nLab uniformly continuous map

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Uniformly continuous maps

Context

Analysis

Topology

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

see also differential topology, 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

Uniformly continuous maps

Idea

Recall that a continuous map ff between spaces XX and YY has the property that ff maps nearby points to nearby points, which may be formalised by first picking one point, then considering how nearby you want the points to be, then picking another point sufficiently nearby.

The concept of uniformly continuous map ff is based on the same intuition but a different formalisation: first you pick how nearby you want the points to be, then you pick two points sufficiently nearby. This results in a stronger criterion, definable in a less general context.

Traditionally this is formalized

but the definition makes sense more generally

and in fact

Definitions

Definition

(uniform continuous map between uniform spaces)

Let XX and YY be uniform spaces, each defined as a set equipped with a collection of entourages. A uniformly continuous map from XX to YY is a function between their underlying sets such that, given any entourage EE on YY, there is an entourage DD on XX such that f(a)f(a) and f(b)f(b) are EE-close in YY whenever aa and bb are DD-close in XX:

βˆ€E:𝒰Y,βˆƒD:𝒰X,βˆ€a,b:X,aβ‰ˆ Dbβ‡’f(a)β‰ˆ Ef(b). \forall\, E\colon \mathcal{U}Y,\; \exists\, D\colon \mathcal{U}X,\; \forall\, a, b\colon X,\; a \approx_D b \;\Rightarrow\; f(a) \approx_E f(b) .

A definition may also be given in terms of uniform covers.

Remark

The definition is exactly like the definition of continuous map between uniform spaces, except for the order of the quantifiers βˆƒD\exists\, D and βˆ€a\forall\, a.

Definition

(uniformly continuous map between quasiuniform spaces)

The definition in terms of entourages extends immediately to quasiuniform spaces, in which case we may speak of quasiuniformly continuous maps since some authors use β€˜uniformly continuous’ for a map which is uniformly continuous between the spaces' symmetrisations.

Definition

(antiuniformly continuous map)

An antiuniformly continuous map, is defined as a uniform map, but such that the order in which the points are compared is reversed:

βˆ€E:𝒰Y,βˆƒD:𝒰X,βˆ€a,b:X,aβ‰ˆ Dbβ‡’f(b)β‰ˆ Ef(a). \forall\, E\colon \mathcal{U}Y,\; \exists\, D\colon \mathcal{U}X,\; \forall\, a, b\colon X,\; a \approx_D b \;\Rightarrow\; f(b) \approx_E f(a) .
Remark

Between uniform spaces viewed as symmetric quasiuniformly continuous spaces, quasiuniformly continuous maps (def. ), antiuniformly continuous maps (def. ), and uniformly continuous maps (def. ) are the same.

In the particular case of metric spaces, it is common to see this definition in elementary form:

Definition

(uniformly continuous map between metric spaces)

Given metric spaces XX and YY, a uniformly continuous map from XX to YY is a function f:X→Yf\colon X\to Y between their underlying sets such that, given any positive real number ϡ\epsilon, there is a positive number δ\delta such that the distance in YY between f(a)f(a) and f(b)f(b) is less than ϡ\epsilon whenever the distance in XX between aa and bb is less than δ\delta:

βˆ€Ο΅>0,βˆƒΞ΄>0,βˆ€a,b:X,d X(a,b)<Ξ΄β‡’d Y(f(a),f(b))<Ο΅. \forall\, \epsilon \gt 0,\; \exists\, \delta \gt 0,\; \forall\, a, b\colon X,\; d_X(a, b) \lt \delta \;\Rightarrow\; d_Y(f(a), f(b)) \lt \epsilon .

Again, this is exactly like the definition of continuous map between metric spaces, except for the order of the quantifiers βˆƒΞ΄\exists\, \delta and βˆ€a\forall\, a.

Definition

(uniformly continuous map between Archimedean fields)

Given Archimedean fields XX and YY, a uniformly continuous map from XX to YY is a function between their underlying sets such that, given any positive element Ο΅>0\epsilon \gt 0, there is a positive element Ξ΄>0\delta \gt 0 such that the maximum of f(a)βˆ’f(b)f(a) - f(b) and f(b)βˆ’f(a)f(b) - f(a) is less than Ο΅\epsilon whenever the maximum of aβˆ’ba - b and bβˆ’ab - a is less than Ξ΄\delta:

βˆ€Ο΅>0,βˆƒΞ΄>0,βˆ€a,b:X,max(aβˆ’b,bβˆ’a)<Ξ΄β‡’max(f(a)βˆ’f(b),f(b)βˆ’f(a))<Ο΅. \forall\, \epsilon \gt 0,\; \exists\, \delta \gt 0,\; \forall\, a, b\colon X,\; \max(a - b, b - a) \lt \delta \;\Rightarrow\; \max(f(a) - f(b), f(b) - f(a)) \lt \epsilon .

Again, this is exactly like the definition of continuous map between Archimedean fields, except for the order of the quantifiers βˆƒΞ΄\exists\, \delta and βˆ€a\forall\, a.

Definition

A uniform homeomorphism is a uniformly continuous bijection whose inverse is also uniformly continuous (which is not automatic). Two (quasi)uniform spaces are uniformly homeomorphic if there exists a uniform homeomorphism between them. We may also speak of antiuniform homeomorphisms between antiuniformly homeomorphic quasiuniform spaces.

As structure

In dependent type theory, one could change the universal quantifiers and existential quantifiers in the definition of uniformly continuous function into dependent product types and dependent sum types.

