Namespaces
Variants
Actions

Pre-compact space

From Encyclopedia of Mathematics
Revision as of 17:08, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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://encyclopediaofmath.org/index.php?title=Pre-compact_space&oldid=31955
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article