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).
|[En]||R. Engelking, "General topology", Heldermann (1989) MR1039321 Zbl 0684.54001|
Pre-compact space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Pre-compact_space&oldid=33622