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Difference between revisions of "Quasi-compact space"

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A [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076420/q0764201.png" /> in which every [[Filter|filter]] has at least one point of adherence. Equivalent to this condition are the following three: 1) every family of closed sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076420/q0764202.png" /> with empty intersection contains a finite subfamily with empty intersection; 2) every [[Ultrafilter|ultrafilter]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076420/q0764203.png" /> is convergent; and 3) every open covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076420/q0764204.png" /> contains a finite open subcovering of this space (the Borel–Lebesgue condition). A quasi-compact space is called compact (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076420/q0764205.png" />-compact) if it is separated (or Hausdorff). For example, every space in which there are only a finite number of open sets is a quasi-compact space. In particular, any finite space is quasi-compact. A continuous image of a quasi-compact space is quasi-compact. A topological product of any number of quasi-compact spaces is quasi-compact (Tikhonov's theorem).
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A [[Topological space|topological space]] $X$ in which every [[Filter|filter]] has at least one point of adherence. Equivalent to this condition are the following three: 1) every family of closed sets in $X$ with empty intersection contains a finite subfamily with empty intersection; 2) every [[Ultrafilter|ultrafilter]] in $X$ is convergent; and 3) every open covering of $X$ contains a finite open subcovering of this space (the Borel–Lebesgue condition). A quasi-compact space is called compact (or $T_2$-compact) if it is separated (or Hausdorff). For example, every space in which there are only a finite number of open sets is a quasi-compact space. In particular, any finite space is quasi-compact. A continuous image of a quasi-compact space is quasi-compact. A topological product of any number of quasi-compact spaces is quasi-compact (Tikhonov's theorem).
  
 
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Revision as of 09:09, 27 April 2014

A topological space $X$ in which every filter has at least one point of adherence. Equivalent to this condition are the following three: 1) every family of closed sets in $X$ with empty intersection contains a finite subfamily with empty intersection; 2) every ultrafilter in $X$ is convergent; and 3) every open covering of $X$ contains a finite open subcovering of this space (the Borel–Lebesgue condition). A quasi-compact space is called compact (or $T_2$-compact) if it is separated (or Hausdorff). For example, every space in which there are only a finite number of open sets is a quasi-compact space. In particular, any finite space is quasi-compact. A continuous image of a quasi-compact space is quasi-compact. A topological product of any number of quasi-compact spaces is quasi-compact (Tikhonov's theorem).

References

[1] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)


Comments

Often a quasi-compact space is called compact and a space called compact here is explicitly called compact Hausdorff. See also Compact space.

How to Cite This Entry:
Quasi-compact space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-compact_space&oldid=16621
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article