# H-closed space > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/H-closed_space > Markdown URL: https://mediated.wiki/source/H-closed_space.md > Source: https://en.wikipedia.org/wiki/H-closed_space > Source revision: 1000724252 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) In [mathematics](/source/mathematics), a [Hausdorff space](/source/Hausdorff_space) is said to be '''H-closed''', or '''Hausdorff closed''', or '''absolutely closed''' if it is closed in every [Hausdorff space](/source/Hausdorff_space) containing it as a subspace. This property is a generalization of [compactness](/source/compact_spaces), 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](/source/Pavel_Alexandrov) and [P. Urysohn](/source/Pavel_Samuilovich_Urysohn). ==Examples and equivalent formulations== * The unit interval [0,1], endowed with the smallest topology which refines the euclidean topology, and contains Q \cap [0,1] as an open set is H-closed but not compact. * Every [regular](/source/regular_space) 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](/source/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) --- Adapted from the Wikipedia article [H-closed space](https://en.wikipedia.org/wiki/H-closed_space) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/H-closed_space?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.