n x}) → 0,
where we have used that Φ(h)(κ x ) = κ hx for any h ∈ Aut(X), which easily follows from (6.1) above. We conclude that g n → g in Aut(X).

With this lemma established, we can show that Aut(U) is a universal Polish group. Theorem 6.3.3. Aut(U) is a universal Polish group.

Proof. First, any Polish group G can be embedded into the isometry group of its left completion ( G L , d L ) where d L is a left-invariant metric. We turn G L into a diversity by using the diameter diversity δ L associated to d L , i.e. δ L (A) is the diameter of A. Then Aut( G L , δ L ) is still just Iso( G L , d L ) so G embeds into this group.

Given any separable diversity X, we let X 1 denote E(X, ω) and for any n ∈ N we let X n denote E(X n-1 , ω). By X ω we denote the union X n . In the lemma above we saw that Aut(X i ) embeds into Aut(X i+1 ) for every i. Hence we obtain a chain of embeddings Aut(X)

ϕ 0 -→ Aut(X 1 ) ϕ 1 -→ Aut(X 2 ) ϕ 2 -→ . . . ,
where ϕ i (g) extends g ∈ Aut(X i ). It follows that we obtain a map

Φ : Aut(X) → Aut(X ω ), where Φ(g) ∈ Aut(X ω ) is defined by Φ(g)(a) = Φ n (g)(a), where Φ n denotes ϕ n-1 • • • • • ϕ 0 for n ∈ N such that a ∈ X n for a ∈ X ω . Note that this is well defined since if a ∈ X m as well for m < n we must have Φ m (g)(a) = Φ n (g)(a) because Φ n (g)(a) = (ϕ n-1 • • • • • ϕ 0 )(g)(a) = (ϕ n-1 • • • • • ϕ m • Φ m )(g)(a) = Φ m (g)(a).
It is easy to check that Φ is a topological group embedding and therefore it follows that Aut(X) embeds into Aut(X ω ).

Finally, by [START_REF] Bryant | A universal separable diversity[END_REF]Theorem 19], the completion of X ω is isomorphic to U. Therefore, it follows from uniform continuity of δ (cf. Lemma 6.1.3) that Aut(X ω ) embeds into Aut(U).

In conclusion, we have seen that given any Polish group G, we can embed G into Aut(G L , δ L ), which in turn may be embedded into Aut(U) using the construction above. Hence, Aut(U) is a universal Polish group, which was what we wanted.

A dense conjugacy class in Aut(U)

In this section, we will show that Aut(U) has a dense conjugacy class (cf. Corollary 6.4.17 below). We will do this by first showing that the automorphism group of the rational Urysohn diversity U Q has a dense conjugacy class and then show that this group embeds densely into Aut(U) (cf. Corollary 6.4.12 and Theorem 6.4.16). Here, rational simply means that the diversity map only attains rational values. Of course in order to do that, the first thing we must show is that U Q exists. To do that, we will need to introduce some classical Fraïssé theory before defining U Q as the Fraïssé limit of the Fraïssé class of finite diversities with rational values, denoted by D. We will therefore recall the basics of Fraïssé theory in the first subsection, where we also provide a useful free amalgamation of diversities (cf. Definition 6.4.7). This amalgamation is a generalisation of the free amalgamation of metric spaces, and we will use it several times below. With these preliminaries done, it is straightforward to verify that D is a Fraïssé class and therefore that U Q exists. Moreover, we will show that U Q satisfies the approximate extension property of Definition 6.1.12 above. It follows that the completion of U Q is isomorphic to U. Thus, we obtain another way of constructing U.

With the existence of U Q established, it is straightforward to show that Aut(U Q ) has a dense conjugacy class by applying a result of Kechris and Rosendal from [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF] (cf. Theorem 6.4.11 below). This result characterises exactly when the automorphism group of a Fraïssé limit has such a conjugacy class. Afterwards, we show that U and U Q have a property referred to as propinquity in [START_REF] Zielinski | Locally roelcke precompact polish groups[END_REF]. From this it follows that Aut(U Q ) embeds densely into Aut(U). Therefore, Aut(U) has a dense conjugacy class as well (Theorem 6.4.16 and Corollary 6.4.17).

Fraïssé theory

Recall that given two structures A and B we denote by A ∼ -→ B that A embeds into B. An embedding between classical first-order structures is defined just as in Definition 2.1.6: it is an injective map commuting with the interpretations of the symbols of the signature. Fraïssé classes in relational signatures are then defined as follows: Definition 6.4.1. Let L be a countable relational signature for a (classical) first-order language and let K be a class of finite L-structures. Then K is a Fraïssé class if it has the following properties:

(i) (HP) K is hereditary, i.e. if B ∈ K and A ∼ -→ B, then A ∈ K.
(ii) (JEP) K has the joint embedding property, i.e. if A, B ∈ K, then there is some We call such a structure D an amalgam of B and C over A.

C ∈ K such that A, B ∼ -→ C. (
(iv ) K contains countably many structures (up to isomorphism), and contains structures of arbitrarily large (finite) cardinality.

Remark 6.4.2. Note that if the class K above contains the empty structure, then AP implies JEP.

The main reason for studying Fraïssé classes is that any Fraïssé class K has a so-called Fraïssé limit K which is universal and ultrahomogeneous. Ultrahomogeinity was already defined previously: any isomorphism between finite substructures extends to an automorphism. Universality in this case means that the class of all finite structures that embeds into K is K. This class is the so-called age of K and is denoted Age(K). Fraïssé's theorem now reads: Theorem 6.4.3 (Fraïssé,[START_REF] Fraïssé | Theory of relations[END_REF][START_REF] Fraïssé | Sur l'extension aux relations de quelques propriétés des ordres[END_REF], cf. also [START_REF] Hodges | Model theory[END_REF]Theorem 7.1.2]). Let L be a countable relational signature and let K be a Fraïssé class of L-structures. Then there exists a unique (up to isomorphism) countable structure K satisfying:

(i) K is ultrahomogeneous. (ii) Age(K) = K.
The structure K in the theorem above is called the Fraïssé limit of K. Using this theorem, we will show that there is a universal ultrahomogeneous countable rational diversity. First we need a couple of definitions and an amalgamation lemma to make it simpler for us to verify the AP for the class of finite rational diversities. Definition 6.4.4. Let Y be a set and let

X ⊆ Y . A connected cover of X is a collection {E i } of subsets of Y such that X ⊆ E i and such that the intersection graph G defined on {E i } by E i GE j ⇐⇒ E i ∩ E j = ∅ is connected.
Remark 6.4.5. If (Y, δ) is a diversity and X ⊆ Y is finite, then for any finite connected cover {E i } of X with each E i finite we have that δ(X) ≤ δ(E i ). This inequality is the main reason why we are interested in connected covers.

