Baire space - Revision history

Vipul: /* Relation with other properties */

2010-10-20T16:16:19Z

Relation with other properties

← Older revision		Revision as of 16:16, 20 October 2010
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	===Stronger properties===		===Stronger properties===

	* [[~~Compact~~ Hausdorff space]]~~: {{proofat~~\|[[compact Hausdorff implies Baire]]}}		{\| class="sortable" border="1"
	* [[~~Locally~~ compact Hausdorff space]]~~: {{proofat~~\|[[locally compact Hausdorff implies Baire]]}}		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[~~Completely~~ metrizable space]]~~: {{proofat~~\|[[completely metrizable implies Baire]]}}		\|-
			\| [[Weaker than::compact Hausdorff space]] \|\| [[compact space\|compact]] and [[Hausdorff space\|Hausdorff]] \|\| [[compact Hausdorff implies Baire]] \|\| \|\| {{intermediate notions short\|compact Hausdorff space\|Baire space}}
			\|-
			\| [[Weaker than::locally compact Hausdorff space]] \|\| [[locally compact space\|locally compact]] and [[Hausdorff space\|Hausdorff]] \|\| [[locally compact Hausdorff implies Baire]] \|\| \|\| {{intermediate notions short\|locally compact Hausdorff space\|Baire space}}
			\|-
			\| [[Weaker than::completely metrizable space]] \|\| arises from a [[complete metric space]] \|\| [[completely metrizable implies Baire]] \|\| \|\| {{intermediate notions short\|completely metrizable space\|Baire space}}
			\|}

			===Weaker properties===

			{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::Volterra space]] \|\| \|\| \|\| \|\|
			\|}

	==Metaproperties==		==Metaproperties==

Vipul: /* Metaproperties */

2008-07-20T22:03:03Z

Metaproperties

← Older revision		Revision as of 22:03, 20 July 2008
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	{{open subspace-closed}}		{{open subspace-closed}}

			{{proofat\|[[Baire property is open subspace-closed]]}}

	==References==		==References==
	===Textbook references===		===Textbook references===
	* {{booklink\|Munkres}}, Page 296 (formal definition)		* {{booklink\|Munkres}}, Page 296 (formal definition)

Vipul: 4 revisions

2008-05-11T19:31:54Z

4 revisions

← Older revision	Revision as of 19:31, 11 May 2008
(No difference)

Vipul at 21:43, 21 April 2008

2008-04-21T21:43:53Z

← Older revision		Revision as of 21:43, 21 April 2008
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	{{open subspace-closed}}		{{open subspace-closed}}

			==References==
			===Textbook references===
			* {{booklink\|Munkres}}, Page 296 (formal definition)

Vipul: /* Relation with other properties */

2008-01-24T02:26:37Z

Relation with other properties

← Older revision		Revision as of 02:26, 24 January 2008
Line 12:		Line 12:
	===Stronger properties===		===Stronger properties===

	* [[Compact Hausdorff space]]		* [[Compact Hausdorff space]]: {{proofat\|[[compact Hausdorff implies Baire]]}}
	* [[Locally compact Hausdorff space]]		* [[Locally compact Hausdorff space]]: {{proofat\|[[locally compact Hausdorff implies Baire]]}}
	* [[Completely metrizable space]]		* [[Completely metrizable space]]: {{proofat\|[[completely metrizable implies Baire]]}}

	==Metaproperties==		==Metaproperties==

	{{open subspace-closed}}		{{open subspace-closed}}

Vipul: /* Stronger properties */

2007-10-27T22:23:26Z

Stronger properties

← Older revision		Revision as of 22:23, 27 October 2007
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	* [[Compact Hausdorff space]]		* [[Compact Hausdorff space]]
	* [[Locally compact Hausdorff space]]		* [[Locally compact Hausdorff space]]
	* [[~~Topologically complete~~ space]]		* [[Completely metrizable space]]

	==Metaproperties==		==Metaproperties==

	{{open subspace-closed}}		{{open subspace-closed}}

Vipul at 12:35, 18 August 2007

2007-08-18T12:35:44Z

New page

{{topospace property}}

==Definition==

A [[topological space]] is termed a '''Baire space''' if it satisfies the following equivalent conditions:

* A countable intersection of [[open subset|open]] [[dense subset]]s is dense
* A countable union of [[closed subset|closed]] [[nowhere dense subset]]s is nowhere dense

==Relation with other properties==

===Stronger properties===

* [[Compact Hausdorff space]]
* [[Locally compact Hausdorff space]]
* [[Topologically complete space]]

==Metaproperties==

{{open subspace-closed}}