# 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).