With this terminology established, we can define a free amalgamation of two diversities sharing a common sub-diversity. This is a diversity version of the free amalgamation of metric spaces. Recall first that if A and B are structures, then A Ă ∼ B denotes that A ⊆ B and that the inclusion is an embedding. Definition 6.4.6. Let (A, δ A ), (B, δ B ) and (C, δ C ) be non-empty finite diversities such that A = B ∩ C and such that A Ă ∼ B, C. The free amalgam of B and C over A is the diversity (D, δ D ) where, D = B ∪ C and where, for X ⊆ D, δ D (X) is given by the minimum over sums i δ(E i ) for a connected cover {E i : i ≤ n} of X such that for each i, either

E i ⊆ B or E i ⊆ C.
Note that if X has elements from both B and C, the definition of δ D (X) requires the connected cover to include elements from A. Hence, if we restrict δ D to pairs, we obtain the usual free amalgamation of metric spaces, i.e. Of course, it is not necessarily evident that δ D above defines a diversity map and that both (B, δ B ) and (C, δ C ) embeds into (D, δ D ). Let us therefore verify this. Proof. First we show δ D agrees with δ B and δ C on B and C, respectively. Suppose therefore X ⊆ B (the other case is similar). Then {X} is a connected cover of X so δ D (X) ≤ δ B (X). To show equality, let {E i } be a connected cover of X. We can assume E i ⊆ B. By monotonicity of δ B , we have

δ B (X) ≤ δ B ( E i ).
By connectivity, we have

δ B ( E i ) ≤ δ B (E i ). We conclude that δ B (X) ≤ δ D (X) as well, so in fact δ D (X) = δ B (X). In particular, δ D (X) = 0 if |X| ≤ 1.
Next, we show monotonicity. Let therefore X ⊆ Y ⊆ B ∪ C. Then any connected cover of Y whose elements are contained in either B or C must also cover X. Hence, δ D (X) ≤ δ D (Y ).

Lastly we show connected sublinearity. Suppose therefore that X ∩ Y = ∅. Let {E i } and {F j } be connected covers realising δ D (X) and δ D (Y ), respectively. Then, since Xand Y intersect, we have that {E i , F j } is a connected cover of X ∪ Y whose elements are either contained in B or C. Hence, we must have

δ D (X ∪ Y ) ≤ δ * (E i ) + δ * (F j ) = δ D (X) + δ D (Y ),
where δ * denotes either δ B or δ C depending on which diversity E i and F j are contained in. Thus, (D, δ D ) is a diversity such that B, C Ă ∼ D, which was what we wanted.

Note that if the diversities A, B and C above are all rational, then the amalgam D will also be a rational diversity. It follows that the class of finite rational diversities, denoted D, has the AP, and hence that this class is a Fraïssé class.

Proposition 6.4.8. D is a Fraïssé class with limit U Q . Moreover, the completion of U Q is (isomorphic to) the Urysohn diversity.

Proof. We first note that there are clearly rational diversities of arbitrarily large finite cardinality. Moreover, up to isomorphism, there are only countably many possible finite rational diversities. Hence, D satisfies property (iv ) above. We verify that D has the three properties HP, JEP and AP. HP is clearly satisfied, since if B ∈ D and A ∼ -→ B, then A must be a finite rational diversity so A ∈ D.

JEP follows from AP, which we show below, since clearly ∅ ∈ D. AP follows from Lemma 6. We conclude that D is a Fraïssé class and hence that it has a Fraïssé limit U Q .

The 'moreover'-part follows since U Q has the approximate extension property: if F ⊆ U Q is finite, f ∈ E(F ) is admissible and ε > 0, we can find an admissible map f with rational values such that |f (A) -f (A)| < ε. Then f defines a rational diversity on F ∪ {z} for some new element z. By universality and ultrahomogeneity of U Q , we find x ∈ U Q such that for all A ⊆ F we have

|δ(A ∪ {x}) -f (A)| = |f (A) -f (A)| < ε.
It now follows from Proposition 6.1.13 above that the completion of U Q has the extension property. Moreover, from Theorem 6.1.14 it follows that this completion is isomorphic to U as claimed.

A dense conjugacy class

With the existence of U Q established, we set out to show that Aut(U Q ) has a dense conjugacy class. First recall that the conjugacy action of a group on itself is given by g • h := ghg -1 . Having a dense conjugacy class is then defined as follows: Definition 6.4.9. A Polish group G is said to have a dense conjugacy class if there is some element of G whose orbit under the conjugacy action of G on itself is dense.

In [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF], Kechris and Rosendal characterise when the automorphism group of a Fraïssé limit of a class K has a dense conjugacy class. They do this in terms of the JEP, not for K itself, but for the class of all K-systems defined as follows: Definition 6.4.10. Let K be a Fraïssé class. A K-system consists of a structure A in K together with a substructure A 0 Ă ∼ A and a partial automorphism f : A 0 → A. Such a system is denoted A = (A, (f, A 0 )). The class of all K-systems is denoted K p .

An embedding of a K-system A = (A, (f, A 0 )) into another

K-system B = (B, (g, B 0 )) is a map Φ : A → B that embeds A into B, A 0 into B 0 and f (A 0 ) into g(B 0 ) such that Φ • f ⊆ g • Φ. In diagram form: A 0 f (A 0 ) B 0 g(B 0 ) f Φ Φ g
Kechris and Rosendal then obtain the following characterisation of having a dense conjugacy class: Theorem 6.4.11 ([46, Theorem 2.1]). Let K be a Fraïssé class with limit K. Then the following are equivalent:

(i) There is a dense conjugacy class in Aut(K). (ii) K p has the JEP.

As an immediate corollary to this, we obtain that Aut(U Q ) has a dense conjugacy class. Recall that denotes disjoint union. Corollary 6.4.12. D p has the JEP. Hence, Aut(U Q ) has a dense conjugacy class.

Proof. Let A = (A, (f, A 0 )) and B = (B, (g, B 0 )) be D-systems. Then we define a system C = (C, (h, C 0 )) by setting C = A B, C 0 = A 0 B 0 and h = f ∪ g, and where the diversity map δ C is defined to be δ A on A, δ B on B and on subsets with elements from both A and B, δ C is constant, equal to some N > δ A (A), δ B (B). It is easy to check that C is in K p and that both A and B embeds into C.

We now wish to show the same thing for the automorphism group of the full Urysohn diversity. In order to do that, we will show that Aut(U Q ) embeds densely into Aut(U). This will follow from a homogeneity-like property that the rational and complete Urysohn diversities and metric spaces all share. In [START_REF] Zielinski | Locally roelcke precompact polish groups[END_REF], the author refers to this property for metric spaces as pair propinquity.

To emphasise that we are working with diversities, we will call this property diversity propinquity. In short, the property says that if two finite subspaces are close to being isomorphic, then we can find an isomorphic copy of one space close to the other space. It is defined as follows: Definition 6.4.13. Let (X, δ X ) be a diversity and let ā = (a i ) and b = (b i ) be two tuples of length n ∈ N of the same cardinality. For ε > 0 we say that ā and b are ε-isomorphic if we have |δ X (ā J ) -δ X ( bJ )| < ε for all J ⊆ n, where bJ := (b j ) j∈J . Definition 6.4.14. Let (X, δ X ) be a diversity. We say that (X, δ X ) has diversity propinquity if for all ε > 0, there is an ε > 0 such that for all ε -isomorphic finite tuples ā and b in X, there is ā isomorphic to ā and pointwise within ε of b, i.e. max i δ X (a i , b i ) < ε.

We now have the following lemma, the proof of which is modelled on the proof of the corresponding fact for the Urysohn metric space in [54, Lemma 6.5]. Lemma 6.4.15. U and U Q both have diversity propinquity. Moreover, the ε of the definition may simply be chosen to be the given ε.

Proof. The proof for the two diversities is the same. In the rational case all one needs to check is that the diversity maps defined below are rational. However, since we are dealing with finite sets, this is easily verified.

