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[parent] uniform continuity (Definition)

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. $ \qedsymbol$

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$. $ \qedsymbol$

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



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See Also: uniformly continuous

Also defines:  uniformly continuous

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Cross-references: even, entourage, uniform neighborhood, open, uniform topologies, continuous, equality, inclusion, converse, iff, equivalent, function, uniformity, component, uniform spaces, metric spaces, uniformly continuous function
There are 19 references to this entry.

This is version 4 of uniform continuity, born on 2007-02-20, modified 2007-08-06.
Object id is 8941, canonical name is UniformContinuity.
Accessed 3428 times total.

Classification:
AMS MSC: 54E15 (General topology :: Spaces with richer structures :: Uniform structures and generalizations)
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