# What are interesting properties of totally bounded uniform spaces?

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.)

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By the way, here is a book “Uniform spaces” by J.R. Isbell (MSM012, AMS, 1964) –  Alex Ravsky May 27 at 6:33

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.

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The way round: uniformities are a generalization of proximities –  porton May 25 at 13:37
How (1) follows from the Wikipedia's definition of total boundness? en.wikipedia.org/wiki/Totally_bounded_space –  porton May 25 at 19:48
OK, (1) is proved. But I have some trouble proving the reverse implication. –  porton May 27 at 16:56