Definition

Let R +β‰”βˆ‘ x:ℝϡ>0\mathrm{R}_+ \coloneqq \sum_{x:\mathbb{R}} \epsilon \gt 0 denote the positive real numbers. Given metric spaces (X,d X)(X, d_X) and (Y,d Y)(Y, d_Y), a uniformly continuous function between XX and YY is a function f:Xβ†’Yf:X \to Y between their underlying sets with a dependent function which says:

Given any positive real number Ο΅>0\epsilon \gt 0, there is as structure a positive real number Ξ΄>0\delta \gt 0 such that for all elements a:Xa:X and b:Xb:X, Ξ΄ Y(f(a),f(b))\delta_Y(f(a), f(b)) is less than Ο΅\epsilon whenever Ξ΄ X(a,b)\delta_X(a, b) is less than Ξ΄\delta

∏ Ο΅:R +βˆ‘ Ξ΄:ℝ +∏ a:X∏ b:X(Ξ΄ X(a,b)<Ξ΄)β†’(Ξ΄ Y(f(a),f(b))<Ο΅)\prod_{\epsilon:\mathrm{R}_+} \sum_{\delta:\mathbb{R}_+} \prod_{a:X} \prod_{b:X} (\delta_X(a, b) \lt \delta) \to (\delta_Y(f(a), f(b)) \lt \epsilon)

By the type theoretic axiom of choice, which is simply the distributivity of dependent function types over dependent sum types, this is the same as saying that

There exists as structure a function on the positive real numbers Ο‰:R +β†’R +\omega:\mathrm{R}_+ \to \mathrm{R}_+ such that for all positive real numbers Ο΅>0\epsilon \gt 0 and for all elements a:Xa:X and b:Xb:X, Ξ΄ Y(f(a),f(b))\delta_Y(f(a), f(b)) is less than Ο΅\epsilon whenever Ξ΄ X(a,b)\delta_X(a, b) is less than Ο‰(Ο΅)\omega(\epsilon)

βˆ‘ Ο‰:R +β†’R +∏ Ο΅:R +∏ a:X∏ b:X(Ξ΄ X(a,b)<Ο‰(Ο΅))β†’(Ξ΄ Y(f(a),f(b))<Ο΅)\sum_{\omega:\mathrm{R}_+ \to \mathrm{R}_+} \prod_{\epsilon:\mathrm{R}_+} \prod_{a:X} \prod_{b:X} (\delta_X(a, b) \lt \omega(\epsilon)) \to (\delta_Y(f(a), f(b)) \lt \epsilon)

There exists a similar definition for uniform spaces:

Definition

Given uniform spaces (X,𝒰(X),β‰ˆ)(X, \mathcal{U}(X), \approx) and (Y,𝒰(Y),β‰ˆ)(Y, \mathcal{U}(Y), \approx), a uniformly continuous function between XX and YY is a function f:Xβ†’Yf:X \to Y with a dependent function which says:

Given any entourage E:𝒰(Y)E:\mathcal{U}(Y), there is as structure an entourage D:𝒰(X)D:\mathcal{U}(X) such that for all elements a:Xa:X and b:Xb:X, f(a)β‰ˆ Ef(b)f(a) \approx_{E} f(b) whenever aβ‰ˆ Dba \approx_{D} b

∏ E:𝒰(Y)βˆ‘ D:𝒰(X)∏ a:X∏ b:X(aβ‰ˆ Db)β†’(f(a)β‰ˆ Ef(b))\prod_{E:\mathcal{U}(Y)} \sum_{D:\mathcal{U}(X)} \prod_{a:X} \prod_{b:X} (a \approx_{D} b) \to (f(a) \approx_{E} f(b))

By the type theoretic axiom of choice, which is simply the distributivity of dependent function types over dependent sum types, this is the same as saying that

There exists as structure a function Ο‰:𝒰(Y)→𝒰(X)\omega:\mathcal{U}(Y) \to \mathcal{U}(X) between the sets of entourages such that for all entourages E:𝒰(Y)E:\mathcal{U}(Y) and for all elements a:Xa:X and b:Xb:X, f(a)β‰ˆ Ef(b)f(a) \approx_{E} f(b) whenever aβ‰ˆ Ο‰(E)ba \approx_{\omega(E)} b

βˆ‘ Ο‰:𝒰(Y)→𝒰(X)∏ E:𝒰(Y)∏ a:X∏ b:X(aβ‰ˆ Ο‰(E)b)β†’(f(a)β‰ˆ Ef(b))\sum_{\omega:\mathcal{U}(Y) \to \mathcal{U}(X)} \prod_{E:\mathcal{U}(Y)} \prod_{a:X} \prod_{b:X} (a \approx_{\omega(E)} b) \to (f(a) \approx_{E} f(b))

Properties

Every uniformly continuous map between uniform spaces is continuous (between the underlying topological spaces) and in fact Cauchy continuous (between the underlying Cauchy spaces). Also, every uniformly continuous or antiuniformly continuous map between quasiuniform spaces is Cauchy continuous. Conversely, every short or even Lipschitz map between metric spaces (or Lipschitz manifolds) is uniformly continuous.

A composite of uniformly continuous maps is uniformly continuous, as is any identity function between (quasi)uniform spaces. The composite of two antiuniformly continuous maps is uniformly continuous. Thus uniform spaces are the objects of a category whose morphisms are the uniformly continuous maps as morphisms, and quasiuniform spaces are the objects of two categories: one with uniformly continuous maps as morphisms and one with both uniformly continuous maps and antiuniformly continuous maps as morphisms (so that quasiuniform spaces are the objects of an β„³\mathcal{M}-category).

Examples

Last revised on June 24, 2024 at 10:57:20. See the history of this page for a list of all contributions to it.