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.
gollark: It's 1, or the nice neat recursive factorial calculation algorithms would stop working.
gollark: It's not an example, this seems to be true in all cases.
gollark: Oh, they said they don't need to be different, so square numbers are fine I guess.
gollark: I mean, you know it has 2 as a factor, and you know it divided by 2 isn't prime, implying it must have multiple prime factors (actually, *is* that the case given square numbers' existence? hmmm.)
gollark: Well, if 0 = 1 then obviously 2 = 3.
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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