text:
General topology

In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.

The fundamental concepts in point-set topology are continuity, compactness, and connectedness: The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.
 * Continuous functions, intuitively, take nearby points to nearby points.
 * Compact sets are those that can be covered by finitely many sets of arbitrarily small size.
 * Connected sets are sets that cannot be divided into two pieces that are far apart.

Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.

History
General topology grew out of a number of areas, most importantly the following:

 * the detailed study of subsets of the real line (once known as the topology of point sets; this usage is now obsolete)
 * the introduction of the manifold concept
 * the study of metric spaces, especially normed linear spaces, in the early days of functional analysis.

General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics.

A topology on a set
Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:

 * 1) Both the empty set and X are elements of τ
 * 2) Any union of elements of τ is an element of τ
 * 3) Any intersection of finitely many elements of τ is an element of τ

If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation Xτ may be used to denote a set X endowed with the particular topology τ.

The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (i.e., its complement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itself are always both closed and open.

Basis for a topology
A base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because many topologies are most easily defined in terms of a base that generates them.

Subspace and quotient
Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can be given the product topology, which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space.

A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : X→ Y is a surjective function, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. In other words, the quotient topology is the finest topology on Y for which f is continuous. A common example of a quotient topology is when an equivalence relation is defined on the topological space X. The map f is then the natural projection onto the set of equivalence classes.

Examples of topological spaces
A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space.

Discrete and trivial topologies
Any set can be given the discrete topology, in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given the trivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must be Hausdorff spaces where limit points are unique.

Cofinite and cocountable topologies
Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T1 topology on any infinite set.

Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations.

Topologies on the real and complex numbers
There are many ways to define a topology on R, the set of real numbers. The standard topology on R is generated by the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the Euclidean spaces Rn can be given a topology. In the usual topology on Rn the basic open sets are the open balls. Similarly, C, the set of complex numbers, and Cn have a standard topology in which the basic open sets are open balls.

The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals [ a, b). This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology.  This example shows that a set may have many distinct topologies defined on it.

The metric topology
Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is the same for all norms.

Further examples

 * There exist numerous topologies on any given finite set. Such spaces are called finite topological spaces. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.
 * Every manifold has a natural topology, since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from Rn.
 * The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety. On Rn or Cn, the closed sets of the Zariski topology are the solution sets of systems of polynomial equations.
 * A linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices and edges.
 * Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.
 * Any local field has a topology native to it, and this can be extended to vector spaces over that field.
 * The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of computation and semantics.
 * If Γ is an ordinal number, then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals (a, b), [0, b) and (a, Γ) where a and b are elements of Γ.

Continuous functions
Continuity is expressed in terms of neighborhoods: $f$ is continuous at some point $x ∈ X$ if and only if for any neighborhood $V$ of $f(x)$, there is a neighborhood $U$ of $x$ such that $f(U) ⊆ V$. Intuitively, continuity means no matter how "small" $V$ becomes, there is always a $U$ containing $x$ that maps inside $V$ and whose image under $f$ contains $f(x)$. This is equivalent to the condition that the preimages of the open (closed) sets in $Y$ are open (closed) in $X$. In metric spaces, this definition is equivalent to the ε–δ-definition that is often used in analysis.

An extreme example: if a set $X$ is given the discrete topology, all functions
 * $$f\colon X \rightarrow T$$

to any topological space $T$ are continuous. On the other hand, if $X$ is equipped with the indiscrete topology and the space $T$ set is at least T0, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.

Alternative definitions
Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.

Neighborhood definition
Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms of neighborhoods: f is continuous at some point x ∈ X if and only if for any neighborhood V of f(x), there is a neighborhood U of x such that f(U) ⊆ V. Intuitively, continuity means no matter how "small" V becomes, there is always a U containing x that maps inside V.

If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f(x) instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance.

Note, however, that if the target space is Hausdorff, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous.

Sequences and nets
In several contexts, the topology of a space is conveniently specified in terms of limit points. In many instances, this is accomplished by specifying when a point is the limit of a sequence, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, known as nets. A function is continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.

In detail, a function f: X → Y is sequentially continuous if whenever a sequence (xn) in X converges to a limit x, the sequence (f(xn)) converges to f(x). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If X is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if X is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.