Let n ∈ N and let ε > 0. The first thing we need is to introduce some notation for dealing with the various diversities one may assign to an n-tuple. Thus, let D x be the set of all diversity assignments to the n-tuple x = (x 0 , . . . , x n-1 ). That is, if we denote {x i : i ∈ I} by xI , then D x is the set of those maps on the power set of x, r : P(x) → R (or into Q for the rational case), such that Of course, any r ∈ D x corresponds to an element of R 2 n that we will also denote by r. Thus, we will use the notation r(I) for r(x I ). This will be convenient below.

Let now d ∞ denote the maximum metric on D x, i.e. Next, we define another metric on D x that measures how close together we can embed two diversities with n elements into a third diversity. To define this metric, let ȳ be another n-tuple of elements disjoint from x. Then define d 1 to be the metric given by Moreover, we claim that this will imply the proposition, but let us do one thing at a time. Let therefore r1 , r2 ∈ D x be two different diversity assignments and set c = d ∞ (r 1 , r2 ). We need to define some r ∈ D x∪ȳ such that r x = r1 , r ȳ = r2 and such that max r(x i , y i ) ≤ c. In order to define such an r we need to introduce some notation. Given a subset s = {y i 1 , . . . , y i k } ⊆ ȳ, we denote the corresponding set {x i 1 , . . . x i k } ⊆ x by s . A collection of subsets {E i } of x or ȳ is said to be connected if the intersection graph on {E i } forms a connected graph. Let now r be the diversity assignment where for each s ⊆ x ∪ ȳ, r(s) is defined to be the minimum over sums of the form i r1 (E i ) + j r2 (F j ) + c/2, where

• E i ⊆ x, • F j ⊆ ȳ, • {E i , F j } is connected, • s ∩ x ⊆ E i , • s ∩ ȳ ⊆ F j .
Let us argue why r is a diversity assignment. If s 1 ⊆ s 2 , then any collection satisfying the properties of the minimum above for s 2 will also satisfy the properties for s 1 . Hence, r(s 1 ) ≤ r(s 2 ). If s 1 ∩ s 2 = ∅, we let {E 1 i }, {F 1 j } realise r(s 1 ) and {E 2 l }, {F 2 k } realise r(s 2 ). Then it is easy to check that {E 1 i , E 2 l }, {F 1 j , F 2 k } satisfiy the properties of the minimum for s 1 ∪ s 2 . Therefore, r(s 1 ∪ s 2 ) ≤ r(s 1 ) + r(s 2 ) as required. We conclude that r is in fact a diversity assignment. Moreover, we see that sup i r(x i , y i ) = c/2, since the singletons {x i } and {y i } satisfy the properties of the minimum. This shows that d 1 (r 1 , r2 ) ≤ c/2 < d ∞ (r 1 , r 2 ) as we claimed.

It now follows that both U and U Q have diversity propinquity. Since the argument for both diversities is the same, we only provide it for U. Let n ∈ N and ε > 0 be given. Then we claim that ε works as the ε of Definition 6.4.14 above. To see this, let ā and b be n-tuples of elements of U and suppose sup I⊆n |δ(ā I ) -δ( bI )| < ε. Let rā and rb be the diversity assignments corresponding to ā and b. Then d ∞ (r ā, rb) < ε and so d 1 (r ā, rb) < ε as well. Therefore, we find a diversity assignment r on ā ∪ b such that restricted to ā we get rā and restricted to b we get rb and such that sup i r(a i , b i ) < ε. By universality of U, we find ā , b ∈ U n isomorphic as diversities to ā and b, respectively, such that sup δ U (a i , b i ) < ε. By ultrahomogeneity, we find an automorphism g of U such that δ U (a i , g • b i ) < ε, which was what we wanted.

We can now show that Aut(U Q ) embeds densely into Aut(U). Theorem 6.4.16. Aut(U Q ) continuously embeds as a dense subgroup into Aut(U).

Proof. Recall that U Q is dense in U by Proposition 6.4.8. Furthermore, since the diversity map defines uniformly continuous maps on finite powers of U (cf. Lemma 6.1.3 above), it follows that any g ∈ Aut(U Q ) uniquely extends to an autoversity of U. Thus, Aut(U Q ) embeds into Aut(U). Moreover, this embedding must be continuous by Pettis' theorem (cf. [START_REF] Pettis | On continuity and openness of homomorphisms in topological groups[END_REF] or Theorem 1.3.4).

We move on to show that Aut(U Q ) is dense in Aut(U). Recall that the topology on Aut(U) is the pointwise convergence topology for which a basis at the identity is given by sets of the form U ā,r := {g ∈ Aut(U) : δ(g(ā), ā) < r} for a tuple ā = (a 1 , . . . , a n ) of elements of U and some r > 0. In each of these sets, we must find an autoversity extending a rational autoversity. Let therefore U ā,r be given and let g ∈ U ā,r . Set ε := r -max i δ(g(a i ), a i ) > 0 and find a tuple x of n elements of U Q with max i δ(a i , x i ) < ε/4. Let moreover ȳ be an n-tuple of elements of U Q such that max i δ(y i , g(x i )) < ε/(4n). Note that g(x i ) is not necessarily in U Q -hence this approximation. By Lemma 6.1.3, it follows that (ȳ, δ) is ε/4-isomorphic to (x, δ) and therefore, by propinquity and ultrahomogeneity of U Q , we find an autoversity g 0 of U Q such that max i δ(g 0 (x i ), y i ) < ε/4. We claim that the extension of g 0 to U is in U ā,r . Let therefore g0 denote this extension. We have δ(a i , g0 (a i )) ≤ δ(a i , g(a i )) + δ(g(a i ), g(x i )) + δ(g(x i ), y i ) + δ(y i , g0 (x i )) + δ(g 0 (x i ), g0 (a i ))

< r -ε + ε/4 + ε/(4n) + ε/4 + ε/4 ≤ r.

We conclude that g0 ∈ U ā,r and hence that Aut(U Q ) is a dense subgroup of Aut(U).

As an immediate corollary, we obtain that Aut(U) has a dense conjugacy class. Corollary 6.4.17. Aut(U) has a dense conjugacy class.

Proof. This follows easily since Aut(U Q ) has a dense conjugacy class and is densely embedded into Aut(U).

6.5 Ample generics of Aut(U Q )

In this section, we will show that Aut(U Q ) has ample generics. Ample generics is a powerful property introduced by Hodges, Hodkinson, Lascar and Shelah in [START_REF] Hodges | The small index property for ω-stable ω-categorical structures and for the random graph[END_REF] in order to study the small index property for automorphism groups of certain countable structures. This property was also studied by Ivanov in [START_REF] Ivanov | Generic expansions of ω-categorical structures and semantics of generalized quantifiers[END_REF], where he describes when ℵ 0 -categorical classical structures have automorphisms with comeagre conjugacy classes. Later, Kechris and Rosendal studied ample generics further in [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF], where they gave equivalent conditions for when the automorphism group of a Fraïssé limit has ample generics. These conditions are the joint embedding property and the weak amalgamation property of the so-called n-systems in the given Fraïssé class. They also show that ample generics have very strong implications such as the automatic continuity property, uncountable cofinality of non-open subgroups, unique Polish group topology and the small index property. All of these concepts will of course be explained below.

Ample generics of Aut(U Q ) follows from an extension result inspired by Solecki's result in [START_REF] Solecki | Extending partial isometries[END_REF]. This will allow us to extend any partial isoversity of a finite diversity to a full autoversity of some larger finite diversity containing the original one.

