In mathematics, a Hausdorff space is said to be '''H-closed''', or '''Hausdorff closed''', or '''absolutely closed''' if it is closed in every Hausdorff space containing it as a subspace. This property is a generalization of compactness, since a compact subset of a Hausdorff space is closed. Thus, every compact Hausdorff space is H-closed. The notion of an H-closed space has been introduced in 1924 by P. Alexandroff and P. Urysohn.

==Examples and equivalent formulations==

* The unit interval <math>[0,1]</math>, endowed with the smallest topology which refines the euclidean topology, and contains <math>Q \cap [0,1]</math> as an open set is H-closed but not compact. * Every regular Hausdorff H-closed space is compact. * A Hausdorff space is H-closed if and only if every open cover has a finite subfamily with dense union.

==See also==

*Compact space

==References==

* K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), ''Encyclopedia of General Topology'', Chapter d20 (by Jack Porter and Johannes Vermeer)

Category:Properties of topological spaces Category:Compactness (mathematics)