nLab
continuous map

Context

Topology

topology (point-set topology, point-free topology)

see also algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Continuous maps

Idea

A function f:XYf \colon X \to Y is called continuous if its values f(x)f(x) do not “jump” with variation of its argument xx, unless xx itself “jumps”. Roughly speaking, if x 1x 2x_1 \approx x_2, then f(x 1)f(x 2)f(x_1) \approx f(x_2). (This can be made into a precise definition in nonstandard analysis if care is taken about the domains of these variables.)

In order to make this precise (in standard analysis) one needs some concept of neighbourhoods of elements of XX and YY.

For instance if XX and YY carry structure of metric spaces, then one may say that ff is continuous if for every point xXx \in X and for every small open ball around its image f(x)f(x) in YY, there exists a sufficiently small open ball around xXx \in X which is still mapped by ff into that target open ball. This definition turns out to have more elegant formulation that needs to mention neither the points of xx nor the radii of open balls around points: the metric induces a concept of open subsets and ff is continuous precisely if preimages under ff of open subsets in YY are still open subsets in XX.

This then is the general definition of continuity of a function ff between topological spaces:

A function between topological spaces is continuous precisely if its preimages of open subsets are again open subsets.

Continuous maps are the homomorphisms between topological spaces. In other words, the collection of topological spaces forms a category, often denoted Top, whose morphisms are the continuous functions.

Further generalization of the concept of continuity exists, for instance to locales (and then to toposes) or to convergence spaces. (See also at continuous space.)

Definitions

The epsilontic definition for metric spaces

We state the definition of continuity in terms of epsilontic analysis, definition 3 below. First recall the relevant concepts:

Definition

A metric space is

  1. a set XX (the “underlying set”);

  2. a function d:X×X[0,)d \;\colon\; X \times X \to [0,\infty) (the “distance function”) from the Cartesian product of the set with itself to the non-negative real numbers

such that for all x,y,zXx,y,z \in X:

  1. d(x,y)=0x=yd(x,y) = 0 \;\Leftrightarrow\; x = y

  2. (symmetry) d(x,y)=d(y,x)d(x,y) = d(y,x)

  3. (triangle inequality) d(x,y)+d(y,z)d(x,z)d(x,y)+ d(y,z) \geq d(x,z).

Example

Every normed vector space (V,)(V, {\Vert -\Vert}) becomes a metric space according to def. 1 by setting

d(x,y)xy. d(x,y) \coloneqq {\Vert x-y\Vert} \,.
Definition

Let (X,d)(X,d), be a metric space. Then for every element xXx \in X and every ϵ +\epsilon \in \mathbb{R}_+ a positive real number, write

B x (ϵ){yX|d(x,y)<ϵ} B^\circ_x(\epsilon) \;\coloneqq\; \left\{ y \in X \;\vert\; d(x,y) \lt \epsilon \right\}

for the open ball of radius ϵ\epsilon around xx.

Definition

(epsilontic definition of continuity)

For (X,d X)(X,d_X) and (Y,d Y)(Y,d_Y) two metric spaces (def. 1), then a function

f:XY f \;\colon\; X \longrightarrow Y

is said to be continuous at a point xXx \in X if for every ϵ>0\epsilon \gt 0 there exists δ>0\delta\gt 0 such that

d X(x,y)<δd Y(f(x),f(y))<ϵ d_X(x,y) \lt \delta \;\Rightarrow\; d_Y(f(x), f(y)) \lt \epsilon

or equivalently such that

f(B x (δ))B f(x) (ϵ) f(B_x^\circ(\delta)) \subset B^\circ_{f(x)}(\epsilon)

where B B^\circ denotes the open ball (definition 2).

The function ff is called just continuous if it is continuous at every point xXx \in X.

This definition is equivalent to a more abstract one, which does not explicitly refer to points or radii anymore:

Definition

Let (X,d)(X,d) be a metric space (def. 1). Say that

  1. A neighbourhood of a point xXx \in X is a subset xUXx \in U \subset X which contains some open ball B x (ϵ)B_x^\circ(\epsilon) around xx (def. 2).

  2. An open subset of XX is a subset UXU \subset X such that for every for xUx \in U it also contains a neighbourhood of xx.

Remark

The collection of open subsets in def. 4 constitutes a topology on the set XX, making it a topological space. This is called the metric topology. Stated more concisely: the open balls in a metric space constitute the basis of a topology for the metric topology.

Proposition

A function f:XYf \colon X \to Y between metric spaces (def. 1) is continuous in the epsilontic sense of def. 3 precisely if it has the property that its pre-images of open subsets of YY (in the sense of def. 4) are open subsets of XX.

Proof

First assume that ff is continuous in the epsilontic sense. Then for O YYO_Y \subset Y any open subset and xf 1(O Y)x \in f^{-1(O_Y)} any point in the pre-image, we need to show that there exists a neighbourhood of xx in UU. But by assumption there exists an open ball B x (ϵ)B_x^\circ(\epsilon) with f(B X (ϵ))O Yf(B_X^\circ(\epsilon)) \subset O_Y. Since this is true for all xx, by definition this means that f 1(O Y)f^{-1}(O_Y) is open in XX.

