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 [0,1], endowed with the smallest topology which refines the euclidean topology, and contains \( {\displaystyle Q\cap [0,1]} \) 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.