# Uniformly continuous maps

## Idea

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.

A 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.

## Definitions

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 $f\left(a\right)$ and $f\left(b\right)$ are $E$-close in $Y$ whenever $a$ and $b$ are $D$-close in $X$:

$\forall \phantom{\rule{thinmathspace}{0ex}}E:𝒰Y,\phantom{\rule{thickmathspace}{0ex}}\exists \phantom{\rule{thinmathspace}{0ex}}D:𝒰X,\phantom{\rule{thickmathspace}{0ex}}\forall \phantom{\rule{thinmathspace}{0ex}}a,b:X,\phantom{\rule{thickmathspace}{0ex}}a{\approx }_{D}b\phantom{\rule{thickmathspace}{0ex}}⇒\phantom{\rule{thickmathspace}{0ex}}f\left(a\right){\approx }_{E}f\left(b\right).$\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) .

Note that this definition is exactly like the definition of continuous map between uniform spaces, except for the order of the quantifiers $\exists \phantom{\rule{thinmathspace}{0ex}}D$ and $\forall \phantom{\rule{thinmathspace}{0ex}}a$.

The definition with 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. We also have the option of considering an antiuniformly continuous map, in which the order in which the points are compared is reversed:

$\forall \phantom{\rule{thinmathspace}{0ex}}E:𝒰Y,\phantom{\rule{thickmathspace}{0ex}}\exists \phantom{\rule{thinmathspace}{0ex}}D:𝒰X,\phantom{\rule{thickmathspace}{0ex}}\forall \phantom{\rule{thinmathspace}{0ex}}a,b:X,\phantom{\rule{thickmathspace}{0ex}}a{\approx }_{D}b\phantom{\rule{thickmathspace}{0ex}}⇒\phantom{\rule{thickmathspace}{0ex}}f\left(b\right){\approx }_{E}f\left(a\right).$\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) .

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

For uniform spaces, a definition can also be given in terms of uniform covers.

In the particular case of metric spaces, it is common to see this definition in elementary form: Given metric spaces $X$ and $Y$, a uniformly continuous map from $X$ to $Y$ is a function between their underlying sets such that, given any positive number $ϵ$, there is a positive number $\delta$ such that the distance in $Y$ between $f\left(a\right)$ and $f\left(b\right)$ is less than $ϵ$ whenever the distance in $X$ between $a$ and $b$ is less than $\delta$:

$\forall \phantom{\rule{thinmathspace}{0ex}}ϵ>0,\phantom{\rule{thickmathspace}{0ex}}\exists \phantom{\rule{thinmathspace}{0ex}}\delta >0,\phantom{\rule{thickmathspace}{0ex}}\forall \phantom{\rule{thinmathspace}{0ex}}a,b:X,\phantom{\rule{thickmathspace}{0ex}}{d}_{X}\left(a,b\right)<\delta \phantom{\rule{thickmathspace}{0ex}}⇒\phantom{\rule{thickmathspace}{0ex}}{d}_{Y}\left(a,b\right)<ϵ.$\forall\, \epsilon \gt 0,\; \exists\, \delta \gt 0,\; \forall\, a, b\colon X,\; d_X(a, b) \lt \delta \;\Rightarrow\; d_Y(a, b) \lt \epsilon .

Again, this is exactly like the definition of continuous map between metric spaces, except for the order of the quantifiers $\exists \phantom{\rule{thinmathspace}{0ex}}\delta$ and $\forall \phantom{\rule{thinmathspace}{0ex}}a$.

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.

## 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 $ℳ$-category).

Revised on November 14, 2011 06:13:06 by Toby Bartels (139.55.238.24)