Closure operator definition
Instead of specifying the open subsets of a topological space, the topology can also be determined by a closure operator (denoted cl), which assigns to any subset A ⊆ X its closure, or an interior operator (denoted int), which assigns to any subset A of X its interior. In these terms, a function
 * $$f\colon (X,\mathrm{cl}) \to (X' ,\mathrm{cl}')\, $$

between topological spaces is continuous in the sense above if and only if for all subsets A of X
 * $$f(\mathrm{cl}(A)) \subseteq \mathrm{cl}'(f(A)).$$

That is to say, given any element x of X that is in the closure of any subset A, f(x) belongs to the closure of f(A). This is equivalent to the requirement that for all subsets A ' of X '
 * $$f^{-1}(\mathrm{cl}'(A')) \supseteq \mathrm{cl}(f^{-1}(A')).$$

Moreover,
 * $$f\colon (X,\mathrm{int}) \to (X' ,\mathrm{int}') \, $$

is continuous if and only if
 * $$f^{-1}(\mathrm{int}'(A)) \subseteq \mathrm{int}(f^{-1}(A))$$

for any subset A of X.

Properties
If f: X → Y and g: Y → Z are continuous, then so is the composition g ∘ f: X → Z. If f: X → Y is continuous and
 * X is compact, then f(X) is compact.
 * X is connected, then f(X) is connected.
 * X is path-connected, then f(X) is path-connected.
 * X is Lindelöf, then f(X) is Lindelöf.
 * X is separable, then f(X) is separable.

The possible topologies on a fixed set X are partially ordered: a topology τ1 is said to be coarser than another topology τ2 (notation: τ1 ⊆ τ2) if every open subset with respect to τ1 is also open with respect to τ2. Then, the identity map
 * idX: (X, τ2) → (X, τ1)

is continuous if and only if τ1 ⊆ τ2 (see also comparison of topologies). More generally, a continuous function
 * $$(X, \tau_X) \rightarrow (Y, \tau_Y)$$

stays continuous if the topology τY is replaced by a coarser topology and/or τX is replaced by a finer topology.

Homeomorphisms
Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. In fact, if an open map f has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. Given a bijective function f between two topological spaces, the inverse function f−1 need not be continuous. A bijective continuous function with continuous inverse function is called a homeomorphism.

If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.

Defining topologies via continuous functions
Given a function
 * $$f\colon X \rightarrow S, \,$$

where X is a topological space and S is a set (without a specified topology), the final topology on S is defined by letting the open sets of S be those subsets A of S for which f−1(A) is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on S. Thus the final topology can be characterized as the finest topology on S that makes f continuous. If f is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f.

Dually, for a function f from a set S to a topological space, the initial topology on S has as open subsets A of S those subsets for which f(A) is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus the initial topology can be characterized as the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X.

A topology on a set S is uniquely determined by the class of all continuous functions $$S \rightarrow X$$ into all topological spaces X. Dually, a similar idea can be applied to maps $$X \rightarrow S.$$

Compact sets
Formally, a topological space X is called compact if each of its open covers has a finite subcover. Otherwise it is called non-compact. Explicitly, this means that for every arbitrary collection

 * $$\{U_\alpha\}_{\alpha\in A}$$

of open subsets of $X$ such that

 * $$X = \bigcup_{\alpha\in A} U_\alpha,$$

there is a finite subset $J$ of $A$ such that

 * $$X = \bigcup_{i\in J} U_i.$$

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact. A compact set is sometimes referred to as a compactum, plural compacta.

Every closed interval in R of finite length is compact. More is true: In Rn, a set is compact if and only if it is closed and bounded. (See Heine–Borel theorem).

Every continuous image of a compact space is compact.

A compact subset of a Hausdorff space is closed.

Every continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism.

Every sequence of points in a compact metric space has a convergent subsequence.

Every compact finite-dimensional manifold can be embedded in some Euclidean space Rn.

Connected sets
A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice.

For a topological space X the following conditions are equivalent:

 * 1) X is connected.
 * 2) X cannot be divided into two disjoint nonempty closed sets.
 * 3) The only subsets of X that are both open and closed (clopen sets) are X and the empty set.
 * 4) The only subsets of X with empty boundary are X and the empty set.
 * 5) X cannot be written as the union of two nonempty separated sets.
 * 6) The only continuous functions from X to {0,1}, the two-point space endowed with the discrete topology, are constant.

Every interval in R is connected.

The continuous image of a connected space is connected.

Connected components
The maximal connected subsets (ordered by inclusion) of a nonempty topological space are called the connected components of the space. The components of any topological space X form a partition of X: they are disjoint, nonempty, and their union is the whole space. Every component is a closed subset of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the rational numbers are the one-point sets, which are not open.

