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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.

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)

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

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