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Interior (topology)

In mathematics, specifically in topology, the interior of a subset $x$ of a topological space $S$ is the union of all subsets of $y$ that are open in $S$. A point that is in the interior of $S$ is an interior point of $X$.

The interior of $S$ is the complement of the closure of the complement of $X$. In this sense interior and closure are dual notions.

The exterior of a set $S$ is the complement of the closure of $S$; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty).

The interior and exterior of a closed curve are a slightly different concept; see the Jordan curve theorem.

Interior point
If $$S$$ is a subset of a Euclidean space, then $$x$$ is an interior point of $$S$$ if there exists an open ball centered at $$x$$ which is completely contained in $$S.$$ (This is illustrated in the introductory section to this article.)

This definition generalizes to any subset $$S$$ of a metric space $$X$$ with metric $$d$$: $$x$$ is an interior point of $$S$$ if there exists a real number $$r > 0,$$ such that $$y$$ is in $$S$$ whenever the distance $$d(x, y) < r.$$

This definition generalizes to topological spaces by replacing "open ball" with "open set". If $$S$$ is a subset of a topological space $$X$$ then $$x$$ is an of $$S$$ in $$X$$ if $$x$$ is contained in an open subset of $$X$$ that is completely contained in $$S.$$ (Equivalently, $$x$$ is an interior point of $$S$$ if $$S$$ is a neighbourhood of $$x.$$)

Interior of a set
The interior of a subset $$S$$ of a topological space $$X,$$ denoted by $$\operatorname{int}_X S$$ or $$\operatorname{int} S$$ or $$S^\circ,$$ can be defined in any of the following equivalent ways: If the space $$X$$ is understood from context then the shorter notation $$\operatorname{int} S$$ is usually preferred to $$\operatorname{int}_X S.$$
 * 1) $$\operatorname{int} S$$ is the largest open subset of $$X$$ contained in $$S.$$
 * 2) $$\operatorname{int} S$$ is the union of all open sets of $$X$$ contained in $$S.$$
 * 3) $$\operatorname{int} S$$ is the set of all interior points of $$S.$$

Examples

 * In any space, the interior of the empty set is the empty set.
 * In any space $$X,$$ if $$S \subseteq X,$$ then $$\operatorname{int} S \subseteq S.$$
 * If $$X$$ is the real line $$\Reals$$ (with the standard topology), then $$\operatorname{int} ([0, 1]) = (0, 1)$$ whereas the interior of the set $$\Q$$ of rational numbers is empty: $$\operatorname{int} \Q = \varnothing$$
 * If $$X$$ is the complex plane $$\Complex,$$ then $$\operatorname{int} (\{z \in \Complex : |z| \leq 1\}) = \{z \in \Complex : |z| < 1\}.$$
 * In any Euclidean space, the interior of any finite set is the empty set.

On the set of real numbers, one can put other topologies rather than the standard one:

 * If $$X$$ is the real numbers $$\Reals$$ with the lower limit topology, then $$\operatorname{int} ([0, 1]) = [0, 1).$$
 * If one considers on $$\Reals$$ the topology in which every set is open, then $$\operatorname{int} ([0, 1]) = [0, 1].$$
 * If one considers on $$\Reals$$ the topology in which the only open sets are the empty set and $$\Reals$$ itself, then $$\operatorname{int} ([0, 1])$$ is the empty set.

These examples show that the interior of a set depends upon the topology of the underlying space. The last two examples are special cases of the following.

 * In any discrete space, since every set is open, every set is equal to its interior.
 * In any indiscrete space $$X,$$ since the only open sets are the empty set and $$X$$ itself, $$\operatorname{int} X = X$$ and for every proper subset $$S$$ of $$X,$$ $$\operatorname{int} S$$ is the empty set.

Properties
Let $$X$$ be a topological space and let $$S$$ and $$T$$ be subsets of $$X.$$

 * $$\operatorname{int} S$$ is open in $$X.$$
 * If $$T$$ is open in $$X$$ then $$T \subseteq S$$ if and only if $$T \subseteq \operatorname{int} S.$$
 * $$\operatorname{int} S$$ is an open subset of $$S$$ when $$S$$ is given the subspace topology.
 * $$S$$ is an open subset of $$X$$ if and only if $$\operatorname{int} S = S.$$
 * : $$\operatorname{int} S \subseteq S.$$
 * Idempotent: $$\operatorname{int} (\operatorname{int} S) = \operatorname{int} S.$$
 * /: $$\operatorname{int} (S \cap T) = (\operatorname{int} S) \cap (\operatorname{int} T).$$
 * However, the interior operator does not distribute over unions since only $$\operatorname{int} (S \cup T) ~\supseteq~ (\operatorname{int} S) \cup (\operatorname{int} T)$$ is guaranteed in general and equality might not hold. For example, if $$X = \Reals, S = (-\infty, 0],$$ and $$T = (0, \infty)$$ then $$(\operatorname{int} S) \cup (\operatorname{int} T) = (-\infty, 0) \cup (0, \infty) = \Reals \setminus \{0\}$$ is a proper subset of $$\operatorname{int} (S \cup T) = \operatorname{int} \Reals = \Reals.$$
 * /: If $$S \subseteq T$$ then $$\operatorname{int} S \subseteq \operatorname{int} T.$$

Other properties include:

 * If $$S$$ is closed in $$X$$ and $$\operatorname{int} T = \varnothing$$ then $$\operatorname{int} (S \cup T) = \operatorname{int} S.$$

Relationship with closure

The above statements will remain true if all instances of the symbols/words
 * "interior", "int", "open", "subset", and "largest"

are respectively replaced by
 * "closure", "cl", "closed", "superset", and "smallest"

and the following symbols are swapped: For more details on this matter, see interior operator below or the article Kuratowski closure axioms.
 * 1) "$$\subseteq$$" swapped with "$$\supseteq$$"
 * 2) "$$\cup$$" swapped with "$$\cap$$"

Interior operator
The interior operator $$\operatorname{int}_X$$ is dual to the closure operator, which is denoted by $$\operatorname{cl}_X$$ or by an overline —, in the sense that $$\operatorname{int}_X S = X \setminus \overline{(X \setminus S)}$$ and also $$\overline{S} = X \setminus \operatorname{int}_X (X \setminus S),$$ where $$X$$ is the topological space containing $$S,$$ and the backslash $$\,\setminus\,$$ denotes set-theoretic difference. Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators, by replacing sets with their complements in $$X.$$

In general, the interior operator does not commute with unions. However, in a complete metric space the following result does hold:

$S$

The result above implies that every complete metric space is a Baire space.

Exterior of a set
The exterior of a subset $$S$$ of a topological space $$X,$$ denoted by $$\operatorname{ext}_X S$$ or simply $$\operatorname{ext} S,$$ is the largest open set disjoint from $$S,$$ namely, it is the union of all open sets in $$X$$ that are disjoint from $$S.$$ The exterior is the interior of the complement, which is the same as the complement of the closure; in formulas, $$\operatorname{ext} S = \operatorname{int}(X \setminus S) = X \setminus \overline{S}.$$

Similarly, the interior is the exterior of the complement: $$\operatorname{int} S = \operatorname{ext}(X \setminus S).$$

The interior, boundary, and exterior of a set $$S$$ together partition the whole space into three blocks (or fewer when one or more of these is empty): $$X = \operatorname{int} S \cup \partial S \cup \operatorname{ext} S,$$ where $$\partial S$$ denotes the boundary of $$S.$$ The interior and exterior are always open, while the boundary is closed.