Let $$\Gamma_x$$ be the connected component of x in a topological space X, and $$\Gamma_x'$$ be the intersection of all open-closed sets containing x (called quasi-component of x.) Then $$\Gamma_x \subset \Gamma'_x$$ where the equality holds if X is compact Hausdorff or locally connected.

Disconnected spaces
A space in which all components are one-point sets is called totally disconnected. Related to this property, a space X is called totally separated if, for any two distinct elements x and y of X, there exist disjoint open neighborhoods U of x and V of y such that X is the union of U and V. Clearly any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbers Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.

Path-connected sets

A path from a point x to a point y in a topological space X is  a continuous function f from the unit interval [0,1] to X with f(0) = x and f(1) = y. A path-component of X is an equivalence class of X under the equivalence relation, which makes x equivalent to y if there is a path from x to y. The space X is said to be path-connected (or pathwise connected or 0-connected) if there is at most one path-component; that is, if there is a path joining any two points in X. Again, many authors exclude the empty space.

Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended long line L* and the topologist's sine curve.

However, subsets of the real line R are connected if and only if they are path-connected; these subsets are the intervals of R. Also, open subsets of Rn or Cn are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for finite topological spaces.

Products of spaces
Given X such that

 * $$X := \prod_{i \in I} X_i,$$

is the Cartesian product of the topological spaces Xi, indexed by $$i \in I$$, and the canonical projections pi : X → Xi, the product topology on X is defined as the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonoff topology.

The open sets in the product topology are unions (finite or infinite) of sets of the form $$\prod_{i\in I} U_i$$, where each Ui is open in Xi and Ui ≠ Xi only finitely many times. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the Xi gives a basis for the product $$\prod_{i\in I} X_i$$.

The product topology on X is the topology generated by sets of the form pi−1(U), where i is in I   and U is an open subset of Xi. In other words, the sets {pi−1(U)} form a subbase for the topology on X. A subset of X is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form pi−1(U). The pi−1(U) are sometimes called open cylinders, and their intersections are cylinder sets.

In general, the product of the topologies of each Xi forms a basis for what is called the box topology on X. In general, the box topology is finer than the product topology, but for finite products they coincide.

Related to compactness is Tychonoff's theorem: the (arbitrary) product of compact spaces is compact.

Separation axioms
Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms; for example, the meanings of "normal" and "T4" are sometimes interchanged, similarly "regular" and "T3", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous.

Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles.

In all of the following definitions, X is again a topological space.

 * X is T0, or Kolmogorov, if any two distinct points in X are topologically distinguishable. (It is a common theme among the separation axioms to have one version of an axiom that requires T0 and one version that doesn't.)
 * X is T1, or accessible or Fréchet, if any two distinct points in X are separated. Thus, X is T1 if and only if it is both T0 and R0. (Though you may say such things as T1 space, Fréchet topology, and Suppose that the topological space X is Fréchet, avoid saying Fréchet space in this context, since there is another entirely different notion of Fréchet space in functional analysis.)
 * X is Hausdorff, or T2 or separated, if any two distinct points in X are separated by neighbourhoods. Thus, X is Hausdorff if and only if it is both T0 and R1. A Hausdorff space must also be T1.
 * X is T2½, or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. A T2½ space must also be Hausdorff.
 * X is regular, or T3, if it is T0 and if given any point x and closed set F in X such that x does not belong to F, they are separated by neighbourhoods. (In fact, in a regular space, any such x and F is also separated by closed neighbourhoods.)
 * X is Tychonoff, or T3½, completely T3, or completely regular, if it is T0 and if f, given any point x and closed set F in X such that x does not belong to F, they are separated by a continuous function.
 * X is normal, or T4, if it is Hausdorff and if any two disjoint closed subsets of X are separated by neighbourhoods. (In fact, a space is normal if and only if any two disjoint closed sets can be separated by a continuous function; this is Urysohn's lemma.)
 * X is completely normal, or T5 or completely T4, if it is T1 and if any two separated sets are separated by neighbourhoods. A completely normal space must also be normal.
 * X is perfectly normal, or T6 or perfectly T4, if it is T1 and if any two disjoint closed sets are precisely separated by a continuous function. A perfectly normal Hausdorff space must also be completely normal Hausdorff.

The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.

Countability axioms
An axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist.

