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]].
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==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 11:30, 4 February 2014

THE HAIR
Rebecca Gonsalves: The higher the hair, the closer to God is not a tenet you'd expect Ms Margulies to live by, but it's all the better to show off those wonderful eyebrows.

Hugh Montgomery: We personally liked the Nurse Hathaway curls but hey, even as we hum TLC's "Waterfalls", we appreciate you can't live in 1995 forever.
THE SUIT
Rebecca: Kinda naff, but the fact it's accessorised with the happiest smile ever makes my Grinch-y fashion heart grow three sizes.

Hugh: Surely everyone knows that if you wear a white tux, you need to wear it backwards, Celine Dion-style, or be found wanting.
THE BAG
Rebecca: Not exactly an earth-shattering choice, but perfect for accentuating her dance moves should she come over with a case of 'Saturday Night Fever'.
Hugh: The kind of uselessly prosaic accessory that makes us lose faith in this vocation of ours.
THE JEWELLERY
Rebecca: Chandelier earrings are a surprisingly romantic choice that are steered into 'maid of honour' territory by such a plump bun.

Hugh: From 'Good Wife' to Pat Butcher? No. Too obvious. They're great.

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