We begin our endeavour by defining the notion of ample generics. Definition 6.5.1. A Polish group G has ample generics if for each n ∈ N, there is a comeagre orbit for the diagonal conjugacy action of G on G n defined by g • (g 1 , . . . , g n ) = (gg 1 g -1 , . . . , gg n g -1 ).

Before explaining the consequences of ample generics mentioned above, let us mention a few examples of groups that are known to have ample generics.

Examples. The following groups have ample generics:

(i) The automorphism group of the random graph, [START_REF] Hrushovski | Extending partial isomorphisms of graphs[END_REF], cf. also [START_REF] Hodges | The small index property for ω-stable ω-categorical structures and for the random graph[END_REF]. (ii) The automorphism group of the free group on countably many generators, [START_REF] Bryant | The small index property for free groups and relatively free groups[END_REF]. (iii) The group of measure preserving homeomorphisms of the Cantor space, [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF]. (iv ) The automorphism group of N <ω seen as the infinitely splitting regular rooted tree, [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF]. (v ) The isometry group of the rational Urysohn space, [START_REF] Solecki | Extending partial isometries[END_REF].

In [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF], where these examples are taken from, Kechris and Rosendal show, as mentioned, a number of interesting consequences of ample generics. We have collected the most important ones in the theorem below. Theorem 6.5.2 (Kechris-Rosendal [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF]). Let G be a Polish group with ample generics. Then G has the following properties:

(1) Automatic continuity property, i.e. any homomorphism from G to a separable group is continuous. Another important result from [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF] is a characterisation of when the automorphism group of a Fraïssé limit has ample generics in terms of the JEP and a weak form of the AP. This weaker form of amalgamation is, naturally enough, called the weak amalgamation property (or WAP for short) and is defined as follows: Definition 6.5.3. Let K be a class of finite structures. Then K has the weak amalgamation property (WAP) if for any A 0 ∈ K, there is A ∈ K and an embedding f 0 : A 0 → A such that whenever However, it is not the Fraïssé class itself that must have the WAP and the JEP in order for the automorphism group to have ample generics, but the class of so-called n-systems for n ≥ 1. This is the class of finite structures A, together with n substructures of A and n partial automorphisms of A defined on these substructures. The exact definition is as follows: Definition 6.5.4. Let K be a Fraïssé class and let n ≥ 1 be given. An nsystem in K consists of a structure A in K together with n substructures A 1 , . . . , A n Ă ∼ A and n embeddings f 1 : A 1 → A, . . . , f n : A n → A. We denote such a system by A = (A, (f i , A i ) i≤n ). The class of all n-systems in K is denoted K n p . An embedding of an n-system A = (A, (f i , A i )) into another n-system B = (B, (g i , B i )) is a map Φ : A → B that embeds A into B, A i into B i and f i (A i ) into g i (B i ) such that Φ • f i ⊆ g i • Φ for each i ≤ n. In diagram form, for each i ≤ n:

A i f i (A i ) B i g i (B i ) f i Φ Φ g i
Note that since we have defined embeddings between n-systems, we can talk about the WAP and the JEP for the class K n p . In [START_REF] Kechris | Turbulence, amalgamation, and generic automorphisms of homogeneous structures[END_REF], Kechris and Rosendal show that these two properties for K n p actually characterise ample generics. Theorem 6.5.5 ([46, Theorem 6.2]). Let K be a Fraïssé class and let K denote its limit. Then the following are equivalent:

(i) Aut(K) has ample generics. (ii) For all n ≥ 1, K n p has the JEP and the WAP.

Using this theorem, we can show that Aut(U Q ) has ample generics. The proof uses the following extension result inspired by [START_REF] Solecki | Extending partial isometries[END_REF]Theorem 2.1]. Recall that an isoversity is an isomorphism of diversities. An isoversity of a diversity to itself is called an autoversity. A partial isoversity of a diversity A is simply an isoversity f : A 0 → B defined on a subset A 0 ⊆ A into some diversity B.

A partial autoversity of A is a partial isoversity into A. We now have the following theorem: Theorem 6.5.6. Let (A, δ A ) be a finite diversity. Then there is a finite diversity (B, δ B ) containing A as a subdiversity and such that any partial autoversity of A extends to a full autoversity of B.

The proof of this theorem uses a theorem due to Herwig and Lascar from [START_REF] Herwig | Extending partial automorphisms and the profinite topology on free groups[END_REF].

For the convenience of the reader, we have included it here. However, before we can state it properly, we need to make a few definitions. Definition 6.5.7. A class of structures K has the extension property for partial automorphisms (EPPA for short) if for any finite A ∈ K and any B ∈ K such that A Ă ∼ B and any partial automorphisms g 1 , . . . , g n of A that extend to automorphisms of B, there is a finite C ∈ K and automorphisms h 1 , . . . , h n of C such that h i extends g i .

Loosely speaking, this property says that if it is possible to extend partial automorphisms to full automorphisms of some structure, then there is a finite structure where the partial automorphisms also extend. Definition 6.5.8. Let L be a finite relational classical signature. A map between two L-structures h : A → B is a weak homomorphism if for all predicates R ∈ L and all tuples ā of elements from A such that A R(ā), we have B R(h(ā)). Definition 6.5.9. Let L be a finite relational classical signature and let T be a set of L-structures. An L-structure A is said to be T -free if there is no structure T ∈ T and weak homomorphism h : T → A.

Herwig and Lascar's theorem then reads: We are now ready to prove Theorem 6.5.6.

Proof of Theorem 6.5.6. We can of course assume without loss of generality that |A| ≥ 2. Let D be the set D := {(δ A (X), |X|) : X ⊆ A} \ {(0, 1), (0, 0)}.

That is, D is all pairs of the non-zero values of δ A together with the size of the set the value comes from. For each (r, n) ∈ D, we let R (r,n) be an nary relation symbol and let L be the (finite) relational language consisting of these symbols. We call a tuple of elements of D, α = ((r 0 , n 0 ), . . . , (r k , n k )), a configuration if we have that k i=1 r i < r 0 and 1 + k i=1

(n i -1) ≥ n 0 .

Given a configuration α = ((r i , n i )) let Y 0 , Y 1 , . . . , Y k be sets such that

(i) Y 0 ⊆ k i=1 Y i , (ii) |Y i | = n i , (
iii) The intersection graph on {Y 1 , . . . , Y k } is connected.

We call such a family of sets {Y i : 0 ≤ i ≤ k} an α-family. Note that since α is a configuration it is always possible to find at least one α-family. Moreover, we note that there are only finitely many α-families. Given a configuration α = ((r i , n i ) : 0 ≤ i ≤ k) and an α-family β = {Y i }, we define an L-structure M α,β with universe Y i by declaring that the only relations satisfied by M α,β are the following:

M α,β R (r i ,n i ) (σ(Y i ))
for any permutation σ of the elements of Y i (considered here as an ordered tuple and not just a set). The permutations merely ensure that the relations are symmetric and do not really serve any other purpose. Let T denote the family of all M α,β for all configurations α and all α-families β. Note that T is finite.

Any diversity (X, δ X ) is naturally also an L-structure by letting X R (r,n) (Y ) ⇐⇒ δ X (Y ) = r and |Y | = n for any finite subset/tuple Y of elements of X, meaning that we are considering Y as a subset on the right-hand side and as an ordered tuple on the left-hand side above. Note, however, that the order we choose on Y is not important.

