Difference between revisions of "Quasi-compact space"
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| − | A [[Topological space|topological space]] | + | {{TEX|done}} |
| + | 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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Often a quasi-compact space is called compact and a space called compact here is explicitly called compact Hausdorff. See also [[Compact space|Compact space]]. | Often a quasi-compact space is called compact and a space called compact here is explicitly called compact Hausdorff. See also [[Compact space|Compact space]]. | ||
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| + | [[Category:General topology]] | ||
Latest revision as of 20:48, 16 October 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.
Quasi-compact space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-compact_space&oldid=16621