Persistence length - Revision history

en>Arrow50311 at 22:22, 19 August 2014

2014-08-19T22:22:42Z

← Older revision		Revision as of 00:22, 20 August 2014
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	~~In [[mathematics]], in~~ the ~~area of [[algebraic topology]]~~, ~~the '''homotopy extension property''' indicates which~~ [~~[homotopy\|homotopies]] defined on a [[subspace topology\|subspace]] can be extended to a homotopy defined on a larger space.~~		Hi there. Allow me start by introducing the writer, her title is Sophia Boon but she psychic phone ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 Click at fashionlinked.com]) by no means truly favored that name. Doing ballet is something she would by no means give up. Alaska is the only location I've been residing in but now I'm contemplating other best psychic readings ([http://chungmuroresidence.com/xe/reservation_branch2/152663 Click at fashionlinked.com]) options. Since he was eighteen he's been working as an info officer but he plans on altering it.<br><br>Here is my web page :: love psychic readings ([http://fashionlinked.com/index.php?do=/profile-13453/info/ click the next site])

	~~==Definition==~~

	~~Let <math>X\,\!<~~/~~math> be a [[topological space]], and let <math>A \subset X<~~/~~math>~~.
	~~We say that the pair <math>(X,A)\,\!<~~/~~math> has the '''homotopy extension property''' if, given a homotopy <math>f_t\colon A \rightarrow Y</math> and a map <math>F_0\colon X \rightarrow Y</math> such that <math>F_0 \|_A~~ = ~~f_0</math>, there exists an ''extension'' of <math>F_0</math> to a homotopy <math>F_t\colon X \rightarrow Y</math> such that~~
	~~<math>F_t\|_A~~ = ~~f_t</math>~~. ~~<ref>A~~. ~~Dold, ''Lectures on Algebraic Topology'', pp~~. ~~84, Springer ISBN 3-540-58660-1</ref>~~

	~~That~~ is, the ~~pair <math>(X,A)\,\!</math> has the homotopy extension property if any map~~
	~~<math>G\colon (X\times \{0\} \cup A\times~~ I~~) \rightarrow Y</math>~~
	~~can be extended to a map <math>G~~'~~\colon X\times~~ I ~~\rightarrow Y</math>~~ (~~i.e. <math>G\,\!<~~/~~math> and <math>G'\,\!<~~/~~math> agree on their common domain)~~.

	~~If the pair has this property only for a certain [[codomain]] <math>Y\,\!<~~/~~math>, we say that <math>(X,A)\,\!<~~/~~math> has the homotopy extension property with respect to <math>Y\,\!<~~/~~math>~~.

	~~==Visualisation==~~
	~~The homotopy extension property is depicted in the following diagram~~

	~~[[Image:Homotopy_extension_property.svg\|175px\|center~~]]

	~~If the above diagram (without the dashed map~~) ~~commutes, which is equivalent to the conditions above, then there exists a map <math> \tilde{f}</math> which makes the diagram commute~~. ~~By [[currying]], note that a map <math> \tilde{f} \colon X \to Y^I</math> is the same~~ as ~~a map <math> \tilde{f} \colon X\times I \to Y </math>~~.

	~~Also compare this to the visualization of the [[Homotopy_lifting_property#Formal_definition\|homotopy lifting property]].~~

	~~==Properties==~~
	* If <~~math~~>~~X\,\!~~<~~/math~~> is ~~a [[cell complex]] and <math>A\,\!</math> is a subcomplex of <math>X\,\!</math>, then the pair <math>(X,A)\,\!</math> has the homotopy extension property.~~

	* A pair <math>(X,A)\,\!</math> has the homotopy extension property if and only if <math>(X\times \{0\} \cup A\times I)</math> is a [[Deformation retract\|retract]] of <math>X\times I.</math>

	~~==Other==~~

	~~If <math>\mathbf{\mathit{(X,A)}}</math> has the homotopy extension property, then the simple inclusion map <math>i~~: ~~A \to X</math> is a [[cofibration]].~~

	~~In fact, if you consider any [[cofibration]] <math>i~~: Y \to Z</math>, then we have that <math>\mathbf{\mathit{Y}}</math> is [[homeomorphic]] to its image under <math>\mathbf{\mathit{i}}</math>. This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.

