View source for Pre-compact space

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''totally-bounded space''

A [[Uniform space|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)|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|Completion of a uniform space]]).



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====References====
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|valign="top"|{{Ref|En}}||valign="top"|  R. Engelking, "General topology", Heldermann  (1989)  {{MR|1039321}} {{ZBL|0684.54001}}
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[[Category:Topological spaces with richer structure]]