What are interesting properties of totally bounded uniform spaces?

Question

I work on reloids, a generalization of uniform spaces.

As such I am interested about properties of totally bounded uniform spaces.

My question: What are interesting properties of totally bounded uniform spaces? (The more theorems/propositions you give, the better.)

Answer 1

As far as I remember, these are equivalent for a uniform space $(X,\mathcal D)$:

1. $(X,\mathcal D)$ is totally bounded, that is, for each entourage $D\in \mathcal D$, there are $x_1,...,x_n\in X$ with $D[x_1]\cup...\cup D[x_n]=X$.
2. $(X,\mathcal D)$ is precompact, that is, it has a compact completion.
3. Every completion of $(X,\mathcal D)$ is compact.
4. Every net in $X$ has a Cauchy subnet.

Also every Tychonoff space has a compatible totally bounded uniformity.