Pre-compact space

totally-bounded space

A uniform space $X$ for all entourages $U$ of which there exists a finite covering of $X$ by sets of $U$. In other words, for every entourage $U\subset X$ there is a finite subset $F\subset X$ such that $X\subset U(F)$. A uniform space is pre-compact if and only if every net (cf. Net (of sets in a topological space)) in $X$ has a Cauchy subnet. Therefore, for $X$ to be a pre-compact space it is sufficient that some completion of $X$ is compact, and it is necessary that every completion of it is compact (cf. Completion of a uniform space).

References

[En]	R. Engelking, "General topology", Heldermann (1989) MR1039321 Zbl 0684.54001