Important countability axioms for topological spaces:
 * sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set
 * first-countable space: every point has a countable neighbourhood basis (local base)
 * second-countable space: the topology has a countable base
 * separable space: there exists a countable dense subspace
 * Lindelöf space: every open cover has a countable subcover
 * σ-compact space: there exists a countable cover by compact spaces

Relations:
 * Every first countable space is sequential.
 * Every second-countable space is first-countable, separable, and Lindelöf.
 * Every σ-compact space is Lindelöf.
 * A metric space is first-countable.
 * For metric spaces second-countability, separability, and the Lindelöf property are all equivalent.

Metric spaces
A metric space is an ordered pair $$(M,d)$$ where $$M$$ is a set and $$d$$ is a metric on $$M$$, i.e., a function

 * $$d \colon M \times M \rightarrow \mathbb{R}$$

such that for any $$x, y, z \in M$$, the following holds:

 * 1) $$d(x,y) \ge 0$$     (non-negative),
 * 2) $$d(x,y) = 0\,$$ iff $$x = y\,$$     (identity of indiscernibles),
 * 3) $$d(x,y) = d(y,x)\,$$     (symmetry) and
 * 4) $$d(x,z) \le d(x,y) + d(y,z)$$     (triangle inequality).

The function $$d$$ is also called distance function or simply distance. Often, $$d$$ is omitted and one just writes $$M$$ for a metric space if it is clear from the context what metric is used.

Every metric space is paracompact and Hausdorff, and thus normal.

The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.

Baire category theorem
The Baire category theorem says: If X is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty.

Any open subspace of a Baire space is itself a Baire space.

Continuum theory
A continuum (pl continua) is a nonempty compact connected metric space, or less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua. These objects arise frequently in nearly all areas of topology and analysis, and their properties are strong enough to yield many 'geometric' features.

Dynamical systems
Topological dynamics concerns the behavior of a space and its subspaces over time when subjected to continuous change. Many examples with applications to physics and other areas of math include fluid dynamics, billiards and flows on manifolds. The topological characteristics of fractals in fractal geometry, of Julia sets and the Mandelbrot set arising in complex dynamics, and of attractors in differential equations are often critical to understanding these systems.

Pointless topology
Pointless topology (also called point-free or pointfree topology) is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann. The ideas of pointless topology are closely related to mereotopologies, in which regions (sets) are treated as foundational without explicit reference to underlying point sets.

Dimension theory
Dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.

Topological algebras
A topological algebra A over a topological field K is a topological vector space together with a continuous multiplication

 * $$\cdot :A\times A \longrightarrow A$$
 * $$(a,b)\longmapsto a\cdot b$$

that makes it an algebra over K. A unital associative topological algebra is a topological ring.

The term was coined by David van Dantzig; it appears in the title of his doctoral dissertation (1931).

Metrizability theory
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space $$(X,\tau)$$ is said to be metrizable if there is a metric
 * $$d\colon X \times X \to [0,\infty)$$

such that the topology induced by d is $$\tau$$. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.

Set-theoretic topology
Set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC). A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.

title: General topology


text:
Moral example

Moral example is trust in the moral code of a role model.

It is the case that since the exact circumstances and decisions of the lives of such moral examples cannot be reproduced or repeated, followers are often reduced to following their etiquette and customs, e.g. in ancestor worship.

Storytelling can take a central role in any culture built on moral example, particularly when the provider of the moral example does not refer to an explicit ethical theory or philosophy as the basis for their behavior. A complex culture built on such stories can fall prey to a clique of experts who interpret them for the lay public. This has led in the past to institutions that sort through anecdotes to decide which of them are true, e.g. isnad in Islam by which the hadith are validated.

Christians still see Jesus as the ultimate divine example- in Epistle to the Hebrews: "looking unto Jesus the author and finisher of our faith" (12:2).

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

title: Moral example


text:
Role model

A role model is a person whose behaviour, example, or success serves as a model to be emulated by others, especially by younger people. The term role model is credited to sociologist Robert K. Merton, who hypothesized that individuals compare themselves with reference groups of people who occupy the social role to which the individual aspires, an example of which is the way young fans may idolize and imitate professional athletes or entertainment artists.

In the second half of the twentieth century, U.S. advocates for workplace equity popularized the term and concept of role models as part of a larger social capital lexicon—which also includes terms such as glass ceiling, networking, mentoring, and gatekeeper—serving to identify and address the problems barring non-dominant groups from professional success. Mainstream business literature subsequently adopted the terms and concepts, promoting them as pathways to success for all career climbers. In 1970 these terms were not in the general American vocabulary; by the mid-1990s they had become part of everyday speech. Although the term role model has been criticized more recently as "outdated", the term and its associated responsibility remains prominent in the public consciousness as a commonly used phrase, and a "powerful presence" in the entertainment industry and media.