Some of the properties of the exterior operator are unlike those of the interior operator:
 * The exterior operator reverses inclusions; if $$S \subseteq T,$$ then $$\operatorname{ext} T \subseteq \operatorname{ext} S.$$
 * The exterior operator is not idempotent. It does have the property that $$\operatorname{int} S \subseteq \operatorname{ext}\left(\operatorname{ext} S\right).$$

Interior-disjoint shapes

Two shapes $$a$$ and $$b$$ are called interior-disjoint if the intersection of their interiors is empty. Interior-disjoint shapes may or may not intersect in their boundary.

title: Interior (topology)


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Alexandroff extension

In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alexandroff. More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c : X → X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any topological space (but provides an embedding exactly for Tychonoff spaces).

Example: inverse stereographic projection
A geometrically appealing example of one-point compactification is given by the inverse stereographic projection. Recall that the stereographic projection S gives an explicit homeomorphism from the unit sphere minus the north pole (0,0,1) to the Euclidean plane. The inverse stereographic projection $$S^{-1}: \mathbb{R}^2 \hookrightarrow S^2$$ is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point $$\infty = (0,0,1)$$. Under the stereographic projection latitudinal circles $$z = c$$ get mapped to planar circles $r = \sqrt{(1+c)/(1-c)}$. It follows that the deleted neighborhood basis of $$(0,0,1)$$ given by the punctured spherical caps $$c \leq z < 1$$ corresponds to the complements of closed planar disks $r \geq \sqrt{(1+c)/(1-c)}$. More qualitatively, a neighborhood basis at $$\infty$$ is furnished by the sets $$S^{-1}(\mathbb{R}^2 \setminus K) \cup \{ \infty \}$$ as K ranges through the compact subsets of $$\mathbb{R}^2$$. This example already contains the key concepts of the general case.

Motivation
Let $$c: X \hookrightarrow Y$$ be an embedding from a topological space X to a compact Hausdorff topological space Y, with dense image and one-point remainder $$\{ \infty \} = Y \setminus c(X)$$. Then c(X) is open in a compact Hausdorff space so is locally compact Hausdorff, hence its homeomorphic preimage X is also locally compact Hausdorff. Moreover, if X were compact then c(X) would be closed in Y and hence not dense. Thus a space can only admit a Hausdorff one-point compactification if it is locally compact, noncompact and Hausdorff. Moreover, in such a one-point compactification the image of a neighborhood basis for x in X gives a neighborhood basis for c(x) in c(X), and—because a subset of a compact Hausdorff space is compact if and only if it is closed—the open neighborhoods of $$\infty$$ must be all sets obtained by adjoining $$\infty$$ to the image under c of a subset of X with compact complement.

The Alexandroff extension
Let $$X$$ be a topological space. Put $$X^* = X \cup \{\infty \},$$ and topologize $$X^*$$ by taking as open sets all the open sets in X together with all sets of the form $$V = (X \setminus C) \cup \{\infty \}$$ where C is closed and compact in X. Here, $$X \setminus C$$ denotes the complement of $$ C$$ in $$X.$$ Note that $$V$$ is an open neighborhood of $$\infty,$$ and thus any open cover of  $$\{\infty \}$$ will contain all except a compact subset $$C$$ of $$X^*,$$ implying that $$X^*$$ is compact.

The space $$X^*$$ is called the Alexandroff extension of X (Willard, 19A). Sometimes the same name is used for the inclusion map $$c: X\to X^*.$$

The properties below follow from the above discussion:

 * The map c is continuous and open: it embeds X as an open subset of $$X^*$$.
 * The space $$X^*$$ is compact.
 * The image c(X) is dense in $$X^*$$, if X is noncompact.
 * The space $$X^*$$ is Hausdorff if and only if X is Hausdorff and locally compact.
 * The space $$X^*$$ is T1 if and only if X is T1.

The one-point compactification
In particular, the Alexandroff extension $$c: X \rightarrow X^*$$ is a Hausdorff compactification of X if and only if X is Hausdorff, noncompact and locally compact. In this case it is called the one-point compactification or Alexandroff compactification of X.

Recall from the above discussion that any Hausdorff compactification with one point remainder is necessarily (isomorphic to) the Alexandroff compactification. In particular, if $$X$$ is a compact Hausdorff space and $$p$$ is a limit point of $$X$$ (i.e. not an isolated point of $$X$$), $$X$$ is the Alexandroff compactification of $$X\setminus\{p\}$$.

Let X be any noncompact Tychonoff space. Under the natural partial ordering on the set $$\mathcal{C}(X)$$ of equivalence classes of compactifications, any minimal element is equivalent to the Alexandroff extension (Engelking, Theorem 3.5.12). It follows that a noncompact Tychonoff space admits a minimal compactification if and only if it is locally compact.

Non-Hausdorff one-point compactifications
Let $$(X,\tau)$$ be an arbitrary noncompact topological space. One may want to determine all the compactifications (not necessarily Hausdorff) of $$X$$ obtained by adding a single point, which could also be called one-point compactifications in this context. So one wants to determine all possible ways to give $$X^*=X\cup\{\infty\}$$ a compact topology such that $$X$$ is dense in it and the subspace topology on $$X$$ induced from $$X^*$$ is the same as the original topology. The last compatibility condition on the topology automatically implies that $$X$$ is dense in $$X^*$$, because $$X$$ is not compact, so it cannot be closed in a compact space. Also, it is a fact that the inclusion map $$c:X\to X^*$$ is necessarily an open embedding, that is, $$X$$ must be open in $$X^*$$ and the topology on $$X^*$$ must contain every member of $$\tau$$. So the topology on $$X^*$$ is determined by the neighbourhoods of $$\infty$$. Any neighborhood of $$\infty$$ is necessarily the complement in $$X^*$$ of a closed compact subset of $$X$$, as previously discussed.

The topologies on $$X^*$$ that make it a compactification of $$X$$ are as follows:
 * The Alexandroff extension of $$X$$ defined above. Here we take the complements of all closed compact subsets of $$X$$ as neighborhoods of $$\infty$$.  This is the largest topology that makes $$X^*$$ a one-point compactification of $$X$$.
 * The open extension topology. Here we add a single neighborhood of $$\infty$$, namely the whole space $$X^*$$.  This is the smallest topology that makes $$X^*$$ a one-point compactification of $$X$$.
 * Any topology intermediate between the two topologies above. For neighborhoods of $$\infty$$ one has to pick a suitable subfamily of the complements of all closed compact subsets of $$X$$; for example, the complements of all finite closed compact subsets, or the complements of all countable closed compact subsets.

Compactifications of discrete spaces

 * The one-point compactification of the set of positive integers is homeomorphic to the space consisting of K = {0} U {1/n | n is a positive integer} with the order topology.
 * A sequence $$\{a_n\}$$ in a topological space $$X$$ converges to a point $$a$$ in $$X$$, if and only if the map $$f\colon\mathbb N^*\to X$$ given by $$f(n) = a_n$$ for $$n$$ in $$\mathbb N$$ and $$f(\infty) = a$$ is continuous. Here $$\mathbb N$$ has the discrete topology.
 * Polyadic spaces are defined as topological spaces that are the continuous image of the power of a one-point compactification of a discrete, locally compact Hausdorff space.

Compactifications of continuous spaces

 * The one-point compactification of n-dimensional Euclidean space Rn is homeomorphic to the n-sphere Sn. As above, the map can be given explicitly as an n-dimensional inverse stereographic projection.
 * The one-point compactification of the product of $$\kappa$$ copies of the half-closed interval [0,1), that is, of $$[0,1)^\kappa$$, is (homeomorphic to) $$[0,1]^\kappa$$.
 * Since the closure of a connected subset is connected, the Alexandroff extension of a noncompact connected space is connected. However a one-point compactification may "connect" a disconnected space: for instance the one-point compactification of the disjoint union of a finite number $$n$$ of copies of the interval (0,1) is a wedge of $n$ circles.
 * The one-point compactification of the disjoint union of a countable number of copies of the interval (0,1) is the Hawaiian earring. This is different from the wedge of countably many circles, which is not compact.
 * Given $$X$$ compact Hausdorff and $$C$$ any closed subset of $$X$$, the one-point compactification of $$X\setminus C$$ is $$X/C$$, where the forward slash denotes the quotient space.
 * If $$X$$ and $$Y$$ are locally compact Hausdorff, then $$(X\times Y)^* = X^* \wedge Y^*$$ where $$\wedge$$ is the smash product. Recall that the definition of the smash product:$$A\wedge B = (A \times B) / (A \vee B)$$ where $$A \vee B$$ is the wedge sum, and again, / denotes the quotient space.

