Countably compact space - Revision history

Vipul: /* Formalisms */

2010-10-20T16:23:55Z

Formalisms

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	# Every countable open [[cover]] has a finite [[subcover]]. In other words, given a countable collection of open subsets whose union is the whole space, there is a finite subcollection whose union is again the whole space.		# Every countable open [[cover]] has a finite [[subcover]]. In other words, given a countable collection of open subsets whose union is the whole space, there is a finite subcollection whose union is again the whole space.
	# Every point-finite open cover has a finite subcover.		# Every point-finite open cover has a finite subcover.
			===Equivalence of definitions===

			{{further\|[[equivalence of definitions of countably compact space]]}}

	==Formalisms==		==Formalisms==

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	It is also an instance of the [[countably qualifier]] applied to compactness-like properties.		It is also an instance of the [[countably qualifier]] applied to compactness-like properties.

			Another refinement formal expression is:

			Point-finite open <math>\to</math> Finite open

	==Relation with other properties==		==Relation with other properties==

Vipul at 16:22, 20 October 2010

2010-10-20T16:22:14Z

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	===Symbol-free definition===		===Symbol-free definition===

	A [[topological space]] is said to be '''countably compact''' if ~~every countable open [[cover]] has a finite [[subcover]]. In other words, given a countable collection of open subsets whose union is~~ the ~~whole space, there is a finite subcollection whose union is again the whole space.~~		A [[topological space]] is said to be '''countably compact''' if it satisfies the following equivalent conditions:

			# Every countable open [[cover]] has a finite [[subcover]]. In other words, given a countable collection of open subsets whose union is the whole space, there is a finite subcollection whose union is again the whole space.
			# Every point-finite open cover has a finite subcover.
	==Formalisms==		==Formalisms==

Vipul: 5 revisions

2008-05-11T19:42:48Z

5 revisions

← Older revision	Revision as of 19:42, 11 May 2008
(No difference)

Vipul at 22:07, 21 April 2008

2008-04-21T22:07:50Z

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	Switching to a coarser topology preserves countable compactness.		Switching to a coarser topology preserves countable compactness.

			==References==
			===Textbook references===
			* {{booklink\|Munkres}}, Page 181, Exercise 4 (definition introduced in exercise)

Vipul: /* Stronger properties */

2007-12-20T23:52:23Z

Stronger properties

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	* [[Compact space]]		* [[Compact space]]
			* [[Sequentially compact space]]

	===Weaker properties===		===Weaker properties===

Vipul: /* Relation with other properties */

2007-08-20T10:01:25Z

Relation with other properties

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	* [[Compact space]]		* [[Compact space]]

			===Weaker properties===

			* [[Limit point-compact space]]

			==Metaproperties==

			{{DP-closed}}

			The proof of this follows from a version of the [[tube lemma]].

			{{closed subspace-closed}}

			Any [[closed subset]] of a countably compact space is countably compact, when endowed with the [[subspace topology]].

			{{coarsening-preserved}}

			Switching to a coarser topology preserves countable compactness.

Vipul: /* Definition */

2007-08-20T09:59:55Z

Definition

← Older revision		Revision as of 09:59, 20 August 2007
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	==Definition==		==Definition==

	===Symbol-free definition====		===Symbol-free definition===

	A [[topological space]] is said to be '''countably compact''' if every countable open [[cover]] has a finite [[subcover]]. In other words, given a countable collection of open subsets whose union is the whole space, there is a finite subcollection whose union is again the whole space.		A [[topological space]] is said to be '''countably compact''' if every countable open [[cover]] has a finite [[subcover]]. In other words, given a countable collection of open subsets whose union is the whole space, there is a finite subcollection whose union is again the whole space.

Vipul at 08:04, 18 August 2007

2007-08-18T08:04:03Z

New page

{{topospace property}}

{{variationof|compactness}}

==Definition==

===Symbol-free definition====

A [[topological space]] is said to be '''countably compact''' if every countable open [[cover]] has a finite [[subcover]]. In other words, given a countable collection of open subsets whose union is the whole space, there is a finite subcollection whose union is again the whole space.

==Formalisms==

===Refinement formal expression===

In the [[refinement]] formalism, a [[refinement formal expression]] is:

Countable open <math>\to</math> Finite open

viz, every countable open cover has a finite open refinement.

It is also an instance of the [[countably qualifier]] applied to compactness-like properties.

==Relation with other properties==

===Stronger properties===

* [[Compact space]]