Observe that any partial autoversity of X is also a partial automorphism of X as an L-structure. Suppose now we are given a configuration α = ((r i , n i ) : 0 ≤ i ≤ k) and an α-family β = {Y 0 , . . . , Y k }. We then claim that X is T -free, i.e. that there are no weak homomorphisms h : M α,β → X. To see this, suppose h is such a map. Then since β is an α-family, we have that

h(Y 0 ) ⊆ k i=1 h(Y i )
and so, by the monotonicity of the diversity map,

δ X (h(Y 0 )) ≤ δ X ( k i=1 h(Y i )).
Since the intersection graph on {Y i : 1 ≤ i ≤ k} is connected, it follows that the intersection graph on the images {h(Y i ) : 1 ≤ i ≤ k} is connected too. Therefore, we can find Y i 0 such that the intersection graph on the family {h(Y i ) : i = i 0 } remains connected (this is always possible for finite connected graphs). Hence, by connected sublinearity of the diversity map, it follows that

δ X ( k i=1 h(Y i )) ≤ δ X (h(Y i 0 )) + δ X ( i =i 0 h(Y i )).
By induction, we obtain that

δ X ( k i=1 h(Y i )) ≤ k i=1 δ X (h(Y i )).
However, since h is a weak homomorphism and β is an α-family, we have that X R (r i ,n i ) (h(Y i )). Hence, δ X (h(Y i )) = r i for each i, and so we have that

r 0 = δ X (h(Y 0 )) ≤ k i=1 δ X (h(Y i )) = k i=1 r i < r 0 ,
Proof. We show that for each n ∈ N, the class D n p of n-systems in D has the WAP. Since it clearly has the JEP, it follows from Kechris and Rosendal's Theorem 6.5.5 above that Aut(U Q ) has ample generics.

Let therefore A = (A, (f i , A i )) be an n-system in D n p . By the extension theorem above, we find a rational diversity B containing A, where the partial isoversities of A extend to autoversities of B. Let fi denote the extension of f i to B, and let B denote the resulting n-system in D n p . Suppose now that we are given n-systems C 1 = (C 1 , (g i 1 , C i 1 )) and C 2 = (C 2 , (g i 2 , C i 2 )) and embeddings Φ j : B → C j , j = 1, 2. We need to construct an amalgam of C 1 and C 2 over B.

To do that, we apply the extension theorem to both C 1 and C 2 and get C 1 and C 2 , where the partial isoversities g i 1 and g i 2 extend to full autoversities gi 1 and gi 2 of C 1 and C 2 , respectively. Denote the resulting n-systems by C 1 and C 2 .

As usual, we can assume that B = C 1 ∩ C 2 . Therefore, we can construct the free amalgam D of C 1 and C 2 over B. Moreover, we can define an n-system using D by letting h i be gi 1 ∪ gi 2 , which is an autoversity of D because gi 1 and gi 2 must agree on B. Denote the resulting n-system by D. In diagram form for the n-systems:

C 1 B C 2 C 1 C 2 D A
and in diagram form for j = 1, 2 and each i:

B B A i f i (A i ) C i j g i j (C i j ) C j C j D D fi g i j gi j h i f i
It is easy to check that D is an amalgam of C 1 and C 2 over B. We conclude that D n p has the WAP and hence that Aut(U Q ) has ample generics. * finite subset. Hence, we need some upper bound on the sizes of the subsets we are measuring. Since we measure arbitrarily large finite subsets such a bound does not exist and therefore the argument fails. We invite the reader to study the proof of compact homogeneity for the Urysohn metric space in the section Compact Homogeneity of [START_REF] Melleray | Some geometric and dynamical properties of the Urysohn space[END_REF] for more details. If compact homogeneity were to fail for the Urysohn diversity, it would be the first example of a property true for the Urysohn metric space but false for the Urysohn diversity.

Another interesting question related to what we have done so far is the following:

Question A.4. Does the automorphism group of the Urysohn diversity have the automatic continuity property? What about other consequences of ample generics? Again, this is true for the Urysohn metric space, which was shown by Sabok in [START_REF] Sabok | Automatic continuity for isometry groups[END_REF] and later also by Malicki in [START_REF] Malicki | Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures[END_REF] by similar methods. However, we discovered that there was a small gap in both proofs. Fortunately, Sabok has recently communicated to us a possible fix for this small gap in both papers. It would be interesting to see if the new proofs can be adapted to the diversity case as well.

We would also like to study simplicity of the automorphism group of the Urysohn diversity along the lines of what Tent and Ziegler did for the Urysohn metric space in [START_REF] Tent | The isometry group of the bounded Urysohn space is simple[END_REF] and [START_REF] Tent | On the isometry group of the Urysohn space[END_REF]. In [START_REF] Tent | The isometry group of the bounded Urysohn space is simple[END_REF], the authors show that the isometry group of the bounded Urysohn space is simple. In the diversity case the first thing to do is simply to show that the bounded Urysohn diversity exists. An easy way to do that is by using Fraïssé theory as we did in Chapter 6 above. With the existence established, we can then ask the following: Question A.5. Is the automorphism group of the bounded Urysohn diversity simple?

Lastly, we would like to investigate the question of extreme amenability. Pestov showed in [START_REF] Pestov | Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups[END_REF] that the isometry group of the Urysohn space is extremely amenable. It was recently brought to our attention by Matěj Konečný, that using results from [START_REF] Hubička | All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)[END_REF] it should be possible to show that the automorphism group of the Urysohn diversity is extremely amenable. However, we have not yet worked out all the details, so we would like to end this appendix by posing it as a question.

Question A.6. Is the automorphism group of the Urysohn diversity extremely amenable?

Proposition 1 . 1 . 5 .

 115 (i) Closed subspaces of Polish spaces are Polish.

Proposition 2 . 3 . 4 .

 234 Given a signature L, there are L-definable predicates d ω and d ω⊕ω in F such that for any L-structure M, the maps d M ω and d M ω⊕ω defined in Definition 2.1.16 above are exactly equal to the interpretations of d ω and d ω⊕ω in M.

Definition 2 . 4 . 2 .

 242 Let T be a complete theory and suppose m < α ≤ ω. Define a map π α m : S α (T ) → S m (T ) by π α m (p)(ϕ) = p(ϕ). Define moreover maps π ω , π ω : S ω⊕ω (T ) → S ω (T ) by

Definition 2 .

 2 4.4 ([15, Definition 4.15]). A theory T has the back-and-forth property if for all n ∈ N and any two ω-saturated models of T , M and N , and any two tuples ā ∈ M n , b ∈ N n , and any singleton

Proposition 2 . 5 . 8 .

 258 For any model M of the complete theory T and any countable I, the set S M I (T ) of I-types realised in M is τ -dense in S I (T ).

  Refinement. Let [ϕ < r] be a basic τ -open set and let p ∈ [ϕ < r]. Let ε > 0 be such that p(ϕ) + ε < r and find δ > 0 such that for any M T and ā, b ∈ M |v| with d

Lemma 3 .

 3 3.18 ([11, Lemma 7.4]). Let p ∈ S v(T ). Then the following are equivalent:(i) p is principal. (ii) τ S v(T ) and τ ∂ S v(T ) coincide at p.

Proposition 4 . 1 . 1 (

 411 [START_REF] Melleray | A note on Hjorth's oscillation theorem[END_REF] Theorem 6]). M G is an approximately ultrahomogeneous separable atomic metric structure with Aut(M G ) = G.