	~~==See also==~~
	* [[~~Homotopy lifting property]]~~

	~~==References==~~
	~~{{Reflist}}~~

	*{{cite book \| first = Allen \| last = Hatcher \| authorlink = Allen Hatcher \| year = 2002 \| title = Algebraic Topology \| publisher = Cambridge University Press \| isbn = 0-521-79540-0 \| url = http://~~www~~.~~math~~.~~cornell.edu~~/~~~hatcher~~/AT/~~ATpage.html}}~~

	* {{planetmath reference\|id=1600\|title=Homotopy extension property}}

	~~[[Category:Homotopy theory]]~~
	~~[[Category:Algebraic topology]~~]

5.147.205.155: removed double use and left the specific one for semantical clarity

2014-01-23T12:15:58Z

removed double use and left the specific one for semantical clarity

← Older revision		Revision as of 14:15, 23 January 2014
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	~~Alyson Meagher~~ is the ~~title her mothers~~ and ~~fathers gave her but she doesn~~'~~t like when people use her complete name~~. ~~Office supervising~~ is ~~my profession~~. ~~Doing ballet~~ is ~~something she would never give up~~. I~~'ve always cherished residing in Alaska~~.<br><br>~~Feel free~~ to ~~surf~~ to ~~my web site: tarot card readings (~~[http://~~help~~.~~ksu~~.edu~~.sa~~/~~node~~/~~65129 ksu~~.~~edu.sa~~])		In [[mathematics]], in the area of [[algebraic topology]], the '''homotopy extension property''' indicates which [[homotopy\|homotopies]] defined on a [[subspace topology\|subspace]] can be extended to a homotopy defined on a larger space.

			==Definition==

			Let <math>X\,\!</math> be a [[topological space]], and let <math>A \subset X</math>.
			We say that the pair <math>(X,A)\,\!</math> has the '''homotopy extension property''' if, given a homotopy <math>f_t\colon A \rightarrow Y</math> and a map <math>F_0\colon X \rightarrow Y</math> such that <math>F_0 \|_A = f_0</math>, there exists an ''extension'' of <math>F_0</math> to a homotopy <math>F_t\colon X \rightarrow Y</math> such that
			<math>F_t\|_A = f_t</math>. <ref>A. Dold, ''Lectures on Algebraic Topology'', pp. 84, Springer ISBN 3-540-58660-1</ref>

			That is, the pair <math>(X,A)\,\!</math> has the homotopy extension property if any map
			<math>G\colon (X\times \{0\} \cup A\times I) \rightarrow Y</math>
			can be extended to a map <math>G'\colon X\times I \rightarrow Y</math> (i.e. <math>G\,\!</math> and <math>G'\,\!</math> agree on their common domain).

			If the pair has this property only for a certain [[codomain]] <math>Y\,\!</math>, we say that <math>(X,A)\,\!</math> has the homotopy extension property with respect to <math>Y\,\!</math>.

			==Visualisation==
			The homotopy extension property is depicted in the following diagram

			[[Image:Homotopy_extension_property.svg\|175px\|center]]

			If the above diagram (without the dashed map) commutes, which is equivalent to the conditions above, then there exists a map <math> \tilde{f}</math> which makes the diagram commute. By [[currying]], note that a map <math> \tilde{f} \colon X \to Y^I</math> is the same as a map <math> \tilde{f} \colon X\times I \to Y </math>.

			Also compare this to the visualization of the [[Homotopy_lifting_property#Formal_definition\|homotopy lifting property]].

			==Properties==
			* If <math>X\,\!</math> is a [[cell complex]] and <math>A\,\!</math> is a subcomplex of <math>X\,\!</math>, then the pair <math>(X,A)\,\!</math> has the homotopy extension property.

			* A pair <math>(X,A)\,\!</math> has the homotopy extension property if and only if <math>(X\times \{0\} \cup A\times I)</math> is a [[Deformation retract\|retract]] of <math>X\times I.</math>

			==Other==

			If <math>\mathbf{\mathit{(X,A)}}</math> has the homotopy extension property, then the simple inclusion map <math>i: A \to X</math> is a [[cofibration]].

			In fact, if you consider any [[cofibration]] <math>i: Y \to Z</math>, then we have that <math>\mathbf{\mathit{Y}}</math> is [[homeomorphic]] to its image under <math>\mathbf{\mathit{i}}</math>. This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.

			==See also==
			* [[Homotopy lifting property]]

			==References==
			{{Reflist}}

			*{{cite book \| first = Allen \| last = Hatcher \| authorlink = Allen Hatcher \| year = 2002 \| title = Algebraic Topology \| publisher = Cambridge University Press \| isbn = 0-521-79540-0 \| url = http://www.math.cornell.edu/~hatcher/AT/ATpage.html}}

			* {{planetmath reference\|id=1600\|title=Homotopy extension property}}

			[[Category:Homotopy theory]]
			[[Category:Algebraic topology]]

en>Saeedghalati at 08:02, 30 July 2012

2012-07-30T08:02:54Z

New page

Alyson Meagher is the title her mothers and fathers gave her but she doesn't like when people use her complete name. Office supervising is my profession. Doing ballet is something she would never give up. I've always cherished residing in Alaska.<br><br>Feel free to surf to my web site: tarot card readings ([http://help.ksu.edu.sa/node/65129 ksu.edu.sa])