uniform continuity

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uniform continuity

In this entry, we extend the usual definition of a uniformly continuous function between metric spaces to arbitrary uniform spaces.

Let $(X,\mathcal{U}),(Y,\mathcal{V})$ be uniform spaces (the second component is the uniformity on the first component). A function $f:X\to Y$ is said to be uniformly continuous if for any $V\in\mathcal{V}$ there is a $U\in\mathcal{U}$ such that for all $x\in X$ , $U[x]\subseteq f^{{-1}}(V[f(x)])$ .

Sometimes it is useful to use an alternative but equivalent version of uniform continuity of a function:

Proposition 1.

Suppose $f:X\to Y$ is a function and $g:X\times X\to Y\times Y$ is defined by $g(x_{1},x_{2})=(f(x_{1}),f(x_{2}))$ . Then $f$ is uniformly continuous iff for any $V\in\mathcal{V}$ , there is a $U\in\mathcal{U}$ such that $U\subseteq g^{{-1}}(V)$ .

Proof.

Suppose $f$ is uniformly continuous. Pick any $V\in\mathcal{V}$ . Then $U\in\mathcal{U}$ exists with $U[x]\subseteq f^{{-1}}(V[f(x)])$ for all $x\in X$ . If $(a,b)\in U$ , then $b\in U[a]\subseteq f^{{-1}}(V[f(a)])$ , or $f(b)\subseteq V[f(a)]$ , or $g(a,b)=(f(a),f(b))\in V$ . The converse is straightforward. ∎

Remark. Note that we could have picked $U$ so the inclusion becomes an equality.

Proposition 2.

. If $f:X\to Y$ is uniformly continuous, then it is continuous under the uniform topologies of $X$ and $Y$ .

Proof.

Let $A$ be open in $Y$ and set $B=f^{{-1}}(A)$ . Pick any $x\in B$ . Then $y=f(x)$ has a uniform neighborhood $V[y]\subseteq A$ . By the uniform continuity of $f$ , there is an entourage $U\in\mathcal{U}$ with $x\in U[x]\subseteq f^{{-1}}(V[y])\subseteq f^{{-1}}(A)=B$ . ∎

Remark. The converse is not true, even in metric spaces.

Defines:

uniformly continuous

Type of Math Object:

Definition

Major Section:

Reference

Parent:

uniform space

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Mathematics Subject Classification

54E15 Uniform structures and generalizations

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uniform continuity

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uniform continuity

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uniform continuity

Proposition 1.

Proof.

Proposition 2.

Proof.

Mathematics Subject Classification

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