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Difference between revisions of "Pre-compact space"

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''totally-bounded space''
 
''totally-bounded space''
  
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A [[Uniform space|uniform space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074280/p0742801.png" /> for all entourages <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074280/p0742802.png" /> of which there exists a finite covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074280/p0742803.png" /> by sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074280/p0742804.png" />. In other words, for every entourage <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074280/p0742805.png" /> there is a finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074280/p0742806.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074280/p0742807.png" />. A uniform space is pre-compact if and only if every net (cf. [[Net (of sets in a topological space)|Net (of sets in a topological space)]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074280/p0742808.png" /> has a Cauchy subnet. Therefore, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074280/p0742809.png" /> to be a pre-compact space it is sufficient that some completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074280/p07428010.png" /> is compact, and it is necessary that every completion of it is compact (cf. [[Completion of a uniform space|Completion of a uniform space]]).
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A [[Uniform space|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)|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|Completion of a uniform space]]).
  
  

Revision as of 19:07, 27 April 2014

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).


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
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