As a functor
The Alexandroff extension can be viewed as a functor from the category of topological spaces with proper continuous maps as morphisms to the category whose objects are continuous maps $$c\colon X \rightarrow Y$$ and for which the morphisms from $$c_1\colon X_1 \rightarrow Y_1$$ to $$c_2\colon X_2 \rightarrow Y_2$$ are pairs of continuous maps $$f_X\colon X_1 \rightarrow X_2, \ f_Y\colon Y_1 \rightarrow Y_2$$ such that $$f_Y \circ c_1 = c_2 \circ f_X$$. In particular, homeomorphic spaces have isomorphic Alexandroff extensions.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Alexandroff extension


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Talk:Locally compact space

Local base
I believe that the reason that some people use the definition require a local base of compact neighbourhoods is that this has been found to be a useful concept in the nonHausdorff context, whereas the definition requiring only a single compact neighbourhood has not. There is nothing in Wikipedia yet about nonHausdorff locally compact spaces in either sense, but I suggest that we set ourselves up in a position to talk about a useful concept in the future by switching the definition that Wikipedia uses. This is easy to do now; just change this and the topology glossary. — Toby Bartels, Sunday, June 9, 2002

Sure, if you think the other concept is more useful, let's change it. AxelBoldt, Sunday, June 9, 2002

Examples
I moved some of the examples over to compact space.

Is it actually true that all compact spaces are locally compact? Just checking :-)

 * See below.

 * This is the reason that I arranged the examples as I did: I wanted to highlight all the possible contrasting possibilities: spaces that were compact, spaces that were locally compact but not compact, spaces that weakly locally compact but not strongly locally compact, and spaces that weren't even weakly locally compact. Then, given the importance of the separation axioms, I wanted examples in each case of both Tychonoff and nonregular spaces. Thus:

 * I don't think it's a crime to have examples duplicated on more than one page. OTOH, you seem to have a different philosophy towards the examples entirely.

 * — Toby Bartels, Wednesday, June 26, 2002

I moved the following example:
 * Some weakly locally compact spaces that aren't locally compact: the right order topology or left order topology on any unbounded totally ordered set, in particular: the right order topology or left order topology on the set R of real numbers, useful in the study of semicontinuous functions

I believe the left order topology on R is locally compact. If x∈R, then ]-∞, x+1/n] for n∈N is a local neighborhood base for x consisting only of compact sets.

 * Hey, you use the backwards bracket thing too! Any chance we can get that accepted as standard practice on Wikipedia?

 * Well, your argument seems correct, but that's not what I remember from Steen & Seebach; let me check again ....

 * Well, I double checked Steen & Seebach, and it looks like they are using yet another definition of locally compact (their "strongly locally compact", since they use "locally compact" for the old fashioned weak sense). I had read this wrong (it is again equivalent in the Hausdorff case). This puts into doubt the claim that every compact space is locally compact, for which I was relying on Steen & Seebach since I didn't see how to prove it, and also removes any indication that I have of a source that calls our local compactness "strong". (The new definition, BTW, is: every point has an open neighbourhood with compact closure.)

 * Now I want to go back and check everything over again. I apparently don't have a source that covers all 3 definitions at once. I'll report back tomorrow with what I can come up with, or you can think about it — you've got the definitions now too.

 * — Toby Bartels, Wednesday, June 26, 2002

Heh, so much for tomorrow. I'll be gone until Monday evening — then I'll work this mess out. In the meantime, visitors to the main article should be sufficiently warned. — Toby Bartels, Friday, June 28, 2002

Maybe we can avoid confusing the beginner by first explaining the standard locally compact notion for Hausdorff spaces (which is what most people need and where the definitions all agree), with all the common examples and properties, and then have a separate section where the various definitions for non-Hausdorff spaces are compared.

The reason I edited the examples was the same: I find it more important that the reader first gets a good grasp of the concept by studying simple motivating examples, rather than find themselves in an overwhelming glut of tiny distinctions that are really only important for the specialists working in the field. AxelBoldt, Friday, June 28, 2002

OK, I've written a new version now with these principles in mind. (Note that I also added the Hausdorffness condition to Wikipedia's definition of topological vector space; the simplicity that this brought to the discussion of TVSs in this article is characteristic of the reasons for that requirement in the definition.) Have at it. — Toby Bartels, Wednesday, July 3, 2002

Herrings
Some thoughts inspired by Axel's recent changes: I've always felt that the complex numbers were a red herring in functional analysis — when you get down to it, you can do most of this stuff over any commutative real C*algebra, so ¿what's so special about C? — which explains my focus on real-valued functions as the simplest example. But I agree that the complex-valued case is so common that it merits a mention here — especially as regards the Gelfand-Naimark theorem, the real version of which exists but is not quite as I had stated it. I do object to using notation like "C0" for the complex case as a default, but I agree that that's pretty common too. — Toby Bartels, Saturday, July 6, 2002

C* algebras
Is the classification of the commutative C* algebras really called "Gelfand-Naimark Theorem"? I seem to recall that name for the result that any C* algebra is a *-subalgebra of the algebra of linear continuous operators on some Hilbert space. AxelBoldt, Saturday, July 6, 2002

What you are citing is called "Gelfand Naimark Segal". Specifically, the Gelfand Naimark Segal construction takes a C* algebra and an element of the algebra and constructs from that a Hilbert space and a representation of the C* algebra on the Hilbert space. Your statement, the Gelfand Naimark Segal theorem, follows because the element of the algebra can be chosen so that the representation is faithful. A quick web search for "gelfand" AND "naimark" AND "theorem" brings up many references to my theorem but no references to your theorem that don't also have the name "Segal" attached. There do seem to be generalisations of the GN theorem to noncommutative C* algebras, representing the C* algebra as an algebra of functions defined on a noncommutative space (see the introduction to http://nyjm.albany.edu:8000/PacJ/1998/184-1-5.html for a survey), but this is again different from the GNS theorem — although I wouldn't be surprised if the GNS theorem couldn't be made a corollary of some of these generalisations, nor would I be surprised if the generalisations made use of the GNS construction. Still, the basic idea seems clear: a GN theorem represents the C* algebra as bounded functions on a topological space (or generalisation thereof), while a GNS theorem represents the C* algebra as bounded operators on a Hilbert space. — Toby Bartels, Saturday, July 6, 2002

Open naighbourhoods
The first paragraph has a link to "local base," which describes them as a collection of open sets. Doesn't this contradict the parenthetical sentence which asserts that the neigborhoods need not be open? Bill Kielhorn, Mar 31, 2003.