Role models can also be national. for example, Chilean politicians and intellectuals had France as the prime role model during much of the 19th century until they shifted to Germany in the last decades of the century. In short, a role model is a person looked to by others as an example to be imitated.

Effect on career opportunity and choice
According to historian Pamela Laird, a person's chosen role models may have a considerable impact on his or her career opportunities and choices. The suitability of a role model depends, in part, on the admirer's perceived commonality with the model, who should provide an image of an ambitious yet realistic goal. For example, Laird suggests that, Benjamin Franklin served as the role model for countless nineteenth-century white businessmen, including notables such as Thomas Mellon, B.F. Goodrich, and Frederick Weyerhäuser. Laird suggests that the lack of commonalities between potential role models and would-be admirers helped perpetuate barriers to American minorities and women as they tried to advance in a business world dominated by white men, thus spurring late twentieth-century efforts to develop suitable role models for these groups.

Parent role models also significantly influence a person's "education and training aspirations, task self-efficacy, and expectancy for an entrepreneurial career".

Celebrity role models
The ever-widening reach of the media in popular culture has elevated certain celebrities to worldwide acclaim. This boom of media coverage and constant exposure to these individuals resulted in a change of mindset toward celebrities in both adults and youth alike. According to a survey of teachers in the United Kingdom conducted in 2008 by the Association of Teachers and Lecturers, young people most frequently chose sports stars as role models, followed by pop stars. Many, however, simply aspired to be "famous for being famous", believing that fame and fortune could be easily accessed through reality television.

Community role models
According to Rita Pierson, teachers, because of the large amount of time spent with children, have such a huge impact on children that they're being advised to be likeable in order to build strong emotional relationships with children.

Athlete role models
There is significant discussion as to whether athletes should be considered role models. Some athletes have been asked to behave as if they were role models for their local communities, and some such as Hank Greenberg have deliberately tried to set a good example but generally regarding athletes as role models has been criticised due to their appointment often being based solely on sporting ability rather than any morality – it has been suggested that the discipline and control shown continuously by sportspeople on the field leads to a belief from viewers that these same qualities are continuously shown off the field. These and other factors such as the elements of competition, excitement and success are what make people want to emulate them. Charles Barkley has stated that he believes athletes are not the figures that children should be emulating and that it is the parent's responsibility to be role models, that the role is deliberately applied by the media out of jealousy in order to make life more difficult for sportspeople, and that it sets up the sportspeople as an unattainable target for most.

title: Role model


text:
Constance of Burgundy

Constance of Burgundy (8 May 1046 – 1093) was the daughter of Duke Robert I of Burgundy and Helie de Semur-en-Brionnais. She was Queen of Castile and León by her marriage to Alfonso VI of León and Castile. She was the granddaughter of King Robert II of France, the second monarch of the French Capetian dynasty. She was the mother of Urraca of León, who succeeded her father in both Castile and León.

Life
In 1065, Constance married her first husband, Hugh II, Count of Chalon. They were married for fourteen years until Hugh's death in 1079, they had no children.

In late 1079, Constance remarried to Alfonso VI of León and Castile. The marriage appears to have been orchestrated via the Cluniac connections at Alfonso's court. He had previously been married to Agnes of Aquitaine, whom he had either divorced or had been widowed by. The marriage of Constance and Alfonso initially faced papal opposition, apparently due to a kinship between Constance and Agnes.

Constance and Alfonso had six children but only one of these lived to adulthood:
 * Urraca (b. April 1079 – March 8, 1126) Queen of Castile and León in her own right. Married firstly to Raymond of Burgundy. Married secondly to Alfonso the Battler, no issue.

Constance died in 1093 leaving her fourteen-year-old daughter and her husband a widower. He went on to marry three further wives after her death, but only had a son by his Muslim mistress, Zaida of Seville.

Burial
After her death, the corpse of Constance was taken to the town of Sahagún and was buried in the Monastery of St. Facundo and Primitivo, where her husband, King Alfonso VI would be buried along with all his wives.

