View source for Quasi-compact space

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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).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR></table>



====Comments====
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]].