Proposition 4 . 2 . 1 .

 421 Let M be a separable approximately homogeneous metric structure that knows ∂ n for each n ∈ N. Then A ⊆ Aut(M) is Roelcke precompact if and only if for any n ∈ N and any finite n-tuple b ∈ M n , the set {tp( b, g b) : g ∈ A} ⊆ S n⊕n (Th(M)) is ∂-totally bounded. Proof. Suppose that A is Roelcke precompact and let ε > 0 and b ∈ M n be given. Set U := {g ∈ Aut(M) : d M ( b, g b) < ε}. Then U is an open neighbourhood of the identity, so we can find a finite set F ⊆ Aut(M) such that A ⊆ U F U . We claim that for any g ∈ A, the type tp( b, g b) is within ε of some type in the set {tp( b, f b) : f ∈ F }. To see this, let g ∈ A. Then g = uf u for some u, u ∈ U and f ∈ F . Moreover, we have that tp( b, uf u b) = tp(u -1 b, f u b).

Proposition 4 . 2 . 2 .

 422 Suppose M is a separable approximately homogeneous metric structure that knows the type distance ∂ ω⊕ω . Let G := Aut(M) and let ξ ∈ M ω be a dense sequence. Then the map ι : g → tp(ξ, gξ) embeds (G, d L∧R ) isometrically into the type space (S ω⊕ω (T ), ∂ ω⊕ω ). It follows that we can identify G L∧R with the ∂ ω⊕ω -closure of the image of this embedding.

Proposition 4 . 2 . 3 .

 423 Let M be a separable approximately homogeneous metric structure that knows ∂ ω⊕ω and let G denote Aut(M). Then S(G) = {p ∈ S(G) : M realises p}.

Proposition 4 . 2 . 4 .

 424 Let M be a separable atomic structure with theory T and let ξ ∈ M ω be a dense sequence. Then S(G) = {p ∈ S ω⊕ω (T ) : π ω (p) = π ω (p) = tp(ξ)}. (4.1)

Theorem 4 . 3 . 1 .

 431 Let M be an exact homogeneous separable atomic structure with theory T and automorphism group G. Let c be any finite tuple of elements of M and let G c denote the stabiliser of c. Then the Roelcke completion of G c is homeomorphic to the closure of G c in the Roelcke completion of G.

2 . 3 .

 23 Therefore, p is of the form tp(ā, b | c) for two sequences ā, b ∈ M ω with tp(ā | c) = tp( b | c) = tp(ξ | c). Define a map λ : S c(G c) → S x⊕ȳ (T ) by λ(tp(ā, b | c)) = tp(cā, cb ).

Theorem 5 .

 5 1.4 ([49, Theorem 3.4]). Any complete metric space with the approximate extension property has the actual extension property. In particular, U has the extension property. Theorem 5.1.5 ([60],[49, Theorems 3.2, 3.3 and 3.4]

Theorem 5 . 3 . 3 .

 533 Let M be a model of T := T met ∪ T ext , and let E M and ρ M be defined as in Definitions 5.2.2 and 5.2.4 above. Then each E M -class has the extension property with respect to ρ

Corollary 5 . 3 . 4 .

 534 T := T met ∪ T ext admits quantifier elimination.

Corollary 5 . 3 . 8 .

 538 Let a ∈ U be given and let U a denote the L(a)-structure based on U. Then Aut(U a ) is Roelcke precompact. Proof. By Proposition 2.5.14, U a is exact homogeneous and atomic. Therefore, we can identify the Roelcke completion of G a := Aut(U a ) with the ∂-closure of {tp(ξ, gξ | a) : g ∈ G a }. Denote this closure by S a (G a ). By Theorem 4.3.1, (S a (G a ), ∂) is homeomorphic to the ∂-closure of {tp(aξ, agξ) : g ∈ G a }. Denote this closure by S(G a ). By Theorem 4.2.5, (S(G a ), ∂) is homeomorphic to (S(G a ), τ ), where τ denotes the logic topology as usual. We claim that (S(G a ), τ ) is closed, hence compact, rendering (S a (G a ), ∂) compact as well.

Examples.

  The following are examples of diversities: (i) Diameter diversity. Given a metric space (M, d), then δ diam (A) = max a,b∈A d(a, b) defines a diversity map on M called the diameter diversity.

Lemma 6 . 2 . 1 .

 621 For each n ≥ 1 and m ≥ 2 the interpretation D U n,m respects the modulus ∆ n,m .

  b)| < ε for two m-tuples ā = (a 0 , . . . , a m-1 ) and b = (b 0 , . . . , b m-1 ).

  ā) for ā ∈ M m and b ∈ M. (v ) For all tuples ā, b and c (not necessarily of the same lengths) with | b| > 0 we have D M n,|ā|+|c| (ā, c) ≤ D M n,|ā|+| b| (ā, b) + D M n,| b|+|c| ( b, c).

Definition 6 . 2 . 2 .Lemma 6 . 2 . 3 .

 622623 Given a model M T U define a relation E M on M by aE M b ⇐⇒ ∃n D M n,2 (a, b) < n. For any M T U , E M is an equivalence relation with clopen classes. Proof. Symmetry and reflexivity are both clear. If aE M b and bE M c, then we find n, m witnessing this. If we let N = n + m, we see that

•

  r (I) < n for all I ⊆ m. Since M σ r n,m and all subformulas of this sentence except the infimum are non-zero, we must have that inf c∈M max I |D M n,m+1 (ā I , c) -r (I)| = 0. Therefore, we find some b ∈ M such that for all I, we have |D M n,m+1 (ā I , b) -r (I)| < ε/2. By our choice of r , it follows that |D M n,m+1 (ā I , b) -r(I)| < ε. Moreover, for ε small enough, it follows that D M n,m+1 (ā I , b) < n as well and hence D M n,m+1 (ā I , b) = δ M (ā I ∪ {b}). Therefore, ([a] E M , δ M ) has the extension property, which was what we wanted.

  iii) (AP) K has the amalgamation property, i.e. if A, B, C ∈ K and f : A → B and g : A → C are embeddings, then there is D ∈ K and embeddings h B : B → D and h C : C → D such that h B • f = h C • g. In diagram form:

  δ D ({b, c}) = min a∈A {δ B ({b, a}) + δ D ({a, c})} for b ∈ B and c ∈ C.

Lemma 6 .

 6 4.7. δ D defined above is a diversity map on B ∪ C extending both δ B and δ C . It follows that (D, δ D ) is an amalgam of B and C over A.

  4.7 above. To see this, suppose we are given A, B, C ∈ D with A ∼ -→ B, C via embeddings f B and f C . Then we let D = B ∪ A C be the union of B and C where we identify f B (A) with f C (A) while leaving B \ f B (A) and C \ f C (A) disjoint. Identifying A with its image inside D, we now have that A = B ∩ C and therefore Definition 6.4.6 applies. Thus, we obtain an amalgam (D, δ D ) of B and C over A.