The grave that contained the remains of Alfonso VI was destroyed in 1810 during a fire in the Monastery. The remains of the king and several of his wives, including those of Constance, were collected and kept in the abbot's chamber until 1821. When the religious were expelled from the monastery, they were then deposited by Abbot Ramon Joys in a box that was placed on the south wall of the chapel of the Crucifix, until, in January 1835, the remains were collected and placed in another box, being brought to the archive. The purpose was to place all remaining interests in a new sanctuary that was being built then. However, when the monastery of San Benito was disentailed in 1835, the monks gave the two boxes containing the actual remains to the relative of a priest, who hid them until 1902 were found by the professor Zamora Rodrigo Fernández Núñez.

Today, the remains of Alfonso VI are buried in the Royal Monastery of San Benito in Sahagún, at the foot of the temple, in a stone chest covered with smooth, modern marble and in a tomb near equally plain, lie the remains of several of the king's wives, including those of Constance.

title: Constance of Burgundy


text:
Ernst Zermelo

Ernst Friedrich Ferdinand Zermelo (, ; 27 July 1871 – 21 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem. Furthermore, his 1929 work on ranking chess players is the first description of a model for pairwise comparison that continues to have a profound impact on various applied fields utilizing this method.

Life
Ernst Zermelo graduated from Berlin's Luisenstädtisches Gymnasium (now ) in 1889. He then studied mathematics, physics and philosophy at the University of Berlin, the University of Halle, and the University of Freiburg. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations (Untersuchungen zur Variationsrechnung). Zermelo remained at the University of Berlin, where he was appointed assistant to Planck, under whose guidance he began to study hydrodynamics. In 1897, Zermelo went to the University of Göttingen, at that time the leading centre for mathematical research in the world, where he completed his habilitation thesis in 1899.

In 1910, Zermelo left Göttingen upon being appointed to the chair of mathematics at Zurich University, which he resigned in 1916. He was appointed to an honorary chair at the University of Freiburg in 1926, which he resigned in 1935 because he disapproved of Adolf Hitler's regime. At the end of World War II and at his request, Zermelo was reinstated to his honorary position in Freiburg.

Research in set theory
In 1900, in the Paris conference of the International Congress of Mathematicians, David Hilbert challenged the mathematical community with his famous Hilbert's problems, a list of 23 unsolved fundamental questions which mathematicians should attack during the coming century. The first of these, a problem of set theory, was the continuum hypothesis introduced by Cantor in 1878, and in the course of its statement Hilbert mentioned also the need to prove the well-ordering theorem.

Zermelo began to work on the problems of set theory under Hilbert's influence and in 1902 published his first work concerning the addition of transfinite cardinals. By that time he had also discovered the so-called Russell paradox. In 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved the well-ordering theorem (every set can be well ordered). This result brought fame to Zermelo, who was appointed Professor in Göttingen, in 1905. His proof of the well-ordering theorem, based on the powerset axiom and the axiom of choice, was not accepted by all mathematicians, mostly because the axiom of choice was a paradigm of non-constructive mathematics. In 1908, Zermelo succeeded in producing an improved proof making use of Dedekind's notion of the "chain" of a set, which became more widely accepted; this was mainly because that same year he also offered an axiomatization of set theory.

Zermelo began to axiomatize set theory in 1905; in 1908, he published his results despite his failure to prove the consistency of his axiomatic system. See the article on Zermelo set theory for an outline of this paper, together with the original axioms, with the original numbering.

In 1922, Abraham Fraenkel and Thoralf Skolem independently improved Zermelo's axiom system. The resulting system, now called Zermelo–Fraenkel axioms (ZF), is now the most commonly used system for axiomatic set theory.

Zermelo's navigation problem
Proposed in 1931, the Zermelo's navigation problem is a classic optimal control problem. The problem deals with a boat navigating on a body of water, originating from a point O to a destination point D. The boat is capable of a certain maximum speed, and we want to derive the best possible control to reach D in the least possible time.

Without considering external forces such as current and wind, the optimal control is for the boat to always head towards D. Its path then is a line segment from O to D, which is trivially optimal. With consideration of current and wind, if the combined force applied to the boat is non-zero, the control for no current and wind does not yield the optimal path.