 * Yes, thanks for the catch! I'll fix it. AxelBoldt 16:53 Apr 21, 2003 (UTC)

Non-Hausdorff theory
I'm curious about the non-Hausdorff theory. I've checked about a dozen different general topology references (including Munkres, Kelley, Armstrong, Steen and Seebach, McCarty, and others) and none of them use the definition given here (in terms of a local base of compact sets). Everyone uses one of the alternate definitions: However, the article claims that the local base definition is the "best". Can anyone explain why? Or, at least give me some references that use this definition. Mostly, I am curious as to why this definition is in Wikipedia when no one seems to use it.
 * 1) Every point has a compact neighborhood
 * 2) Every point has a relatively compact neighborhood

With the current definition, it is not at all clear to me that a compact space is locally compact (this is true with either of the alternate definitions). Am I wrong, or are there any easy counterexamples? — Fropuff 19:24, 2004 Aug 12 (UTC)

Infinite dim vector spaces
K igor k: Banach-Alaoglu doesn't provide examples. You are mixing up topologies. Like in the Hilbert cube example in the text, the unit ball is not a weak neighborhood. The only way an infdim tvs can be locally compact is if it is really stupid - like having the indiscrete topology. That separated locally compact => finite dim is Chapter 1, section 13, theorem 8 of Grothendieck's TVS, or p. 65 of Edwards' Functional Analysis.--John Z 1 July 2005 05:14 (UTC)

I see, sorry for being a little obtuse. I now see the point of the Hilbert cube example. But I find that example a little misleading. If the Hilbert cube is seen as the set of sequences in [0,1], it is not even a subspace of a Hilbert space (which I think is implied to be $$l_2$$ here). Not all such sequences have finite norm. The Hilbert cube would be the unit ball in $$l_\infty$$ which is a Banach, not a Hilbert, space. -- k_igor_k (Wed Jul 6 09:21:28 EDT 2005)

Yes, you are right. If you want to use Hilbert rather than Banach spaces, the alternate definition from the Hilbert Cube article should be referred to here, sequences with nth term bounded by 1/n, to get it inside $$l_2$$, otherwise it just doesn't make sense. --John Z 7 July 2005 23:28 (UTC)

Definition
Well, I've gone and changed the definition of local compactness again. Actually, the article now mentions all three definitions up front and refuses to commit to any one. I did this because almost all references I've checked were not using our (previous) definition. The sole exception being Willard. I see no good reason to favor Willard's definition above the rest (I welcome arguments either way). If Wikipedia is to be accurate it should reflect the definitions that readers are likely to encounter in the "wild". Unfortunately, this seems to preclude choosing a favored definition here. -- Fropuff 02:49, 17 October 2006 (UTC)

expand
this page never describes, even vaguely, what a compact space is. an article should not require another article for a modicum of understanding. 66.160.66.86 (talk) 16:15, 22 January 2008 (UTC)

"one-point compactification of the rational numbers"
The article says: The one-point compactification of the rational numbers Q is compact and therefore locally compact in senses (1) and (2) but it is not locally compact in sense (3). But one-point compactification is defined only for locally compact spaces, where Q is (as mentioned some not locally compact. So, what does this sentence wants to say? -- Paul Ebermann (talk) 14:36, 8 July 2008 (UTC)
 * The construction should not require the original space to be locally compact. If Y is a topological space, define the Alexandroff extension Y* to be Y union {p}, with neighborhoods of p defined as (Y-L) union {p} where L is compact. Y* is compact, Y is contained as a subspace, Y is open in Y*, Y is dense in Y* if and only if Y is not compact, Y* is Hausdorff if and only if Y is locally compact and Hausdorff.  This is from Willard, 1970, 19A, p. 140. JackSchmidt (talk) 15:45, 8 July 2008 (UTC)
 * The construction does not, but the one-point compactification may no longer be Hausdorff. Indeed, the one-point compactification of the rationals is not Hausdorff, since any neighborhood of the point at infinity is a finite-complement neighborhood.  This should probably be addressed somewhere in the one-point compactification article, but I am reluctant to omit the requirement of local compactness from the definition.  Perhaps it should be mentioned in a separate "generalizations" section to avoid polluting the main results with pathologies.   siℓℓy rabbit  (  talk  ) 16:00, 8 July 2008 (UTC)

Examples Correct?
I am having some issues with some of the examples of "Hausdorff Spaces that are not Locally Compact" on this page, see for instance:

 * * the space Q of rational numbers (endowed with the topology from R), since its compact subsets all have empty interior and therefore are not neighborhoods;

As far as i'm aware, if some subset of $$\mathbb{Q}$$ is taken as a subset of $$\mathbb{R}$$ then sure, its interior is empty. But *within* the sub-space topology the interior of, e.g. [a,b] is just (a,b), no? A valid reason that $$\mathbb{Q}$$ is not locally compact is maybe because sequences of the form $$\left( \left[ r + \frac{\lfloor 2^n \sqrt{2} \rfloor}{2^n},r + \frac{\lfloor 2^n \sqrt{2} \rfloor + 1}{2^n} \right] \right)_{n \in \mathbb{N}, n > N}$$ for suitable rational $$r$$ and number $$N$$ form descending sequences of closed non-empty sets with empty interesection in $$\mathbb{Q}$$, and by suitable choice of $$N$$ and $$r$$ can be placed inside any neighborhood, showing that no neighborhood is compact.

... and on this example:

 * * the subspace {(0,0)} union {(x,y) : x > 0} of R2, since the origin does not have a compact neighborhood;

I'm wondering how this can be? Is there a citation (specific chapter in one of the standard books)? It seems that this is a closed subset of R and hence locally compact? Am i missing something?

199.43.12.100 (talk) 10:45, 25 April 2013 (UTC)

 * In the first example, you are right that if a and b are rationals with a < b, then the interior (in Q) of [a,b] ∩ Q is (a,b) ∩ Q. But [a,b] ∩ Q is not compact, so this doesn't contradict the assertion. As for the other example, it's not a closed subset of R2. --Zundark (talk) 11:46, 25 April 2013 (UTC)

Okay looking at both examples more closely i see that the wording of the article is correct, thanks. 199.43.13.100 (talk) 12:12, 25 April 2013 (UTC)

title: Talk:Locally compact space


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Nurse uniform

A nurse uniform is attire worn by nurses for hygiene and identification. The traditional nurse uniform consists of a dress, apron and cap. It has existed in many variants, but the basic style has remained recognizable.

History
The first nurse uniforms were derived from the nun's habit. Before the 19th century, nuns took care of sick and injured people so it was obvious that trained lay nurses might copy the nun's habit as they have adopted ranks like "Sister". One of Florence Nightingale's first students (Miss van Rensselaer) designed the original uniform for the students at Miss Nightingale's school of nursing. Before the 1940s minor changes occurred in the uniform. The clothing consisted of a mainly blue outfit. Hospitals were free to determine the style of the nurse uniform, including the nurse's cap which exists in many variants.

In Britain, the national uniform (or simply "national") was designed with the advent of the National Health Service (NHS) in 1948, and the Newcastle dress. From the 1960s open necks began to appear. In the 1970s, white disposable paper caps replaced cotton ones; in the 1980s, plastic aprons displaced the traditional ones and outerwear began to disappear. From the 1990s, scrubs became popular in Britain, having first appeared in the USA; however, some nurses in Britain continue to wear dresses, although some NHS trusts have removed them in favour of scrubs as in many other countries.

Standard nurse's uniform
Historically, a typical nurse uniform consisted of a dress, pinafore apron and nurse's cap. In some hospitals, however, student nurses also wore a nursing pin, or the pinafore apron may have been replaced by a cobbler style apron. This type of nurse's dress continues to be worn in many countries.

Traditional uniforms remain common in many countries, but in Western Europe and North America, the so-called "scrubs" or tunics have become more popular. "Scrub dress" is a simpler type of uniform, and is almost always worn in operating rooms and emergency rooms.

Nurses have a variety of roles in the 21st century and therefore have a variety of "uniforms". Nurse scientists may wear a lab coat, while nurse executives may wear a suit. The "uniform" will vary by country and role.

Alternative nurse uniforms
Since the late 1980s, there has been a move towards alternative designs of nursing uniforms in some countries. Newer style nurse's uniform in the United Kingdom consists of either:

 * 1) A tunic-style top and dark blue trousers that are optimally designed to prevent cross-infection, the colour of which depends upon the grade (or, more recently, band) and gender of the nurse – the colour varies between NHS Trusts. The tunics often feature piping around the edges of the uniform.
 * 2) A dress in the same colour as the tunic-style top.

