H-closed space
   HOME

TheInfoList



Click Here for Items Related To - H-closed space
OR:
H-closed space on:   [Wikipedia]   [Google]   [Amazon]

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
is said to be H-closed, or Hausdorff closed, or absolutely closed if it is closed in every
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
containing it as a subspace. This property is a generalization of
compactness In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...
, 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 ,1/math>, endowed with the smallest topology which refines the euclidean topology, and contains Q \cap ,1/math> 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 In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...


References

* K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), ''Encyclopedia of General Topology'', Chapter d20 (by Jack Porter and Johannes Vermeer) Properties of topological spaces Compactness (mathematics)