Publications

 * Jean van Heijenoort, 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press.
 * 1904. "Proof that every set can be well-ordered," 139−41.
 * 1908. "A new proof of the possibility of well-ordering," 183–98.
 * 1908. "Investigations in the foundations of set theory I," 199–215.
 * 1913. "On an Application of Set Theory to the Theory of the Game of Chess" in Rasmusen E., ed., 2001. Readings in Games and Information, Wiley-Blackwell: 79–82.
 * 1930. "On boundary numbers and domains of sets: new investigations in the foundations of set theory" in Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford University Press: 1219–33.
 * 1913. "On an Application of Set Theory to the Theory of the Game of Chess" in Rasmusen E., ed., 2001. Readings in Games and Information, Wiley-Blackwell: 79–82.
 * 1930. "On boundary numbers and domains of sets: new investigations in the foundations of set theory" in Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford University Press: 1219–33.

Works by others:
 * Zermelo's Axiom of Choice, Its Origins, Development, & Influence, Gregory H. Moore, being Volume 8 of Studies in the History of Mathematics and Physical Sciences, Springer Verlag, New York, 1982.

title: Ernst Zermelo


text:
Day fighter

A day fighter is a fighter aircraft equipped only to fight during the day. More specifically, it refers to a multi-purpose aircraft that does not include equipment for fighting at night (such as a radar and specialized avionics), although it is sometimes used to refer to some interceptors as well.

The term is an example of a retronym: before the development of effective dedicated night fighter aircraft early in World War II, in effect, all fighter aircraft that were not specifically modified for night combat were day fighters.

World War II
Examples of planes that were classified as day fighters were the Supermarine Spitfire and Messerschmitt Bf 109. Both were excellent interceptors, but were also found in roles such as fighter-bomber and reconnaissance. However, the weight of the radar systems needed to effectively track down enemy bombers at night was such that these smaller aircraft simply couldn't carry them given the electronics of the day. This led to the use of twin-engine aircraft like the Bristol Beaufighter and Messerschmitt Bf 110 as dedicated night fighters, at which point the term day fighter began to be used.

Some lighter designs like the Hawker Hurricane and Focke-Wulf Fw 190 were also adapted to night fighting after a fashion, lacking radar and relying on searchlights or fires on the ground to illuminate their targets. These were preferred over the Spitfire and Bf 109 largely due to their wide-track landing gear, which made landing at night somewhat safer. Although such designs were in service throughout the war, they proved to have little real effectiveness compared to the dedicated night fighter designs.

Late in the war the considerable advances in radar technology, notably the introduction of the cavity magnetron, allowed for much smaller radar sets. At the same time, aircraft engines had roughly doubled in performance over the same period. This led to the late-war introduction of single-engine night fighters like the Grumman F6F, Vought F4U, as well as more agile twin-engine planes such as the Lockheed P-38, with some success.

Post-war
As the size and weight of night-fighting equipment remained considerable in comparison to airframe size and engine power, the day fighter remained a part of front-line air forces into the 1960s. In the immediate post-war era, designs like the North American F-86 Sabre, Mikoyan-Gurevich MiG-15 and Hawker Hunter typified the day-fighter role. These were followed by similar supersonic designs in the early 1960s, including the Lockheed F-104 Starfighter and Mikoyan-Gurevich MiG-21.

However, as the size of basic fighters grew, along with their engine power, the relative cost of carrying night-fighting equipment fell until most designs carried such equipment. Lightweight fighters without full night equipment such as the F-5 Freedom Fighter/Tiger II are still used by nations that lack the money to acquire advanced fighter aircraft.

The last of the purpose-designed day fighters were the original American General Dynamics F-16 Fighting Falcon and the Northrop YF-17 that competed for the Lightweight Fighter contract, which had the goal of providing the USAF with an inexpensive day fighter that could be purchased in great numbers to establish air superiority. The designs originated in a mid-1960s effort known as the "Advanced Day Fighter". After winning the contract the F-16 quickly matured into a much more capable multi-role aircraft and can no longer be considered a day fighter. Likewise, the YF-17 evolved into the much larger McDonnell Douglas F/A-18 Hornet and Boeing F/A-18E/F Super Hornet, which is used in the 2010s by the United States Navy as the primary all-weather combat aircraft.

title: Day fighter


text:
Talk:Opportunity cost

Wiki Education Foundation-supported course assignment
This article was the subject of a Wiki Education Foundation-supported course assignment, between 14 January 2019 and 2 April 2019. Further details are available on the course page. Student editor(s): Toubizac. Peer reviewers: Toubizac.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 05:46, 17 January 2022 (UTC)

Can we clarify the first paragraph definition to make clear how 'best choice' is typically defined?
Without this important information, non-experts are unlikely to be able to make sense of the general definition. For example, does 'best choice' refer to short-term or long-term planning? Does it already take costs and benefits into account, including 'opportunity costs' of different alternatives? Is opportunity cost dependent on what criteria are used to arrive at 'best choice'?

