A uniform space for all entourages of which there exists a finite covering of by sets of . In other words, for every entourage there is a finite subset such that . A uniform space is pre-compact if and only if every net (cf. Net (of sets in a topological space)) in has a Cauchy subnet. Therefore, for to be a pre-compact space it is sufficient that some completion of is compact, and it is necessary that every completion of it is compact (cf. Completion of a uniform space).
|[a1]||R. Engelking, "General topology" , Heldermann (1989)|
Pre-compact space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Pre-compact_space&oldid=14489