  (i) r(x I ) = 0 if and only if |I| ≤ 1, (ii) For all I 1 , I 2 and all I = ∅, we have r(xI 1 ∪ xI 2 ) ≤ r(x I 1 ∪ xI )+ r(x I ∪ xI 2 ).

d

  ∞ (r, r ) = sup I⊆n {|r(I) -r (I)|}.

d 1 (r 1 ,

 11 r2 ) = inf r max i≤n {r(x i , y i )} : r ∈ D x∪ȳ , r x = r1 , r ȳ = r2 },where r1 , r2 ∈ D x are two different diversity assignments. If r1 = r2 , we set d 1 (r 1 , r1 ) = 0. Of course, here r ȳ = r2 means that the diversity assignment on ȳ given by r2 (i.e. ȳI → r2 (x I )) is equal to r ȳ. That d 1 is in fact a metric follows from[START_REF] Bryant | A universal separable diversity[END_REF] Proposition 10]. We now claim that d 1 (r 1 , r2 ) ≤ d ∞ (r 1 , r2 ).

( 2 )

 2 Small index property, i.e. any subgroup of G of index < 2 ℵ 0 is open. (3) G cannot be the union of countably many non-open subgroups. (4) G has a unique Polish group topology.

  g B : A → B and g C : A → C are embeddings into B, C ∈ K, there is D ∈ K and embeddings h B : B → D and h C : C → D such that h B • g B • f 0 = h C • g C • f 0 . In diagram form:

Theorem 6 . 5 . 10 ([ 35 ,

 651035 Theorem 3.2]). Let L be a finite classical signature and T a finite set of finite L-structures. Then the class of T -free L-structures has the EPPA.

  A ⊆ G is Roelcke precompact if and only if the closure of A in the Roelcke completion G L∧R is compact. Thus, G is Roelcke precompact if and only if G L∧R is compact. However, the analogous statement for locally Roelcke precompact groups is not immediate. Fortunately it is still true, as Zielinski showed in [68]. Theorem 1.4.3 ([68, Theorem 16]). A Polish group G is locally Roelcke precompact if and only if the Roelcke completion of G is locally compact. This theorem is important for us, since we will use it in Chapter 4. It is important to stress that properties of the Roelcke completion of a group G are not necessarily inherited by subgroups of G. In particular, if H is a subgroup of G which is Roelcke precompact as a subset of G, it does not follow that H is a Roelcke precompact group. This is because the Roelcke completion of H as a group and the closure of H inside the Roelcke completion of G need not coincide. Of course, if H is an open subgroup, then H is a Roelcke precompact group if and only if it is a Roelcke precompact subset of G, which follows from a straightforward verification of the definition above. However, in general this is false. We will study this phenomenon further in Chapter 4.

Acknowledgements

glemme.

Definition 5.2.1. For each n ≥ 1 and variables x, y, z define the following sentences:

• sym n := sup x,y |P n (x, y) -P n (y, x)|,

• ref n := sup x P n (x, x),

• tri n := sup x,y,z P n (x, y) .

-(P n (x, z) + P n (z, y)), • incr n := sup x,y P n (x, y) .

-P n+1 (x, y), • stab n := sup x,y min{n -P n (x, y), |P n (x, y) -P n+1 (x, y)|}.

Denote the collection of all these sentences by T met .

We note that the first three sentences above are simply the axioms of a metric -hence their names. The last two ensure that interpretations of the P n 's behave in the way we want them to: incr n says that the P n 's are increasing in n, and stab n says that they stabilise, i.e. if P n (a, b) < n at some n, then P m (a, b) = P n (a, b) for all m ≥ n.

It is easy to check that any metric space (M, d M ) (bounded or not) is naturally an L U -structure by interpreting P n as the map P M n : M 2 → R given by P M n (a, b) = min{d M (a, b), n}. Moreover, it is easy to check that M T met . Furthermore, conversely, if M T met , then we can define a map ρ M that is almost a metric on M. We write "almost" because we do need to allow the value ∞ in the range of ρ M . However, if ρ M never attains this value, then ρ M is actually a metric on M that may or may not be bounded. Thus, T met can be viewed as an axiomatisation of the theory of generalised metric spaces, where the metric is allowed to attain the value ∞. To explain these things precisely, we first need to define an equivalence relation on M using the predicates.

We verify that E M is an equivalence relation below. Moreover, each class is actually clopen. Proof. Reflexivity and symmetry are both clear. Transitivity follows easily as well, since if aE M b and bE M c, we find some n, m ∈ N witnessing this and let N = m + n. Then

Proof. Let M be a model of T . Then, by the Löwenheim-Skolem Theorem, we can find a separable substructure M 0 M. By Theorem 5.3.3, it follows that M 0 is a disjoint union of isomorphic copies of U. We may therefore assume that U ⊆ M 0 . Since T admits quantifier elimination, it follows that this inclusion is an elementary embedding.

We can now show that T met ∪T ext is in fact an axiomatisation of the theory of U. Thus, all results we have shown for T met ∪ T ext also hold for T U .

Corollary 5.3.6. For any L U -structure M we have that M T U if and only if M T := T met ∪ T ext . Thus, all results for T above also hold for T U .

Proof. If M T U , then clearly M T met ∪ T ext . Suppose therefore that M is a model of T met ∪ T ext . Then U M by primeness and so M T U . The conclusions follow immediately.

Finally, we can show that U satisfies the conditions of Theorem 4.2.5, and hence that Iso(U) is locally Roelcke precompact. Of course, as we have mentioned, this is not a new result, but the proof and our approach are new. Corollary 5.3.7. Iso(U) is locally Roelcke precompact.

Proof. Let G := Iso(U) = Aut(U). We begin by noting that U is homogeneous simply by the definition of the Urysohn space. Hence, (U, {P U n }) is a homogeneous metric structure. From Proposition 5.3.6, we know that U is prime and therefore atomic. Thus, U knows ∂ ω⊕ω by Proposition 2.5.12. All that is left to show is therefore that S(G) is τ -open inside S(G).

We claim that S(G) = S(G) \ {p ∞ }, where p ∞ ∈ S(G) is the type defined by p ∞ (P n (x i , y j )) = n for all i, j, n. First of all, by the Compactness Theorem and by quantifier elimination, there is a unique such type. Next, we note that by Proposition 4.

To see this, we observe that if p = p ∞ , we have that for all i, j, there is some n such that p(P n (x i , y j )) < n. Therefore, if M T U is a separable model realising p, it follows by Proposition 5.3.6 that

follows that all elements of ā and b are E M -equivalent. Hence, they come from the same U i 0 . Moreover, by quantifier elimination, it follows that (ā, b) is a realisation in the structure U i 0 . Therefore, p is realised by the Urysohn space, so p ∈ S(G).

We conclude that S(G) = S(G) \ {p ∞ }, which is a τ -open subset of S(G). Theorem 4.2.5 therefore implies that G is locally Roelcke precompact.

We can now show, using Theorem 4.2.5, that Aut(U) is locally Roelcke precompact. Corollary 6.2.9. G := Aut(U, {D U n,m }) is locally Roelcke precompact.

Proof. We will verify that U satisfies the conditions of Theorem 4.2.5. Let therefore S(G) be the embedding of the Roelcke completion of G in the type space S ω⊕ω (T U ).

First of all, simply by definition (or rather construction), U is separable and complete. Moreover, it is easy to check that the automorphism group of the diversity is the same as the automorphism group of the metric structure. Hence, U is homogeneous simply by construction.

