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 , 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.