# Pre-compact space

From Encyclopedia of Mathematics

*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).

#### Comments

#### References

[a1] | R. Engelking, "General topology" , Heldermann (1989) |

**How to Cite This Entry:**

Pre-compact space.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Pre-compact_space&oldid=14489

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article