CW-space - Revision history

Vipul: /* Relation with other properties */

2014-01-29T20:13:04Z

Relation with other properties

← Older revision		Revision as of 20:13, 29 January 2014
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	===Stronger properties===		===Stronger properties===

	* [[Polyhedron]]		{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::Polyhedron]] \|\| occurs as the geometric realization of a [[simplicial complex]] \|\| \|\| \|\| {{intermediate notions short\|CW-space\|polyhedron}}
			\|}

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

	* [[Hereditarily paracompact Hausdorff space]]		{\| class="sortable" border="1"
	* [[~~Paracompact~~ Hausdorff space]]: {{~~proofat~~\|[[CW implies paracompact Hausdorff]]}}		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Perfectly normal space]]~~: {{proofat~~\|[[CW implies perfectly normal]]}}		\|-
	* [[Normal space]]~~: {{proofat~~\|[[CW implies normal]]}}		\| [[Stronger than::Hereditarily paracompact Hausdorff space]] \|\| [[Hausdorff space]] that is also a [[hereditarily paracompact space]]: every subspace (with the [[subspace topology]]) is a [[paracompact space]] \|\| \|\| \|\| {{intermediate notions short\|hereditarily paracompact Hausdorff space\|CW-space}}
	* [[Hausdorff space]]		\|-
	* [[Locally contractible space]]		\| [[Stronger than::Paracompact Hausdorff space]] \|\| \|\| [[CW implies paracompact Hausdorff]] \|\| \|\| {{intermediate notions short\|paracompact Hausdorff space\|CW-space}}
	* [[Locally path-connected space]]~~: {{proofat~~\|[[CW implies locally path-connected]]}}		\|-
	* [[Homotopy-CW-space]]		\| [[Stronger than::Perfectly normal space]] \|\| \|\| [[CW implies perfectly normal]] \|\| \|\| {{intermediate notions short\|perfectly normal space\|CW-space}}
			\|-
			\| [[Stronger than::Normal space]] \|\| \|\| [[CW implies normal]] \|\| \|\| {{intermediate notions short\|normal space\|CW-space}}
			\|-
			\| [[Stronger than::Hausdorff space]] \|\| \|\| [[CW implies Hausdorff]] \|\| \|\| {{intermediate notions short\|Hausdorff space\|CW-space}}
			\|-
			\| [[Stronger than::Locally contractible space]] \|\| every point is contained in a contractible open subset \|\| \|\| \|\| {{intermediate notions short\|locally contractible space\|CW-space}}
			\|-
			\| [[Stronger than::Locally path-connected space]] \|\| \|\| [[CW implies locally path-connected] \|\| \|\| {{intermediate notions short\|locally path-connected space\|CW-space}}
			\|-
			\| [[Stronger than::Homotopy-CW-space]] \|\| [[homotopy-equivalent topological spaces\|homotopy-equivalent]] to a CW-space \|\| (obvious) \|\| any contractible space that is not a Hausdorff space, e.g., the [[line with two origins]] \|\| {{intermediate notions short\|homotopy-CW-space\|CW-space}}
			\|}

Vipul: 8 revisions

2008-05-11T19:32:43Z

8 revisions

← Older revision	Revision as of 19:32, 11 May 2008
(No difference)

Vipul: /* Weaker properties */

2008-01-16T20:16:46Z

Weaker properties

← Older revision		Revision as of 20:16, 16 January 2008
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	* [[Hereditarily paracompact Hausdorff space]]		* [[Hereditarily paracompact Hausdorff space]]
	* [[Paracompact Hausdorff space]]: {{proofat\|[[CW implies paracompact Hausdorff]]}}		* [[Paracompact Hausdorff space]]: {{proofat\|[[CW implies paracompact Hausdorff]]}}
			* [[Perfectly normal space]]: {{proofat\|[[CW implies perfectly normal]]}}
	* [[Normal space]]: {{proofat\|[[CW implies normal]]}}		* [[Normal space]]: {{proofat\|[[CW implies normal]]}}
	* [[Hausdorff space]]		* [[Hausdorff space]]

Vipul: /* Weaker properties */

2008-01-16T20:15:51Z

Weaker properties

← Older revision		Revision as of 20:15, 16 January 2008
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	===Weaker properties===		===Weaker properties===

			* [[Hereditarily paracompact Hausdorff space]]
	* [[Paracompact Hausdorff space]]: {{proofat\|[[CW implies paracompact Hausdorff]]}}		* [[Paracompact Hausdorff space]]: {{proofat\|[[CW implies paracompact Hausdorff]]}}
	* [[Normal space]]: {{proofat\|[[CW implies normal]]}}		* [[Normal space]]: {{proofat\|[[CW implies normal]]}}

Vipul: /* Weaker properties */

2007-12-15T18:02:12Z

Weaker properties

← Older revision		Revision as of 18:02, 15 December 2007
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	===Weaker properties===		===Weaker properties===

			* [[Paracompact Hausdorff space]]: {{proofat\|[[CW implies paracompact Hausdorff]]}}
	* [[Normal space]]: {{proofat\|[[CW implies normal]]}}		* [[Normal space]]: {{proofat\|[[CW implies normal]]}}
	* [[Hausdorff space]]		* [[Hausdorff space]]

Vipul: /* Weaker properties */

2007-12-15T18:00:51Z

Weaker properties

← Older revision		Revision as of 18:00, 15 December 2007
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	===Weaker properties===		===Weaker properties===

	* [[Normal space]]		* [[Normal space]]: {{proofat\|[[CW implies normal]]}}
	* [[Hausdorff space]]		* [[Hausdorff space]]
	* [[Locally contractible space]]		* [[Locally contractible space]]
	* [[Locally path-connected space]]		* [[Locally path-connected space]]: {{proofat\|[[CW implies locally path-connected]]}}
	* [[Homotopy-CW-space]]		* [[Homotopy-CW-space]]

Vipul: /* Weaker properties */

2007-12-03T16:55:26Z

Weaker properties

← Older revision		Revision as of 16:55, 3 December 2007
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	* [[Locally contractible space]]		* [[Locally contractible space]]
	* [[Locally path-connected space]]		* [[Locally path-connected space]]
			* [[Homotopy-CW-space]]

Vipul: /* Relation with other properties */

2007-11-11T00:00:35Z

Relation with other properties

← Older revision		Revision as of 00:00, 11 November 2007
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	==Relation with other properties==		==Relation with other properties==

			===Stronger properties===

			* [[Polyhedron]]

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

Vipul: /* Relation with other properties */

2007-10-26T23:54:39Z

Relation with other properties

← Older revision		Revision as of 23:54, 26 October 2007
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	* [[Hausdorff space]]		* [[Hausdorff space]]
	* [[Locally contractible space]]		* [[Locally contractible space]]
			* [[Locally path-connected space]]

Vipul at 14:23, 22 May 2007

2007-05-22T14:23:19Z

New page

{{topospace property}}

==Definition==

A [[topological space]] is said to be a CW-space if it possesses a CW-decomposition, or in other words, if it can be viewed as the underlying topological space of a [[CW-complex]].

==Relation with other properties==

===Weaker properties===

* [[Normal space]]
* [[Hausdorff space]]
* [[Locally contractible space]]