# Which of the conditions imply that a function is uniformly continuous relative to an uniformity?

Let $$(X, \mathcal{U})$$ be an uniform space. $$f:(X, \mathcal{U})\to (X, \mathcal{U})$$ is called uniformly continuous relative to $$\mathcal{U}$$, if for every entourage $$V\in \mathcal{U}$$, $$(f\times f)^{-1}(V)\in \mathcal{U}$$. It is known that every uniformly continuous function is continuous relative to the uniformly topology and the conversely if $$(X, \mathcal{U})$$ is a compact and $$f$$ is continuous relative to the uniformly topology, then $$f:(X, \mathcal{U})\to (X, \mathcal{U})$$ is uniformly continuous relative to $$\mathcal{U}$$.

Which of conditions for $$f$$ imply that $$f$$ is uniformly continuous?

Please help me to know it.

## 1 Answer

The definition is the condition, I would say.

There are some cases where we have more structure and we get uniform continuity "for free": if $$X$$ is compact (in its topology induced from the uniformity) then any continuous $$f$$ on it to some other uniform space will be uniformly continuous.

If we have a topological vector space in its uniformity induced from some neighbourhood base of $$0$$, the a continuous linear map into another topological vector space is also uniformly continuous.

So compactness and linearity have an effect to promote continuity to uniform continuity.. There are probably more such special cases; these are classic ones I happen to know about.