Conversely, assume that f 1f^{-1} takes open subsets to open subsets. Then for every xXx \in X and B f(x) (ϵ)B_{f(x)}^\circ(\epsilon) an open ball around its image, we need to produce an open ball B x (δ)B_x^\circ(\delta) in its pre-image. But by assumption f 1(B f(x) (ϵ))f^{-1}(B_{f(x)}^\circ(\epsilon)) contains a neighbourhood of xx which by definition means that it contains such an open ball around xx.

For topological spaces

Definition

A function f:XYf \;\colon\; X\to Y between topological spaces is a continuous map (or is said to be continuous) if for every open subset UYU \subset Y, the preimage f 1(U)f^{-1}(U) is an open subset XX.

In nonstandard analysis, this is equivalent to

Definition

A function f:XYf \;\colon\; X\to Y between topological spaces is a continuous map (or is said to be continuous) if for every standard point? x 1x_1 and every hyperpoint? x 2x_2, if x 1x_1 and x 2x_2 are adequal? (infinitely close, or in other words if x 2x_2 is in the halo of x 1x_1), then f(x 1)f(x_1) and f *(x 2)\multiscripts{^*}f{}(x_2) are adequal (where f *\multiscripts{^*}f{} is the nonstandard extension? of ff). Equivalently, ff is continuous iff f *\multiscripts{^*}f{} is microcontinuous?.

Further variants

A function ff between convergence spaces is continuous if for any filter FF such that FxF \to x, it follows that f(F)f(x)f(F) \to f(x), where f(F)f(F) is the filter generated by the filterbase {F(A)|AF}\{F(A) \;|\; A \in F\}.

A continuous map between locales is simply a frame homomorphism in the opposite direction. Equivalently (via the adjoint functor theorem), it may be defined as a homomorphism of inflattices whose left adjoint preserves finitary meets (and hence is a frame homomorphism).

Properties

Since continuity is defined in terms of preservation of property (namely preserving “openness” under preimages), it is natural to ask what other properties they preserve.
Also, when a property is not always preserved it is useful to label those maps which do preserve it for closer study.

Properties preserved

  1. By definition, the preimage of an open set is open.

  2. Similarly, the preimage of an closed set is closed.

  3. The image of a connected subset is again connected.

  4. The image of a compact subset is again compact (see at continuous images of compact subsets are compact?)

Special maps

  1. The preimage of a compact set need not be compact; a continuous map for which this is true is known as a proper map.

  2. The image of an open set need not be open; a continuous map for which this is true is said to be an open map. (Technically, an open map is any function with just this property.)

  3. The image of an closed set need not be closed; a continuous map for which this is true is said to be an closed map. (Technically, a closed map is any function with just this property.)

  4. A continuous map of topological spaces which is invertible as a function of sets is a homeomorphism if the inverse function is a continuous map as well.

Special cases in specific contexts

Although these don’t make sense for arbitrary topological spaces (convergence spaces, locales, etc), they are special kinds of continuous maps in contexts such as metric spaces:

In constructive mathematics

Various notions of continuous function are used in constructive mathematics. A function ff (say real-valued and defined on a real interval) is:

  • pointwise-continuous if it continuous in the usual epsilon-delta (or equivalently open-subset?) sense;
  • uniformly continuous if it uniformly continuous in the usual epsilon-delta (or equivalently entourage-theoretic) sense;
  • Bishop-continuous if it is pointwise continuous and furthermore, the restriction to any closed and bounded interval is uniformly continuous;
  • Bridges-continuous if … (this one's kind of complicated).

In classical mathematics, these are all equivalent when the domain is itself a closed and bounded interval, and all of them except for uniform continuity are equivalent in general. The same equivalences hold in intuitionistic mathematics, thanks to the fan theorem. But no two of these are equivalent in Russian constructivism.

In fact, assuming that \mathbb{R} is defined as the set of located Dedekind cuts, there is the following negative result by Frank Waaldijk (Waaldijk2003): Without the fan theorem, there is no notion of continuity for set-theoretic functions in constructive mathematics, spelled “kontinuity” in the following, such that all of the following desiderata are met:

  • A function [0,1][0,1] \to \mathbb{R} is kontinuous if and only if it is uniformly continuous in the usual sense.
  • The composition of kontinuous functions is kontinuous.
  • The function +,x1/x\mathbb{R}^+ \to \mathbb{R}, x \mapsto 1/x is kontinuous.

The key problem is that a uniformly continuous, positive-valued function defined on [0,1][0,1] might fail to be bounded below by a positive number, since the interval [0,1][0,1] might fail to be compact, yet its reciprocal (if also uniformly continuous) must be bounded above.

Waaldijk’s negative result can be circumvented by dropping the insistence on points and instead working with maps between locales, toposes, or formal spaces as studied in formal topology.

References

  • Frank Waaldijk, On the foundations of constructive mathematics – especially in relation to the theory of continuous functions, 2003 (pdf)

Revised on April 13, 2017 16:48:27 by Urs Schreiber (77.56.60.54)