It seems to me that opportunity cost as economists use it refers not simply to a cost of not taking the benefit of the 'second best alternative', but rather the cost involved in choosing any alternative over another. The whole notion of 'first best' and 'second best' alternatives rests upon criteria that are themselves important to the discussion of opportunity cost.

As an example, if both electricity and water utilities are damaged after a storm, one could ask what is the opportunity cost of focusing resources on prioritizing the repair of one utility over the other. It might be argued that water utilities are more expensive to repair because pipes are underground while electricity utilities are more accessible by being above ground. But repairing electricity utilities first based on the idea that they are the cheaper option doesn't mean that it is the best option, if ongoing drinking water pollution or shortages lead to more illnesses and death than would happen if one were without an electric light bulb or cooker. Of course, water supplies may also rely on electricity supply for pumping stations, but this is meant to be a simplified example.

In other words, if 'best choice' is defined in monetary terms alone, it creates a different set of opportunity costs than if it is defined in terms of 'lives saved' or 'risks minimized' or any other host of criteria. If 'risks minimized' were to be the criterion, then the opportunity cost will be different, for example, accidents that happen as people move around in the dark, as opposed to an opportunity cost of people drinking unclean water in the alternative scenario.

Any thoughts on how to clarify this? I speak as a student of the topic wanting to understand, not an expertMark.helsinki (talk) 23:26, 2 January 2018 (UTC)

Caseyarandela (talk) 15:02, 1 November 2020 (UTC) I did a complete redo of this article, keeping in line with the plagiarism guidelines. This will start off the process of making this article have a higher article rating.

I have adjusted the intro section for greater clarity. 76.65.24.235 (talk) 04:35, 22 August 2022 (UTC)

Economics
Opportunity cost 27.56.234.36 (talk) 05:23, 2 December 2021 (UTC)

Opportunity costs are microeconomic . ShuFanLi (talk) 12:53, 2 May 2022 (UTC)

Text in images
In this article there are images containing text which would be better formatted as tables or quoted text. Crookesmoor (talk) 15:31, 5 July 2022 (UTC)

economic
1. विभिन्न सरकारी नीतियों के कारण पीपीसी पर प्रभाव अदृश्य हाथ (एडम — Preceding unsigned comment added by 2409:4050:E03:480:40A4:9BB5:76FC:3DCE (talk) 10:15, 21 October 2022 (UTC)

Definition unclear
I must admit I cannot understand the concept of opportunity cost from the definitions given in this article.

Let's take a simple scenario: I have to choose between two options, A and B. With option A I must pay $100 now but will receive $130 a year from now. With option B I must pay $100 now and will receive $110 a year from now. What would be the opportunity cost of choosing option A? Would it be $110 (the value of benefit given up) or $10 (the value of benefit given up, understood differently) or -$20 ((returns on best Forgone Option) - (returns on Chosen Option))?

Here's another scenario: I can make $100 per hour working as a lawyer. I have to choose between option A, mowing my own lawn, which takes 2 hours, or option B, hiring someone to mow my lawn, which costs $50, while lawyering for 2 hours and making $200. What is the opportunity cost of option A? I could argue for $150, or $200. AxelBoldt (talk) 00:27, 15 November 2022 (UTC)

Problems and solutions for opportunity cost
Answers 197.186.4.93 (talk) 08:57, 1 December 2022 (UTC)

 * how would you know that the opportunity costs was decreasing 41.121.38.251 (talk) 16:54, 26 May 2024 (UTC)

Economics
Opportunity cost 105.112.213.4 (talk) 14:31, 13 March 2023 (UTC)

Natural resource depletion
Depletion of resources means lost opportunities to use them in the future.

A policy of charging (higher) fees to extract resources could motivate reduced consumption of limited resources in the present, in favor of more availability in the future. Sharing proceeds from extraction fees to all people can ensure that the policy is fair, and that people will not be compelled to reduce use of resources in service of essential needs in the present. John Champagne (talk) 18:35, 5 January 2024 (UTC)

Explicit Costs
The definition as described includes "wages" then the explained scenarios leave out the cost of the wage in the scenario given. This should be edited to reflect the proper cost on the business in the scenario. Otherwise it is not 100% accurate. The wage has to be taken into account for accurate cost scenarios. Jmurphy042000 (talk) 04:12, 2 May 2024 (UTC)

title: Talk:Opportunity cost