Male nursing uniform
In some countries the nurse's uniform is now gender neutral. When this is not the case, male nurses wear a different uniform to their female counterparts. In the UK, male nurses often wear a white tunic with epaulettes in a color or quantity that represents their year of training or grade. However, in many trusts the white uniform is now obsolete and uniforms are non-gender specific.

No Uniform

In some settings, for example in psychiatric inpatient units in the UK, it is common for mental health and learning disorder nurses to not wear a formal uniform.

Nurse uniforms versus scrubs
Beginning in the 1990s, and until the present time, the traditional nurse uniforms have been replaced with the "new" scrub dress in some countries. Most hospitals in the US and Europe argue that the scrub uniform is easier to clean than the old nurse uniforms. The nurses who wear the uniforms are divided into two camps:

 * Those who prefer the new scrubs; disliked the old white nurse dress uniforms.
 * The nurses who liked the old white nurse dress uniforms; they argue that nurses who wear scrubs are seen by the patients as cleaners or surgeons and cannot be identified as nurses.

In many parts of the world, nurses continue to wear a uniform consisting of a dress, pinafore and cap. The traditional white uniform for male nursing staff is now going out of fashion, except for student nurses. A tunic of either the dental surgeon style or a V-neck with a collar is very often used. The colours vary with grade, area of work, and hospital; however, the male equivalent of a sister (that is, charge nurse) tend to be shades of blue or dark green: often, this is the only colour to be recognised by the public as signifying a person in authority.

Nursing jewellery
Nurses were actively discouraged from wearing jewellery which might distract from their purpose and get caught on patient skin during care activity. A fob watch or pendant watch is considered synonymous with nursing. The fob watch frees the nurses' hands for client care and prevents the wrist watch becoming a vector for disease. Watches are sometimes given as a token rite-of-passage gift from parents to young nurses, who are making the transition into nurses' quarters and live away from home for the first time.

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School uniform

A school uniform is a uniform worn by students primarily for a school or otherwise an educational institution. They are common in primary and secondary schools in various countries.

An example of a uniform would be requiring button-down shirts, trousers for boys and blouses, pleated skirts for girls, with both wearing blazers. A uniform can even be as simple as requiring collared shirts, or restricting colour choices and limiting items students are allowed to wear.

Uniform
Although often used interchangeably, there is an important difference between dress codes and school uniforms: according to scholars such as Nathan Joseph, clothing can only be considered a uniform when it "(a) serves as a group emblem, (b) certifies an institution's legitimacy by revealing individual's relative positions and (c) suppresses individuality." Conversely, a dress code is much less restrictive, and focuses "on promoting modesty and discouraging anti-social fashion statements", according to Marian Wilde. Examples of a dress code would be not allowing ripped clothing, no logos or limiting the amount of skin that can be shown.

School uniforms are clothes that are usually used for school, each level of school has a different uniform. Each educational unit has its own distinctive school uniform.

History

It is difficult to trace the origins of the uniform as there is no comprehensive written history, but rather a variety of known influences. School uniforms are believed to be a practice which dates to the 16th century in the United Kingdom. It is believed that the Christ's Hospital School in England in 1552 was the first school to use a school uniform. Students were given a uniform that most notably consisted of a long blue coat and yellow, knee-high socks. An almost identical uniform is still worn by students attending the school today. The earliest documented proof of institutionalized use of a standard academic dress dates back to 1222 when the then Archbishop of Canterbury ordered the wearing of the cappa clausa. This monastic and academic practice evolved into collegiate uniforms in England, particularly in charity schools where uniform dress was often provided for poor children. Universities, primary schools and secondary schools used uniforms as a marker of class and status. Although school uniforms can often be considered conservative and old-fashioned, uniforms in recent years have changed as societal dress codes have changed.

Contemporary

In the United States, a movement toward using uniforms in state schools began when Bill Clinton addressed it in the 1996 State of the Union, saying: "If it means that teenagers will stop killing each other over designer jackets, then our public schools should be able to require their students to wear uniforms." As of 1998 approximately 25% of all U.S. public elementary, middle and junior high schools had adopted a uniform policy or were considering a policy, and two-thirds were implemented between 1995 and 1997. New York City's then-new schools chancellor, Rudy Crew, made it clear that he would not follow Clinton's idea.

There is an abundance of theories and empirical studies looking at school uniforms, making statements about their effectiveness. These theories and studies elaborate on the benefits and also the shortcomings of uniform policies. The issue of nature vs. nurture comes into play, as uniforms affect the perceptions of masculinity and femininity, over-simplify issues of gender classification, and attempt to suppress students' sexuality. Uniforms bring a variety of pros, cons, and major legal implications and controversies.

There are two main empirical findings that are most often cited in the political rhetoric surrounding the uniform debate. One of these, the case study of the Long Beach Unified School District, is most often cited in support of school uniforms and their effectiveness whereas Effects of Student Uniforms on Attendance, Behavior Problems, Substance Use, and Academic Achievement is the most frequently cited research in opposition to the implementation of school uniform policies.

Effects of uniforms on students

The case study of the Long Beach Unified School District was the study of the first large, urban school in the United States to implement a uniform policy. In 1994, mandatory school uniforms were implemented for the districts elementary and middle schools as a strategy to address the students' behavior issues. The district simultaneously implemented a longitudinal study to research the effects of the uniforms on student behavior. The study attributed favorable student behavioral changes and a significant drop in school discipline issues to the mandatory uniform policy. Wearing school uniforms was associated with fewer absences and truancies and fewer referrals to the office for behavior problems. Suspensions and expulsions were reduced by 28% (elementary) and 36% (middle school), crime and vandalism by 74% (elementary) and 18% (middle school). However the school district also added other security measures such as security guards, and metal detectors so the success cannot be solely attributed to the uniforms. The district later removed the uniforms. Other research found that uniforms were not an effective deterrent to decrease truancy, did not decrease behavior problems, decrease substance use, and in fact may be associated with poorer student achievement relative to students not required to wear school uniforms.

Brunsma stated that despite the inconclusiveness of the effects of uniforms, they became more common because "this is an issue of children's rights, of social control, and one related to increasing racial, class and gender inequalities in our schools."

Laws and rulings

As uniforms have become more normalised, there have also been an increasing number of lawsuits brought against school districts. According to David Brunsma, one in four public elementary schools and one in eight public middle and high schools in the United States have policies dictating what a student wears to school. The school code within states' constitutions typically asserts that it allows the board of school directors to make reasonable rules and regulations as they see fit in managing the school's affairs. As of 2008, there are currently 23 states that allow school districts to mandate school uniforms. The constitutional objections usually brought upon school districts tend to fall into one of the following two categories: (1) a violation of the students' First Amendment right to free expression (2) a violation of parents' right to raise their children without government interference. Although up until this point, The Supreme Court has not ruled on a case involving school uniforms directly, in the 1968 decision Tinker v. Des Moines Independent Community School District, the Court ruled that upon entering school, students do not shed their constitutional rights to freedom of speech.

Internationally, there are differing views of school uniforms. In the Australian state of Queensland, Ombudsman Fred Albietz ruled in 1998 that state schools may not require uniforms. In the Philippines, the Department of Education abolished the requirement of school uniforms in public schools. In England and Wales, technically a state school may not permanently exclude students for "breaching school uniform policy", under a policy promulgated by the Department for Children, Schools and Families but students not wearing the correct uniform are asked to go home and change. In Scotland, some local councils (that have responsibility for delivering state education) do not insist on students wearing a uniform as a precondition to attending and taking part in curricular activities. Turkey abolished mandatory uniforms in 2010.

