In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Intuitively, this is a function f where a set of points near f(x) always contain the image of a set of points near x. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage For a general topological space, this means a neighbourhood of f(x) always contains the image of a neighbourhood of x. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space.

In a metric space (for example, the real numbers) this means that the points within a given distance of f(x) always contain the images of all the points within some other distance of x, giving the ε-δ definition. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

Definitions

Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function. In Mathematics, a Topological space is usually defined in terms of Open sets However there are many equivalent characterizations of the Category

Open and closed set definition

The most common notion of continuity in topology defines continuous functions as those functions for which the preimages of open sets are open. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in Similar to the open set formulation is the closed set formulation, which says that preimages of closed sets are closed. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Topology and related branches of Mathematics, a closed set is a set whose complement is open.

Neighborhood definition

Definitions based on preimages are often difficult to use directly. Instead, suppose we have a function f from X to Y, where X,Y are topological spaces. We say f is continuous at x for some $x \in X$ if for any neighborhood V of f(x), there is a neighborhood U of x such that $f(U) \subseteq V$. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. Although this definition appears complex, the intuition is that no matter how "small" V becomes, we can always find a U containing x that will map inside it. If f is continuous at every $x \in X$, then we simply say f is continuous.

In a metric space, it is equivalent to consider the neighbourhood system of open balls centered at x and f(x) instead of all neighborhoods. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Topology and related areas of Mathematics, the neighbourhood system or neighbourhood filter \mathcal{V}(x for a point x is the In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric This leads to the standard ε-δ definition of a continuous function from real analysis, which says roughly that a function is continuous if all points close to x map to points close to f(x). In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output This only really makes sense in a metric space, however, which has a notion of 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). In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space 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 Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated" 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. The limit of a sequence is one of the oldest concepts in Mathematical analysis. In Mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics 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 : XY is sequentially continuous if whenever a sequence (xn) in X converges to a limit x, the sequence (f(xn)) converges to f(x). The limit of a sequence is one of the oldest concepts in Mathematical analysis. Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If X is a first-countable space, then the converse also holds: any function preserving sequential limits is continuous. In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " 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. In Topology and related fields of Mathematics, a sequential space is a Topological space that satisfies a very weak Axiom of countability. ) This motivates the consideration of nets instead of sequences in general topological spaces. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.

Closure operator definition

Given two topological spaces (X,cl) and (X ' ,cl ') where cl and cl ' are two closure operators then a function

$f:(X,\mathrm{cl}) \to (X' ,\mathrm{cl}')$

is continuous if for all subsets A of X

$f(\mathrm{cl}(A)) \subseteq \mathrm{cl}'(f(A)).$

One might therefore suspect that given two topological spaces (X,int) and (X ' ,int ') where int and int ' are two interior operators then a function

$f:(X,\mathrm{int}) \to (X' ,\mathrm{int}')$

is continuous if for all subsets A of X

$f(\mathrm{int}(A)) \subseteq \mathrm{int}'(f(A))$

or perhaps if

$f(\mathrm{int}(A)) \supseteq \mathrm{int}'(f(A));$

however, neither of these conditions is either necessary or sufficient for continuity. A closure operator on a set S is a function cl P ( S) → P ( S) from the Power set of S In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S "

Instead, we must resort to inverse images: given two topological spaces (X,int) and (X ' ,int ') where int and int ' are two interior operators then a function

$f:(X,\mathrm{int}) \to (X' ,\mathrm{int}')$

is continuous if for all subsets A of X '

$f^{-1}(\mathrm{int}'(A)) \subseteq \mathrm{int}(f^{-1}(A)).$

We can also write that given two topological spaces (X,cl) and (X ' ,cl ') where cl and cl ' are two closure operators then a function

$f:(X,\mathrm{cl}) \to (X' ,\mathrm{cl}')$

is continuous if for all subsets A of X '

$f^{-1}(\mathrm{cl}'(A)) \supseteq \mathrm{cl}(f^{-1}(A)).$

Closeness relation definition

Given two topological spaces (X,δ) and (X ' ,δ ') where δ and δ ' are two closeness relations then a function

$f:(X,\delta) \to (X' ,\delta')$

is continuous if for all points x and y of X

$x \delta y \Leftrightarrow f(x)\delta'f(y).$

Useful properties of continuous maps

Some facts about continuous maps between topological spaces:

• If f : XY and g : YZ are continuous, then so is the composition g o f : XZ. In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " A closure operator on a set S is a function cl P ( S) → P ( S) from the Power set of S In Topology and related areas in Mathematics closeness is one of the basic concepts in a Topological space.
• If f : XY is continuous and
• X is compact, then f(X) is compact.
• X is connected, then f(X) is connected. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of
• X is path-connected, then f(X) is path-connected. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of
• The identity map idX : (X, τ2) → (X, τ1) is continuous if and only if τ1 ⊆ τ2 (see also comparison of topologies ). This article is about the Identity Map software design pattern In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation

Other notes

If a set is given the discrete topology, all functions with that space as a domain are continuous. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " If the domain set is given the indiscrete topology and the range set is at least T0, then the only continuous functions are the constant functions. In Topology, a Topological space with the trivial topology is one where the only Open sets are the Empty set and the entire space In Topology and related branches of Mathematics, the T0 spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class Conversely, any function whose range is indiscrete is continuous.

Given a set X, a partial ordering can be defined on the possible topologies on X. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of A continuous function between two topological spaces stays continuous if we strengthen the topology of the domain space or weaken the topology of the codomain space. In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation In Mathematics, the codomain, or target, of a function f: X → Y is the set In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined Thus we can consider the continuity of a given function a topological property, depending only on the topologies of its domain and codomain spaces. In Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is

For a function f from a topological space X to a set S, one defines the final topology on S by letting the open sets of S be those subsets A of S for which f-1(A) is open in X. In General topology and related areas of Mathematics, the final topology ( inductive topology or strong topology) on a set 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. In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation Thus the final topology can be characterized as the finest topology on S which makes f continuous. If f is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" This construction can be generalized to an arbitrary family of functions XS.

Dually, for a function f from a set S to a topological space, one defines the initial topology on S by letting the open sets of S be those subsets A of S for which f(A) is open in X. In General topology and related areas of Mathematics, the initial topology ( projective topology or weak topology) on a set 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 which makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is This construction can be generalized to an arbitrary family of functions SX.

Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets In fact, if an open map f has an inverse, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open.

If a function is a bijection, then it has an inverse function. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B The inverse of a continuous bijection is open, but need not be continuous. If it is, this special function is called a homeomorphism. Topological equivalence redirects here see also Topological equivalence (dynamical systems. If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is automatically a homeomorphism. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space

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