Loewner differential equation: Difference between revisions
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In [[mathematics]], a [[topological space]] ''X'' is said to be '''H-closed''', or '''Hausdorff closed''', or '''absolutely closed''' if it is closed in every [[Hausdorff space]] space containing it as a subspace. This property is a generalization of [[compact spaces|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 [[Pavel Alexandrov|P. Alexandroff]] and [[Pavel Samuilovich Urysohn|P. Urysohn]]. | |||
==Examples and equivalent formulations== | |||
* The unit interval <math>[0,1]</math>, endowed with the smallest topology which refines the euclidean topology, and contains <math>Q \cap [0,1]</math> as an open set is H-closed but not compact. | |||
* Every [[regular space|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) | |||
[[Category:Properties of topological spaces]] | |||
[[Category:Compactness (mathematics)]] | |||
Revision as of 10:25, 26 January 2014
In mathematics, a topological space X is said to be H-closed, or Hausdorff closed, or absolutely closed if it is closed in every Hausdorff space 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.
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
![{\displaystyle Q\cap [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47cafb8d2800273c978e8915173309033702f543)
- 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
References
- K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), Encyclopedia of General Topology, Chapter d20 (by Jack Porter and Johannes Vermeer)