Canady v. Bossier Parish School Board
In the Canady v. Bossier Parish School Board lawsuit in 2000, a Louisiana district court ruled in favour of the school board because it did not see how the free speech rights of the students were being violated due to the school board's uniform policy. Even though the plaintiff appealed the decision, the Fifth Circuit Court also ruled in favour of the school board after implementing a four-step system that is still used today. Firstly, a school board has to have the right to set up a policy. Secondly, the policy must be determined to support a fundamental interest of the board as a whole. Thirdly, the guidelines cannot have been set for the purpose of censorship. Finally, the limits on student expression cannot be greater than the interest of the board. As long as these four policies are in place, then no constitutional violation can be claimed.

Littlefield v. Forney Independent School District
In the Forney Independent School District of Forney, Texas in 2001, the school board decided to implement a school uniform policy allowing the students to wear a polo shirt, oxford shirt or blouse in four possible colours, and blue or khaki trousers or shirts, a skirt or jumper. While there was some flexibility with shoes, certain types were prohibited along with any sort of baggy clothes. The parents of the Littlefield family requested that their son be exempt from the policy, but were denied. In response, the Littlefields filed a lawsuit against the school district, under the pretenses that this uniform mandate infringed on their rights as parents to control how they brought up their children and their education. They even went as far as to cite an infringement on religious freedom, claiming that opting out of the uniforms on the grounds of religion allowed the school to rank the validity of certain religions. Before trial, the District Court dismissed the case, so the family appealed. Ultimately, the Fifth Circuit Court ruled that the students' rights were not being violated even though the claims presented were valid. They ruled that school rules derived from the education would override the parents' right to control their children's upbringing in this specific situation. As far as the religious freedom violation accusations, the court ruled that the policy did not have a religious goal, and thus did not infringe on religious freedom rights.

Jacobs v. Clark County School District
In 2003, Liberty High School, a school of the Clark County School District in Henderson, Nevada, implemented a uniform policy of khakis and red, white or blue polo shirts. A junior by the name of Kimberly Jacobs was suspended a total of five times because she wore a religious shirt to school and got cited for uniform violations. Her family sued the Clark County School District under the claims that her First Amendment rights were being infringed upon and that the uniform policy was causing students to be deprived of due process. The plaintiff's requests were for injunctive relief, the expunging of suspensions from Jacob's school record and awarding of damages. The injunction was granted to the family meaning that the school could no longer discipline her for breaking the uniform policy. At this ruling, the school district appealed. The next court ruled on the side of the school district as it determined that the uniform policy was in fact neutral and constitutional, and it dismissed the claims of the plaintiff.

Frudden v. Washoe County School District
In 2011, a Nevada public elementary school of the Washoe County School District decided to add the school's motto, Tomorrow's Leaders embroidered in small letters on the shirt. In response, Mary and John Frudden, parents of a student sued the school district on the basis of it violating the 1st Amendment. The court ultimately dismissed the case filed by the Fruddens over the uniforms. However, the family appealed, and two years later, a three-judge panel of the 9th U.S. Circuit Court of Appeals heard the case. The court ruled to reverse the previous decision of dismissing the case, and also questioned the apparent policy for students that were part of a nationally recognised group such as Boy Scouts and Girl Scouts who were able to wear the uniforms in place of the school ones on regular meeting days. The 9th circuit panel ruled that the school had not provided enough evidence for why it instituted this policy, and that the family was never given a chance to argue.

Social implications of school uniforms on gender
There are several positive and negative social implications of uniforms on both the students wearing them and society as a whole.

Perceptions of masculinity and femininity
One of the criticisms of uniforms is that it imposes standards of masculinity and femininity from a young age. Uniforms are considered a form of discipline that schools use to control student behavior and often promote conventional gendered dress.

Boys often are required to wear trousers, belts, and closed-toe shoes and have their shirts tucked in at all times. They are also often required to have their hair cut short. Some critics allege that this uniform is associated with the dress of a professional business man, which, they claim, gives boys at a young age the impression that masculinity is gained through business success.

For girls, many uniforms promote femininity by requiring girls to wear skirts. Skirts are seen by some critics as a symbol of femininity because they restrict movement and force certain ways of sitting and playing. Uniforms that include an apron for girls may suggest that the appropriate feminine societal role is a primarily domestic one. Some girls' school uniforms have been criticized as having an uncomfortable design, which prevents girls from freedom of movement and exposes girls to cold during winter.

School uniforms are embedded with gender symbolism. Schools that require students to wear a formal uniform almost universally provide trousers for boys and skirts or dresses for girls. Skirts differentiate the female from the male, thereby confirming traditional gender identities for students who must wear the correct attire corresponding to their sex. Skirts and dresses demand a particular type of feminine gender performance, whereas trousers demand a particular masculine gender performance. By forcing students to wear attire that corresponds with their sex inherently assigns the ways a student must perform their gender. This causes controversy when a student does not want to identify with a gender that does not align with their sex. There are rarely guidelines that allow for students to dress according to their performed gender, but almost always according to their sex assigned at birth.

Sexualization of girls

Around middle or junior school, students begin going through puberty. Uniforms can be seen as a way to restrict the sexualization of girls by taking the focus away from sexuality and focus it on academics in a school setting for girls.

Sometimes the desire to prevent overtly sexualized clothing through uniforms can fail. As an example, miniskirts have been very popular in Japan, where they are common parts of school uniforms and came to be worn within the Kogal culture.

"The pleasure our culture derives from gazing at girls who look feminine conflicts with girls' freedom to run around unselfconsciously and to develop their gross motor talents as boys are encouraged to do" (Collins et al. 1996, p. 170). Schoolgirl uniforms are used in costumes in the context of "Sexy Schoolgirl" and are sold on costume sites year round. The idea of the female school uniform has become sexual and in Britain a new survey from Plan International UK found that a third of girls have been sexually harassed while wearing their school uniform. School uniforms can encourage harassment as children, as some cultures can define the "schoolgirl look" as sexual. Children as young as 8 years old report being victims of, or witnesses to, harassment. Two-thirds of the children questioned in the survey said they have experienced "unwanted sexual attention" in public, and 35 per cent said they have been touched, groped or grabbed without their consent. These experiences teach girls that being harassed by men is just a part of growing up. The perception of schoolgirl uniforms allows for men to harass girls at a young age, causing girls to self-objectify their bodies from the beginning of their schooling experience.

General

In some cultures, the topic of school uniforms has sparked a multitude of controversies and debates over the years. Debates concerning the constitutionality and economic feasibility of uniforms also contribute to the controversy.

In the United States, the implementation of school uniforms began following ten years of research indicating the effectiveness of private schools. Some state-school reformers cited this research to support policies linked to private and Catholic school success. Some public-school administrators hence began implementing uniform policies to improve the overall school environment and academic achievement of the students. This is based on the assumption that uniforms are the direct cause of behavioral and academic outcome changes. However, within the Catholic school literature, school uniforms have never been acknowledged as a primary factor in producing a Catholic school effect.

Another area of controversy regarding school uniform and dress code policies revolve around the issue of gender. Nowadays, more teenagers are more frequently "dressing to articulate, or confound gender identity and sexual orientation", which brings about "responses from school officials that ranged from indifferences to applause to bans".

Positives

Advocates of uniforms have proposed several reasons supporting their implementation and claiming their success in schools. Advocates believe that uniforms affect student safety by decreasing student victimization, gang activity, and fights. There has been no concrete evidence of this, and studies by Ohio State University and others showed that uniforms did not increase test scores, grades, or focus. However, attendance increased by less than half of a day. The students felt even less of a sense of belonging at a school with uniforms.

Kathleen Wade conducted an experiment to see if bullying and gang presence was higher in uniform or non-uniform schools. The research was done with multiple schools where she gave a questionnaire to both students and faculty to see if there was a significant difference. Her results showed that bullying and gang presence significantly decreases with students wearing school uniforms.
 * Differentiating strangers from students in school buildings

For example, in the first year of the mandatory uniform policy in Long Beach, California, officials reported that fighting in schools decreased by more than 50%, assault and battery by 34%, sex offenses by 74%, and robbery by 66%. However the district also added other safety measures like security guards so the success cannot be attributed to the uniforms solely.

