# Pre-compact space

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

totally-bounded space

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).