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Encyclopedia > Continuous function (topology)

In topology and related areas of mathematics a continuous function is a morphism between topological spaces. 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. For a general topological space, this means a neighbourhood of f(x) always contains the image of a neighbourhood of x. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Euclid, detail from The School of Athens by Raphael. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with 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 an 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 between elements of the set is defined. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

Contents


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. ...


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. 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 an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... 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

Definition 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. Although this definition appears complex, the intuition is that no matter how "small" V becomes, we can find a small U containing x that will map inside it. If f is continuous at every x in X, then we simply say f is continuous. This is a glossary of some terms used in the branch of mathematics known as topology. ...

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. 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). This only really makes sense in a metric space, however, which has a notion of distance. Image File history File links Continuity_topology. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In topology and related areas of mathematics, the neighbourhood system or neighbourhood filter for a point x is the collection of all neighbourhoods for the point x. ... A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...


Sequences and nets

In several contexts, the topology of a space is conventiently 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 mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ... Limit of a sequence is one of the oldest concepts in mathematical analysis. ... In mathematics, a directed set is a set A together with a binary relation ≤ having the following properties: a ≤ a for all a in A (reflexivity) if a ≤ b and b ≤ c, then a ≤ c (transitivity) for any two a and b in A, there exists a c in A... In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ...


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). 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 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. Limit of a sequence is one of the oldest concepts in mathematical analysis. ... In topology, a first-countable space is a topological space satisfying the first axiom of countability. Specifically, a space X is said to be first-countable if each point has a countable local base. ... In topology and related fields of mathematics, a sequential space is a topological space which satisfies a very weak axiom of countability. ... In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ...


Closure operator definition

Given two topological spaces (X,cl) and (X ' ,cl ') where cl and cl ' are two closure operators then a function In mathematics, given a partially ordered set (P, ≤), a closure operator on P is a function C : P → P with the following properties: if x ≤ y, then C(x) ≤ C(y), i. ...

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 In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

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.


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 In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

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 In mathematics, given a partially ordered set (P, ≤), a closure operator on P is a function C : P → P with the following properties: if x ≤ y, then C(x) ≤ C(y), i. ...

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 In topology and related areas in mathematics closeness is one of the basic concepts in a topological space. ...

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.
  • If f : XY is continuous and

In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ...

Other notes

If a set is given the discrete topology, all functions with that space as a domain are continuous. 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. Conversely, any function whose range is indiscrete is continuous. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... 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 form a broad class of well behaved topological spaces. ...


Given a set X, a partial ordering can be defined on the possible topologies on X. A continuous functions between two topological spaces stays continuous if we strengthen the topology of the domain space or weaken the topology of the codomain space. 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 mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... 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 to each other. ... A codomain in mathematics is the set of output values associated with (or mapped to) the domain of input arguments in a function. ... 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 to each other. ... In mathematics, the domain of a function is the set of all input values to the function. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ...


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. 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 which makes f continuous. If f is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f. This construction can be generalized to an arbitrary family of functions XS. In topology and related areas of mathematics, the final topology on a set is the strongest topology to make a family of functions into continuous. ... In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set X may stand in relation to each other. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... For quotient spaces in linear algebra, see quotient space (linear algebra). ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...


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. 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. This construction can be generalized to an arbitrary family of functions SX. In topology and related areas of mathematics, the initial topology (projective topology or weak topology) on a set , with respect to a family of functions on , is the coarsest topology on X which makes those functions continuous. ... In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology). ...


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, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. In topology, an open map is a function between two topological spaces which maps open sets to open sets. ...


If a function is a bijection, then it has an inverse function. The inverse of a continuous bijection is open, but need not be continuous. If it is, this special function is called a homeomorphism. If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is automatically a homeomorphism. A bijective function. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...


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