Contractibility is not closure-preserved - Revision history

Vipul at 15:27, 31 May 2016

2016-05-31T15:27:59Z

← Older revision		Revision as of 15:27, 31 May 2016
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	It is possible to have a [[topological space]] <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is a [[contractible space]] in the [[subspace topology]] from <math>X</math>, but <math>\overline{A}</math>, the closure of <math>A</math> in <math>X</math>, is not contractible.		It is possible to have a [[topological space]] <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is a [[contractible space]] in the [[subspace topology]] from <math>X</math>, but <math>\overline{A}</math>, the closure of <math>A</math> in <math>X</math>, is not contractible.

			== Related facts ==

			* [[Contractibility is not interior-preserved]]
			* [[Seifert-van Kampen theorem]]
			* [[Union of two simply connected open subsets with path-connected intersection is simply connected]]

	== Proof ==		== Proof ==

Vipul: /* Proof */

2016-05-31T15:27:03Z

Proof

← Older revision		Revision as of 15:27, 31 May 2016
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	Set <math>X</math> to be a [[circle]] and set <math>A</math> to be the complement of a single point in the circle. Then, <math>A</math> is homeomorphic to the real line, and is contractible. However, it closure <math>\overline{A}</math> equals <math>X</math>, which is not even simply connected, and therefore not contractible.		Set <math>X</math> to be a [[circle]] and set <math>A</math> to be the complement of a single point in the circle. Then, <math>A</math> is homeomorphic to the real line, and is contractible. However, it closure <math>\overline{A}</math> equals <math>X</math>, which is not even simply connected, and therefore not contractible.

	This example can be generalized to the <math>n</math>-[[sphere]] <math>S^n</math> for <math>n \ge 1</math>.		This example can be generalized to the <math>n</math>-[[sphere]] <math>S^n</math> for <math>n \ge 1</math>, where the complement of any point is homeomorphic to <math>\R^n</math> (via, for instance, a stereographic projection).

Vipul: Created page with "{{topospace metaproperty dissatisfaction| property = contractible space| metaproperty = closure-preserved property of topological spaces}} == Definition == It is possible to..."

2016-05-31T15:26:37Z

Created page with "{{topospace metaproperty dissatisfaction| property = contractible space| metaproperty = closure-preserved property of topological spaces}} == Definition == It is possible to..."

New page

{{topospace metaproperty dissatisfaction|
property = contractible space|
metaproperty = closure-preserved property of topological spaces}}

== Definition ==

It is possible to have a [[topological space]] <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is a [[contractible space]] in the [[subspace topology]] from <math>X</math>, but <math>\overline{A}</math>, the closure of <math>A</math> in <math>X</math>, is not contractible.

== Proof ==

Set <math>X</math> to be a [[circle]] and set <math>A</math> to be the complement of a single point in the circle. Then, <math>A</math> is homeomorphic to the real line, and is contractible. However, it closure <math>\overline{A}</math> equals <math>X</math>, which is not even simply connected, and therefore not contractible.

This example can be generalized to the <math>n</math>-[[sphere]] <math>S^n</math> for <math>n \ge 1</math>.