H-closed space

In mathematics, a Hausdorff space X 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 , endowed with the smallest topology which refines the euclidean topology, and contains 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.
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gollark: I "know" "JavaScript", yes.
gollark: You may have accidentally routed all traffic the wrong way, or something.
gollark: They did make an ARM chip one time.

See also

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