Retract - Revision history

Vipul: 7 revisions

2008-05-11T19:57:54Z

7 revisions

← Older revision	Revision as of 19:57, 11 May 2008
(No difference)

Vipul at 21:09, 21 April 2008

2008-04-21T21:09:02Z

← Older revision		Revision as of 21:09, 21 April 2008
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	==Definition==		==Definition==
			===Symbol-free definition===
			A subspace of a topological space is said to be a '''retract''' if it satisfies the following equivalent conditions:

	~~A subspace of a topological space is said to be a '''retract''' if there~~ is a continuous map on the whole topological space that maps everything to within the subspace, and that is identity on the ~~retract~~. ~~In other words, there~~ is a continuous idempotent map whose image-cum-fixed-point space is precisely the given subspace. ~~Such a map is termed~~ a ~~[[retraction]]~~.		* There is a continuous map on the whole topological space that maps everything to within the subspace, and that is the identity map when restricted to the subspace.
			* There is a continuous idempotent map whose image-cum-fixed-point space is precisely the given subspace.
			* The inclusion of the subspace in the whole space has a left inverse.

			Such a map (satisfying any of the three equivalent conditions) is termed a [[retraction]].

			===Definition with symbols===
			{{fillin}}
	==Facts==		==Facts==

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	* In a [[Hausdorff space]], any retract is a [[closed subset]]		* In a [[Hausdorff space]], any retract is a [[closed subset]]
	* Many properties of topological spaces are preserved on taking retracts. Examples are properties like being [[simply connected space\|simply connected]], [[weakly contractible space\|weakly contractible]], [[contractible space\|contractible]], or having the [[fixed-point property]]. For a full list of such properties, refer: [[:Category:Retract-hereditary properties of topological spaces]]		* Many properties of topological spaces are preserved on taking retracts. Examples are properties like being [[simply connected space\|simply connected]], [[weakly contractible space\|weakly contractible]], [[contractible space\|contractible]], or having the [[fixed-point property]]. For a full list of such properties, refer: [[:Category:Retract-hereditary properties of topological spaces]]

			==References==
			===Textbook references===
			* {{booklink\|Munkres}}, Page 223, Exercise 4 (definition introduced in exercise)
			* {{booklink\|Rotman}}, Page 4 (formal definition)
			* {{booklink\|Spanier}}, Page 28 (formal definition)

Vipul: /* Facts */

2007-11-10T22:25:26Z

Facts

← Older revision		Revision as of 22:25, 10 November 2007
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	* In a [[Hausdorff space]], any retract is a [[closed subset]]		* In a [[Hausdorff space]], any retract is a [[closed subset]]
	* ~~Any retract~~ of a space ~~with~~ the [[fixed-point property]] ~~also has the fixed~~-~~point property~~		* Many properties of topological spaces are preserved on taking retracts. Examples are properties like being [[simply connected space\|simply connected]], [[weakly contractible space\|weakly contractible]], [[contractible space\|contractible]], or having the [[fixed-point property]]. For a full list of such properties, refer: [[:Category:Retract-hereditary properties of topological spaces]]

Vipul: /* Weaker properties */

2007-10-27T22:34:58Z

Weaker properties

← Older revision		Revision as of 22:34, 27 October 2007
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	* [[Homologically injective subspace]]		* [[Homologically injective subspace]]
	* [[Weak retract]]		* [[Weak retract]]
			* [[Neighbourhood retract]]

	==Facts==		==Facts==

Vipul: /* Relation with other properties */

2007-10-27T22:34:17Z

Relation with other properties

← Older revision		Revision as of 22:34, 27 October 2007
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	* [[Homotopically injective subspace]]		* [[Homotopically injective subspace]]
	* [[Homologically injective subspace]]		* [[Homologically injective subspace]]
			* [[Weak retract]]

			==Facts==

			* In a [[Hausdorff space]], any retract is a [[closed subset]]
			* Any retract of a space with the [[fixed-point property]] also has the fixed-point property

Vipul: /* Relation with other properties */

2007-10-27T01:45:14Z

Relation with other properties

← Older revision		Revision as of 01:45, 27 October 2007
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	* [[Weak deformation retract]]		* [[Weak deformation retract]]
	* [[Deformation retract]]		* [[Deformation retract]]

			===Weaker properties===

			* [[Homotopically injective subspace]]
			* [[Homologically injective subspace]]

Vipul: /* Relation with other properties */

2007-09-30T17:25:04Z

Relation with other properties

← Older revision		Revision as of 17:25, 30 September 2007
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	* [[Homotopy retract]]		* [[Homotopy retract]]
			* [[Weak deformation retract]]
	* [[Deformation retract]]		* [[Deformation retract]]

Vipul at 02:33, 22 May 2007

2007-05-22T02:33:42Z

New page

{{toposubspace property}}

==Definition==

A subspace of a topological space is said to be a '''retract''' if there is a continuous map on the whole topological space that maps everything to within the subspace, and that is identity on the retract. In other words, there is a continuous idempotent map whose image-cum-fixed-point space is precisely the given subspace. Such a map is termed a [[retraction]].

==Facts==

Clearly the whole space is a retract of itself (the identity map being a retraction) and every one-point subspace is also a retract (the constant map to that one point being the retraction).

==Relation with other properties==

===Stronger properties===

* [[Homotopy retract]]
* [[Deformation retract]]