nLab uniformly continuous map

Idea

Recall that a continuous map $f$ between spaces $X$ and $Y$ has the property that $f$ 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 $f$ 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

• for Archimedean fields (def. below)

• for metric spaces (def. below)

but the definition makes sense more generally

• for uniform spaces (def. below)

and in fact

• for quasi-uniform spaces (def. below).

Definitions

Definition

(uniform continuous map between uniform spaces)

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

$$\forall\, E\colon \mathcal{U}Y,\; \exists\, D\colon \mathcal{U}X,\; \forall\, a, b\colon X,\; a \approx_D b \;\Rightarrow\; f(a) \approx_E f(b) .$$

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 $\exists\, D$ and $\forall\, a$.

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 ‘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:

$$\forall\, E\colon \mathcal{U}Y,\; \exists\, D\colon \mathcal{U}X,\; \forall\, a, b\colon X,\; a \approx_D b \;\Rightarrow\; f(b) \approx_E f(a) .$$

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 $X$ and $Y$, a uniformly continuous map from $X$ to $Y$ is a function $f\colon X\to Y$ between their underlying sets such that, given any positive real number $\epsilon$, there is a positive number $\delta$ such that the distance in $Y$ between $f(a)$ and $f(b)$ is less than $\epsilon$ whenever the distance in $X$ between $a$ and $b$ is less than $\delta$:

$$\forall\, \epsilon \gt 0,\; \exists\, \delta \gt 0,\; \forall\, a, b\colon X,\; d_X(a, b) \lt \delta \;\Rightarrow\; d_Y(f(a), f(b)) \lt \epsilon .$$

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

Definition

(uniformly continuous map between Archimedean fields)

Given Archimedean fields $X$ and $Y$, a uniformly continuous map from $X$ to $Y$ is a function between their underlying sets such that, given any positive element $\epsilon \gt 0$, there is a positive element $\delta \gt 0$ such that the maximum of $f(a) - f(b)$ and $f(b) - f(a)$ is less than $\epsilon$ whenever the maximum of $a - b$ and $b - a$ is less than $\delta$:

$$\forall\, \epsilon \gt 0,\; \exists\, \delta \gt 0,\; \forall\, a, b\colon X,\; \max(a - b, b - a) \lt \delta \;\Rightarrow\; \max(f(a) - f(b), f(b) - f(a)) \lt \epsilon .$$

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

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 $\mathrm{R}_+ \coloneqq \sum_{x:\mathbb{R}} \epsilon \gt 0$ denote the positive real numbers. Given metric spaces $(X, d_X)$ and $(Y, d_Y)$, a uniformly continuous function between $X$ and $Y$ is a function $f:X \to Y$ between their underlying sets with a dependent function which says:

Given any positive real number $\epsilon \gt 0$, there is as structure a positive real number $\delta \gt 0$ such that for all elements $a:X$ and $b:X$, $\delta_Y(f(a), f(b))$ is less than $\epsilon$ whenever $\delta_X(a, b)$ is less than $\delta$

$$\prod_{\epsilon:\mathrm{R}_+} \sum_{\delta:\mathbb{R}_+} \prod_{a:X} \prod_{b:X} (\delta_X(a, b) \lt \delta) \to (\delta_Y(f(a), f(b)) \lt \epsilon)$$

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 $\omega:\mathrm{R}_+ \to \mathrm{R}_+$ such that for all positive real numbers $\epsilon \gt 0$ and for all elements $a:X$ and $b:X$, $\delta_Y(f(a), f(b))$ is less than $\epsilon$ whenever $\delta_X(a, b)$ is less than $\omega(\epsilon)$

$$\sum_{\omega:\mathrm{R}_+ \to \mathrm{R}_+} \prod_{\epsilon:\mathrm{R}_+} \prod_{a:X} \prod_{b:X} (\delta_X(a, b) \lt \omega(\epsilon)) \to (\delta_Y(f(a), f(b)) \lt \epsilon)$$

There exists a similar definition for uniform spaces:

Definition

Given uniform spaces $(X, \mathcal{U}(X), \approx)$ and $(Y, \mathcal{U}(Y), \approx)$, a uniformly continuous function between $X$ and $Y$ is a function $f:X \to Y$ with a dependent function which says:

Given any entourage $E:\mathcal{U}(Y)$, there is as structure an entourage $D:\mathcal{U}(X)$ such that for all elements $a:X$ and $b:X$, $f(a) \approx_{E} f(b)$ whenever $a \approx_{D} b$

$$\prod_{E:\mathcal{U}(Y)} \sum_{D:\mathcal{U}(X)} \prod_{a:X} \prod_{b:X} (a \approx_{D} b) \to (f(a) \approx_{E} f(b))$$

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 $\omega:\mathcal{U}(Y) \to \mathcal{U}(X)$ between the sets of entourages such that for all entourages $E:\mathcal{U}(Y)$ and for all elements $a:X$ and $b:X$, $f(a) \approx_{E} f(b)$ whenever $a \approx_{\omega(E)} b$

$$\sum_{\omega:\mathcal{U}(Y) \to \mathcal{U}(X)} \prod_{E:\mathcal{U}(Y)} \prod_{a:X} \prod_{b:X} (a \approx_{\omega(E)} b) \to (f(a) \approx_{E} f(b))$$

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 $\mathcal{M}$-category).

Examples

• continuous metric space valued function on compact metric space is uniformly continuous

Related concepts

• uniform convergence

• uniform property

• modulus of continuity

• Heine-Cantor theorem