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