H-closed space

In mathematics, a topological 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 $[0,1]$, 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 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)