Advocates also believe that uniforms increase student learning and positive attitudes toward school through:

 * Enhanced learning environments
 * Heightened school pride
 * Increased student achievement
 * High levels of preparedness
 * Conformity to organizational goals

Currently, pros of school uniforms center around how uniforms affect school environments. Advocates say that uniforms may create a safe learning environment for students to help them focus on school work and can lead them to great academic accomplishments. Students who wear school uniforms may not feel anxious or nervous about peer pressure in buying new clothes to fit in or being teased by other classmates. Proponents have found a significant positive impact on school climate, safety, and students' self-perception from the implementation of uniforms.

However, though modern studies and tests prove uniforms did not increase test scores, behavior, bullying, focus and attendance barely increased.

Negatives
The opposing side of uniforms has claimed their ineffectiveness using a variety of justifications, a variety of which have research supporting them. Some of the cons to school uniforms include the following legal, financial, and questionable effectiveness concerns: The primary concern with school uniforms or strict dress codes is that it limits the ability of students to express themselves. While in countries where uniforms are the norm it simply isn't the case. Clothing is viewed as a means of expression – making all students wear the same clothes or limit them to what they can wear can disrupt their sense of identity. One of the main controversies focuses on dress code policies versus freedom of speech. This establishes that students cannot wear the latest trends or clothes that the school finds that interrupt the learning environment. However, students can wear clothing that express their religion. "Both the Constitution and most state laws protect students' rights to wear religious attire... such as the wearing of a turban, yarmulke, or headscarf."

Another negative aspect of school uniforms is that the policy can be sexist. Boys and girls are often not disciplined in the same ways when it comes to dress codes. Girls are more commonly disciplined for certain articles of clothing that are prohibited because they "distract" boys. "Transgender students have been sent home for wearing clothing different from what's expected of their legalness, while others have been excluded from yearbooks."

Uniforms also generally disadvantage students, especially girls, in freedom of movement and comfort. The research was conducted on an Australian independent private school and its uniform. Comfort-wise, for boys, the blazer was too hot/cold and uncomfortable. For girls, the light coloured cotton school dress was restrictive, see-through, hot, uncomfortable, and impractical. Furthermore, the stockings were often cold, grey woolen kilt was too heavy and restrictive of movement, and the wind could cause it to reveal more than the girls wanted. When playing and moving around, for boys, the school tie was a choking hazard, and the trousers had no stretch. For girls, the dress/skirt caused modesty issues (e.g. hard to swing on monkey bars/run around while keeping her privacy, hence stop being active), and the kilts were too big and heavy.

Research on how school uniforms and school dress codes influence the student can be inconclusive. "In the U.S., over half of public schools have a dress code, which frequently outline gender-specific policies."

Students that do not wear uniforms can be just as successful as students who do wear school uniforms. The amount of effort and participation a student does during class determines their academic success, regardless of what they are wearing. Students who wear school uniforms does not grant them academic achievement.
 * Legal concerns
 * Focus on the supposition that requiring a uniform violates children's individual rights (Thomas, 1994; Virginia State Dep’t of Edu, 1992)
 * Mandatory uniform policies are being considered largely for urban school districts, and, hence are being forced on a predominantly minority and poor student population (Thomas, 1994)
 * No effect on social status
 * Many students felt the school uniform policy had little impact on the social dynamic of the school and students found ways to express individuality by making minor alterations to the school uniform (Da Costa, 2006 ). Some parents and students interviewed in a research about the social aspect of school uniforms said that uniforms were a violation of their rights and freedom. "Like adults, children's freedom to choose or to act is also circumscribed by the community – massively so by schools, with their high density, constant supervision and evaluation, lack of privacy, and the obligatory nature of their activities." (Bodine, 2003 )
 * Financial concerns
 * Groups such as the American Civil Liberties Union have voiced concerns about the cost of uniforms, specifically that some disadvantaged parents are unable to afford them (Gursky, 1996)
 * Questionable effectiveness of those policies
 * Strongest opponents of uniform policies charge that no empirical evidence exists to support the numerous and varied claims of uniform proponents (LaPorte, Holoman, & Alleyne, 1992)
 * School uniforms suppress students' individuality by mandating standardization of appearance and removing student expression (Joseph, 1986 )
 * While uniform policies have been linked to school climate, safety, and student self-perception, there is no evidence to indicate that a uniform policy increases academic achievement (Wade & Stafford, 2003 )

According to Marian Wilde, additional opponent arguments include that school uniforms:

 * Are simply a Band-Aid on the issue of school violence
 * Make students a target for bullies from other schools
 * Are an unfair additional expense for parents who pay taxes for a free public education
 * Are difficult to enforce in state/public (government) schools

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Catilinarian orations

The Catilinarian orations (Marci Tullii Ciceronis orationes in Catilinam; also simply the Catilinarians) are four speeches given in 63 BC by Marcus Tullius Cicero, one of the year's consuls. The speeches all related to the discovery, investigation, and suppression of the Catilinarian conspiracy, a plot that year to overthrow the republic. All of the speeches in the form available today were published, probably around 60, as part of Cicero's attempt to justify his actions during the consulship; whether they are accurate reflections of the original speeches in 63 is debated.

The first speech was in the senate, where Cicero accused a senator, Catiline, of leading a plot to overthrow the republic; in response, Catiline withdrew from the city and joined an uprising in Etruria. The next two speeches were before the people, with Cicero justifying his actions as well as relating further news of the conspiracy in Rome itself and the arrest of four conspirators. The fourth speech, supposedly delivered before the Senate, was an intervention in an on-going debate as to the fate of the urban conspirators; Cicero argued in favour of their illegal execution without trial.

Some modern historians suggest that Catiline was a more complex character than Cicero's writings declare, and that Cicero was heavily influenced by a desire to establish a lasting reputation as a great Roman patriot and statesman. The Catilinarian orations, along with Sallust's monograph Bellum Catilinae, make the conspiracy one of the best-documented events from the ancient world; for centuries after their delivery, the Catilinarians were praised as model speeches and taught as part of the standard Latin rhetorical curriculum.

Background
The Catilinarian conspiracy was a plot by the patrician senator Lucius Sergius Catilina (Catiline) to overthrow the Roman republic. He started this plot in 63 BC after being repulsed at elections for consul for the third time; after failing to be elected to the consulships of 65, 63, and 62 BC. The conspirators included various disaffected groups. The aristocrats who joined were largely men who were similarly unsuccessful in elections for high office or were otherwise bankrupt. They were joined by many disaffected Italian farmers – concentrated in Etruria – in two broad groups: farmers dispossessed by Sulla's proscriptions and colonisation programmes as well as Sulla's veterans who had fallen on debt after poor harvests.

The first indications of a plot in 63 BC were in autumn, handed over by Marcus Licinius Crassus on 18 or 19 October. Crassus' letters were corroborated by reports of armed men gathering in support of the conspiracy. In response, the senate passed a decree declaring a tumultus (a state of emergency) and, after receipt of the reports of armed men gathering in Etruria, carried the senatus consultum ultimum instructing the consuls to do whatever it took to respond to the crisis. By 27 October, the senate had received reports that Gaius Manlius, a former centurion and leader of an army there, had taken up arms near Faesulae.

Catiline remained in the city. While named in the anonymous letters sent to Crassus, this was insufficient evidence for incrimination. But after messages from Etruria connected him directly to the uprising, he was indicted under the lex Plautia de vi (public violence) in early November. The conspirators met, probably on 6 November, and found two volunteers to make an attempt on Cicero's life. After the attempts on Cicero's life failed on 7 November 63 BC, he assembled the senate and delivered the First Catilinarian, revealing Catiline's involvement in the plot; Catiline promptly left the city and joined Manlius' men in Etruria shortly thereafter.

