Fixed-point property is retract-hereditary - Revision history

Vipul at 11:22, 8 August 2008

2008-08-08T11:22:45Z

← Older revision		Revision as of 11:22, 8 August 2008
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	{{topospace metaproperty satisfaction}}		{{topospace metaproperty satisfaction\|
			property = fixed-point property\|
			metaproperty = retract-hereditary property of topological spaces}}

	==Statement==		==Statement==

Vipul at 20:00, 20 July 2008

2008-07-20T20:00:25Z

← Older revision		Revision as of 20:00, 20 July 2008
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	* Compose with the retraction to get a self-map of the whole space		* Compose with the retraction to get a self-map of the whole space
	* Find a fixed point, and observe that it must be a fixed point of the original self-map		* Find a fixed point, and observe that it must be a fixed point of the original self-map

			===Proof details===

			'''Given''': A topological space <math>X</math> satisfying the fixed-point property, a retraction <math>r:X \to A</math> where <math>A \subset X</math> and <math>r(a) = a</math> for all <math>a \in A</math>

			'''To prove''': <math>A</math> satisfies the fixed-point property

			'''Proof''': Let <math>i</math> denote the inclusion of <math>A</math> in <math>X</math>.

			Consider any continuous map <math>f:A \to A</math>. We need to show that <math>f</math> has a fixed point in <math>A</math>. Consider the composition <math>g = i \circ f \circ r</math>. This is a map from <math>X</math> to <math>X</math> that first retracts to <math>A</math>, then applies <math>f</math>, and then views the resulting point of <math>A</math> as a point in <math>X</math>. <math>g</math> is a composite of continuous maps, so <math>g</math> is continuous. Since <math>X</math> has the fixed-point property, there exists <math>x \in X</math> such that <math>g(x) = x</math>.

			But by construction, <math>g(x)</math> is actually inside <math>A</math>, so in fact <math>x \in A</math>. But if <math>x \in A</math>, <math>r(x) = x</math>, so we conclude that <math>x = g(x) = f(r(x)) = f(x)</math>. Thus, <math>x \in A</math> is a fixed point of <math>f</math>, completing the proof.

Vipul: 1 revision

2008-05-11T19:44:59Z

1 revision

← Older revision	Revision as of 19:44, 11 May 2008
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Vipul at 01:22, 27 December 2007

2007-12-27T01:22:54Z

New page

{{topospace metaproperty satisfaction}}

==Statement==

===Property-theoretic statement===

The [[property of topological spaces]] called the [[fixed-point property]] is a [[retract-hereditary property of topological spaces]].

===Verbal statement===

Any [[retract]] of a [[topological space]] having the [[fixed-point property]], also has the fixed-point property.

==Definitions used==

===Fixed-point property===

===Retract===

===Subspace topology===

==Proof==

===Proof outline===

* Consider a self-map of the [[retract]]
* Compose with the retraction to get a self-map of the whole space
* Find a fixed point, and observe that it must be a fixed point of the original self-map