Quasi-compact space

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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).


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


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:
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article