Since U is atomic, it follows that it knows the type distance. Thus, what remains to be seen is that

We claim that S(G) = S(G)\{p ∞ } where p ∞ (D n,m (x, ȳ)) = n for all n and m. To see this, first note that there is a unique such type by quantifier elimination. Moreover, p ∞ is not in S(G) by Proposition 4.2.3, since it cannot be realised in U. Conversely, if p is any type different from p ∞ , then there are variables x and ȳ and an n ∈ N such that p(D n,m (x, ȳ)) < n, where m = |x| + |ȳ|. By the properties of the predicates, it follows that p(D n,2 (x i , y j )) < n for any i and j. Let now (ā, b)

p in some separable model M of T U . Then, by Theorem 6.2.5, it follows that M is a disjoint union of isomorphic copies of U. Moreover, since p(D n,2 (x i , y j )) < n, it follows that ā and b belong to the same E M -class, i.e. they belong to the same copy of U. By quantifier elimination it follows that U p. Thus, p ∈ S(G), which was what we wanted.

We finish the section by observing that, by repeating the proof of Corollary 5.3.8, we can show that the stabiliser of any element a ∈ U is Roelcke precompact. Corollary 6.2.10. The stabiliser of any a ∈ U is Roelcke precompact.

Proof. Repeat the proof of Corollary 5.3.8 (almost) verbatim.

Universality of Aut(U)

In this section, we will show that Aut(U) is a universal Polish group. We will do this by adapting Uspenskij's proof of this fact for the isometry group of the Urysohn metric space in [START_REF] Uspenskij | On the group of isometries of the Urysohn universal metric space[END_REF]. In order to do that, we will need the Katětov construction introduced earlier in Section 6. which is of course a contradiction. Next, by universality of (U, δ U ), we can embed (A, δ A ) into (U, δ U ). By ultrahomogeneity, we can extend each partial autoversity of A to an autoversity of U. Note that since D includes all values of δ A , any partial L-automorphism of A is a partial autoversity of A (and vice versa of course). Hence, we can extend any partial L-automorphism of A to a full L-automorphism of U viewed as an L-structure. By Theorem 6.5.10 above, we can find a finite T -free L-structure C containing A as a substructure such that each partial L-automorphism of A extends to an automorphism of C. Given a partial automorphism g of A, we will denote its extension to C by g. By convention, we will assume that the empty map is extended to the identity map.

A sequence of subsets e 1 , . . . , e k ⊆ C is called a connection if the intersection graph on {e i } is connected and if there are (r 1 , n 1 ), . . . , (r n , n k ) ∈ D such that for each i ≤ k

for any permutation σ of e i considered as an ordered tuple.

Given c, c ∈ C, we say that they are connected if there is a connection e 1 , . . . , e k such that c ∈ e 1 and c ∈ e k . Let B ⊆ C be those b ∈ C that are connected to some a ∈ A. Note that any b ∈ B is connected to all a ∈ A, since if b is connected to a ∈ A via the connection e 1 , . . . , e k , then {a, a }, e 1 , . . . , e k is a connection between a and b. Moreover, clearly A ⊆ B, since given a ∈ A, we pick a ∈ A \ {a} (remember that we have assumed |A| ≥ 2) and see that {a, a } is a connection between a and a . Given a partial automorphism g of A, we claim that g(B) = B. To show this, it is enough to show that g(B) ⊆ B, since we are dealing with finite sets. If g is the empty map, then we extend it to the identity and there is nothing to show. If not, pick a in the domain of g and let b ∈ B. Then, as noted above, we can find a connection between a and b. Let e 1 , . . . , e k denote such a connection. Since g is an automorphism, it follows that g(e 1 ), . . . , g(e k ) is a connection between g(a) = g(a) ∈ A and g(b), because clearly the intersection graph on {g(e i ) : 1 ≤ i ≤ k} is connected and g preserves the relations. We conclude that g(b) ∈ B as we claimed.

Define now a diversity δ B on B by letting δ B (X) be 0 if |X| ≤ 1 and otherwise letting it be the minimum over all sums k i=1 r i , where for some connection e 1 , . . . , e k with e i ⊆ B, we have C R (r i ,|e i |) (σ(e i )) for any permutation of e i considered as a tuple and where X ⊆ e i . Note that since X ⊆ B, each element of X is connected to the same element of a ∈ A. Hence, the collection of all these connections, one for each x ∈ X, forms a connection containing X. Therefore, this minimum is not over the empty set and so δ B is well-defined.

We must argue why δ B is a diversity map, i.e. we must show that for each X, Y, Z ⊆ B with Z = ∅ we have

Let {e i } and {r i } realise δ B (X ∪ Z) and let {f j } and {s j } realise δ B (Z ∪ Y ). Then since Z ⊆ X ∪ Z ⊆ e i and Z ⊆ Z ∪ Y ⊆ f j , it follows that the intersection graph on {e i } ∪ {f j } is connected. Hence, {e i } ∪ {f j } forms a connection. Moreover, this connection covers X ∪ Y . Therefore, we have that

as we wanted. Moreover, if g is a partial autoversity of A, it follows that the extension g and its inverse g-1 maps connections to connections. Therefore, we must have that g : B → B is an autoversity with respect to δ B . Finally, we must show that δ B extends δ A . First of all, it is clear that we must have δ B (X) ≤ δ A (X) for all X ⊆ A, since {X} is itself a connection covering X as C R (δ A (X),|X|) (X).

Suppose towards a contradiction that we have δ B (X) < δ A (X). Then let e 1 , . . . , e k be a connection with corresponding values r 1 , . . . , r k witnessing this, i.e. r i < δ A (X). It follows that

is a configuration because X ⊆ e i , so

where the second inequality follows since the first sum counts each element of e i exactly once and the second sum counts each element at least once given that the intersection graph on {e i } is connected. If we denote this configuration by α, then {e i , X : 1 ≤ i ≤ k} is an α-family β. Therefore, M α,β is in T and the identity map on M α,β is a weak homomorphism into C. This contradicts that C is T -free. We conclude that δ B (X) = δ A (X).

All in all, we have extended each partial autoversity of A to an autoversity of (B, δ B ), and this diversity contains (A, δ A ) as a subdiversity. This was what we wanted.

We are now ready to prove that Aut(U Q ) has ample generics. Theorem 6.5.11. Aut(U Q ) has ample generics.

Open problems

We present here a list of open problems related to our work. The first was already mentioned in Chapter 3.

Question A.1. Is the set of realisable types in infinitary continuous logic always complete with respect to the type distance?

In Chapter 3, we showed that this was true for ℵ 0 -categorical theories, and we conjecture that it is true in more generality.

In [START_REF] Zielinski | Locally roelcke precompact polish groups[END_REF], the author claims to show that the group of affine isometries of the Gurarij space is locally Roelcke precompact. However, there is a gap in the proof, since the author seems to assume that this space is homogeneous as a metric space, which it is not. Therefore, the following remains an open question:

Question A.2. Is the group of affine isometries of the Gurarij space locally Roelcke precompact? Since the Gurarij space is approximately homogeneous as a metric structure, our approach to local Roelcke precompactness might be better suited for this problem.

Diversities were defined only a few years ago and so there are many natural questions concerning them. We list some of them here.

Question A.3. Is the Urysohn diversity U compactly homogenous? That is, can every isoversity defined on a compact subset of U be extended to all of U? This is true for the Urysohn metric space. The authors of [START_REF] Bryant | A universal separable diversity[END_REF] claim that the usual proof for the metric space goes through for the diversity as well. Unfortunately, this turned out to be wrong, because the usual proof uses that we only measure distances between pairs of points when we exploit the triangle inequality to do the appropriate approximations. In the general diversity setting we have to add an ε to this approximation every time we add an element to our 143