At this time, Cicero then discovered a plot led by one of the sitting praetors, to bring in the Allobroges, a Gallic tribe, to support the Catilinarians. Using the Allobroges' envoys as double agents, Cicero used them to identify conspirators in the city. After intercepting incriminating letters between the conspirators and the Allobroges, five conspirators were arrested on 2 or 3 December. With the Gallic envoys divulging all they knew and confessions from the five men, there was no doubt of their guilt. After an attempt to rescue the five men from house arrest, the senate debated their fate on 5 December. After a prolonged debate, the Senate, after momentarily being convinced to sentence the men to life imprisonment without trial by Julius Caesar, advised Cicero to have the urban conspirators summarily executed. After the execution of the urban conspirators, most of Catiline's forces melted away; Catiline was eventually defeated and killed in early January 62 BC at the Battle of Pistoria.

At the close of the consular year, Cicero's valedictory speech was vetoed by two tribunes of the plebs. One of the tribunes, Quintus Caecilius Metellus Nepos, also sought to bring Cicero up on charges for executing citizens without trial. Although popular among large portions of the people for having taken decisive action to avoid civil war and suppress the coup attempt, Cicero's legal position came under attack in the coming years. In response, Cicero attempted to shore up his reputation and justify his actions by publishing his consular speeches: the Catilinarian orations were published after some editing in 60 BC as part of this effort.

First Catilinarian

The First Catilinarian is the most famous speech in Latin literature. Its first sentence in particular is carefully crafted so as to have its form support its content. In consequence, it is still widely remembered and used after more than 2000 years:

Also remembered is the famous exasperated exclamation, O tempora, o mores!, used as an exclamation of outrage or indignation as to the state of the republic in Cicero's days.

Structure and context
The First Catilinarian is a denunciation of Catiline, delivered before the Senate in the Temple of Jupiter Stator on 7 or 8 November 63 BC. The Senate met to discuss an attempt on Cicero's life. Whether the speech is entirely historical is not entirely clear: the Second Catilinarian depicts Cicero's first speech as a simple interrogatory rather than the extended denunciation that survives. Unlike the other speeches, most of the speech is directed to Catiline personally with concluding remarks addressed to the Senate.

Categorisation of the speech into one of the genres of ancient rhetoric is difficult. The denunciatory aspects of the speech are couched in the framework of a senatorial address while also largely being delivered to Catiline's person. Scholars disagree as to whether it should be seen as a speech in the genre of the law courts (forensic or prosecutorial) or otherwise in the genre of senatorial rhetoric (deliberative). This difficulty may be due to its original extemporaneous nature, delivered not in as part of a structured meeting but rather on the Catiline's arrival to the senate.

The oration's arguments, somewhat cloudy and meandering, are intended more to influence senatorial opinion than argue in favour of any specific course of action or actually advise Catiline. Cicero, in a letter, later described it as a farewell; Berry, in Cicero's Catilinarians, argues that Cicero had to dress up inaction since, within the bounds of the law, he had limited authority to act against Catiline proactively. A more retrospective interpretation of how it would have played c. 60 BC would instead emphasise how Cicero chose to act slowly and deliberatively rather than, as alleged by his political enemies, cruelly and autocratically.

Content
Cicero starts the speech by informing Catiline that the conspiracy is revealed and that Cicero would be within his rights as consul and justified by precedent to have Catiline killed as a threat to the state. Cicero then connects Catiline to the rebels in Etruria, against which the Senate had already mobilised men; Cicero also disclaims any intention to have Catiline killed since it would be controversial, something possibly inserted in 60 BC to paint Cicero as merciful and rebut allegations of cruelty. Cicero then describes at length the conspiracy before urging Catiline to leave the city with his followers to take command of the Etrurian rebels, something which Cicero asserts Catiline was to do shortly regardless. Catiline likely asked whether Cicero's advice was a command for him to go into exile – the power to exile citizens, relegatio, was within consular authority – but Cicero in the speech insists that he is merely advising Catiline to leave.

Insisting that Catiline is not detained by any business in Rome due to his poor reputation, Cicero then engages in invective, indirectly accusing Catiline of a variety of sexual crimes, imminent bankruptcy, and past plots against the state. Drawing attention to how other senators moved away from Catiline when he entered the senate, Cicero argues then that no formal senatorial vote on Catiline's exile – which Catiline demanded; if it had passed it would have cast Catiline as a victim of senatorial overreach; if it had failed it would have undermined Cicero's position in the senate – was necessary due to the senate's obvious displeasure. This political isolation is then emphasised when Cicero relates that Catiline sought to place himself into voluntary custody to service his reputation but found nobody willing to take him. Such isolation is further illustrated noting how the Senate's did not voice any immediate objections to the idea of exiling Catiline.

Changing tack, Cicero then tells Catiline that if he leaves the city but, contrary to Catiline's existing plans, does not join the rebels in Etruria, Cicero would be seen as having forced an innocent man to go into exile. This argument was meant to paint Cicero in an unselfish light. An outburst of invective against Catiline and his followers, who Cicero brands as corrupt and bankrupt political failures, then follows. The conclusion of the speech notes that Cicero intends to do nothing compulsory at the moment, justified by rejection of arguments to have Catiline summarily executed (placed in the mouth of an abstract personification of Rome). Cicero instead seeks a longer term goals of ensuring that – by allowing Catiline to join the Etrurian rebels – the whole Senate is convinced of Catiline's guilt and that, when the rebels are defeated with Catiline and followers among them, the body politic is improved by their absence. The speech finally concludes with a prayer to Jupiter Stator that Catiline and his followers be defeated.

Second Catilinarian
Cicero informed the citizens of Rome that Catiline had left the city not into exile, as Catiline had said, but to join with his illegal army. He described the conspirators as rich men who were in debt, men eager for power and wealth, Sulla's veterans, ruined men who hoped for any change, criminals, profligates and other men of Catiline's ilk. He assured the people of Rome that they had nothing to fear because he, as consul, and the gods would protect the state. This speech was delivered with the intention of convincing the lower class, or common man, that Catiline would not represent their interests and they should not support him.

Meanwhile, Catiline joined up with Gaius Manlius, commander of the rebel force. When the Senate was informed of the developments, they declared the two of them public enemies. Antonius Hybrida (Cicero's fellow consul), with troops loyal to Rome, followed Catiline while Cicero remained at home to guard the city.

Third Catilinarian
Cicero claimed that the city should rejoice because it had been saved from a bloody rebellion. He presented evidence that all of Catiline's accomplices confessed to their crimes. He asked for nothing for himself but the grateful remembrance of the city and acknowledged that the victory was more difficult than one in foreign lands because the enemies were citizens of Rome.

Fourth Catilinarian

In his fourth and final published argument, which took place in the Temple of Concordia, Cicero establishes a basis for other orators (primarily Cato the Younger) to argue for the execution of the conspirators. As consul, Cicero was formally not allowed to voice any opinion in the matter, but he circumvented the rule with subtle oratory. Although very little is known about the actual debate (except for Cicero's argument, which has probably been altered from its original), the Senate majority probably opposed the death sentence for various reasons, one of which was the nobility of the accused. For example, Julius Caesar argued that exile and disenfranchisement would be sufficient punishment for the conspirators, and one of the accused, Lentulus, was a praetor. However, after the combined efforts of Cicero and Cato, the vote shifted in favor of execution, and the sentence was carried out shortly afterwards.

While some historians agree that Cicero's actions, in particular the final speeches before the Senate, may have saved the Republic, they also reflect his self-aggrandisement and, to a certain extent envy, probably born out of the fact that he was considered a novus homo, a Roman citizen without noble or ancient lineage.

Translations

 * The Conspiracy of Catiline (63 B.C.)

 * At Perseus Project (Latin text, translation and analysis):

title: Catilinarian orations