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{{redirect|Retract|other meanings including concepts in group theory and category theory|Retraction (disambiguation)}}
== from His Holiness to the Lord of the universe ==
In [[topology]], a branch of mathematics, a '''retraction''',<ref>{{cite journal|title=Sur les rétractes|author=K. Borsuk|journal=Fund. Math.|volume=17|year=1931|pages=2–20}}</ref> is a continuous mapping from the entire [[space (mathematics)|space]] into a [[Subspace topology|subspace]] which preserves the position of all points in that subspace. A '''deformation retraction''' is a [[function (mathematics)|map]] which captures the idea of ''[[continuous function|continuously]] shrinking'' a space into a subspace.


== Definitions ==
In East Timor shrine strong clouds can only be regarded as common,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_46.htm オークリー サングラス 登山], is the 'main quasi Universe,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_29.htm ゴルフ サングラス オークリー],' a sacred place ...... because of the limited universe withstand East Timor,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_25.htm オークリー レディース サングラス], in the case of the presence of so many of the Lord of the universe,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_52.htm オークリー サングラス ジョウボーン], the universe they are a group of quasi- Lord simply can not break through,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_56.htm オークリーサングラス画像], not the supply of holy origin of the universe to break.<br><br>from His Holiness to the Lord of the universe,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_56.htm オークリーサングラス画像], the divine power consumption is extremely alarming. No source supply is impossible.<br><br>His Holiness ...... Treasure of the Sierra<br><br>from such an embarrassing primary identity quasi universe, all of a sudden become the most dazzling East Timor holy status of high ...... even more than any master universe,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_73.htm オークリー サングラス 人気], the universe strongest! East Timor ancestor shrine even met him personally, and he even said: 'rhino Wong Anatomy Board also proved your potential ...... even if you never get this heritage, and will also get my holy best cultivation. You do not need to try to pressure ......! '<br><br>'If the performance is good enough for you! unlimited potential,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_52.htm オークリー サングラス ジョウボーン]!'
=== Retract ===
相关的主题文章：
Let ''X'' be a [[topological space]] and ''A'' a [[subspace (topology)|subspace]] of ''X''. Then a continuous map
<ul>
 
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</ul>


:<math>r:X \to A</math>
== even immediately salute. Other Peng work of the Lord ==


is a '''retraction''' if the [[function (mathematics)#Restrictions and extensions|restriction]] of ''r'' to ''A'' is the [[Identity function|identity map]] on ''A''; that is, ''r''(''a'') = ''a'' for all ''a'' in ''A''. Equivalently, denoting by
A point, continue to climb up, the ultimate goal is - gold, wood, water, fire,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_30.htm オークリーサングラス カスタム], earth, wind, lightning, light,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_6.htm オークリー 自転車 サングラス], time, space, this ten basic rules, one to the plenary, two to do are integrated into 'Chaos Theory'. 'Feng Luo His words, revealing the presence of all color shocked, shocked even the ax founders stand up.<br><br>even across founders Theme ax,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_12.htm スポーツサングラス オークリー], a white robe figure appeared out of thin air on another Theme.<br><br>'You say Chaos Theory?' Luo Feng stared at the man in white robes,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_34.htm オークリー サングラス フロッグスキン], with eyes ever light.<br><br>'original ancestors!' ax founders surprised.<br><br>'teacher,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_4.htm 登山 サングラス オークリー].'<br><br>chaos, darkness,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_76.htm オークリー サングラス 調整], and the thrill of climbing three of them surprised, even immediately salute. Other Peng work of the Lord, the Lord of the East Green,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_77.htm オークリーサングラス画像], virtual gold Lord, Lord Dragon, the shortage of primary discriminator is excited,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_52.htm サングラスケース オークリー], are even opening shouted: '! Original ancestors.'<br><br>Luo Feng also amazed watching this white robe man ...... he is passing
 
相关的主题文章：
:<math>\iota : A \hookrightarrow X</math>
<ul>
 
 
the [[Inclusion map|inclusion]], a retraction is a continuous map ''r'' such that
  <li>[http://www.morgen-platz.ch/cgi-bin/glight.cgi http://www.morgen-platz.ch/cgi-bin/glight.cgi]</li>
 
 
:<math>r \circ \iota = id_A,</math>
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that is, the composition of ''r'' with the inclusion is the identity of ''A''. Note that, by definition, a retraction maps ''X'' [[onto]] ''A''. A subspace ''A'' is called a '''retract''' of ''X'' if such a retraction exists. For instance, any space retracts to a point in the obvious way (the constant map yields a retraction). If ''X'' is [[Hausdorff space|Hausdorff]], then ''A'' must be closed.
  <li>[http://forum.radio4.ru/cgi-bin/search.cgi http://forum.radio4.ru/cgi-bin/search.cgi]</li>
 
 
If <math>r:X \to A</math> is a retraction, then the composition <math>\iota \circ r</math> is an [[idempotent]] continuous map from ''X'' to ''X''. Conversely, given any idempotent continuous map <math>s:X\to X</math>, we obtain a retraction onto the image of ''s'' by restricting the codomain.
</ul>
 
A space ''X'' is known as an '''absolute retract''' if for every [[normal space]] ''Y'' that contains ''X'' as a closed subspace, ''X'' is a retract of ''Y''. The [[unit cube]] ''I<sup>n</sup>'' as well as the [[Hilbert cube]] ''I<sup>ω</sup>'' are absolute retracts.
 
=== Neighborhood retract ===
If there exists an [[open set]] ''U'' such that
 
:<math>A \subset U \subset X</math>
 
and ''A'' is a retract of ''U'', then ''A'' is called a '''neighborhood retract''' of ''X''.
 
A space ''X'' is an '''absolute neighborhood retract''' (or '''ANR''') if for every normal space ''Y'' that embeds ''X'' as a closed subset, ''X'' is a neighborhood retract of ''Y''. The ''n''-sphere ''S<sup>n</sup>'' is an absolute neighborhood retract.
 
=== Deformation retract and strong deformation retract===
A continuous map
 
:<math>F:X \times [0, 1] \to X \, </math>
 
is a ''deformation retraction'' of a space ''X'' onto a subspace ''A'' if, for every ''x'' in ''X'' and ''a'' in ''A'',
 
:<math> F(x,0) = x, \; F(x,1) \in A ,\quad \mbox{and} \quad F(a,1) = a \mbox{ for every } a \in A .</math>
 
In other words, a deformation retraction is a [[homotopy]] between a retraction and the identity map on ''X''. The subspace ''A'' is called a '''deformation retract''' of ''X''. A deformation retraction is a special case of [[homotopy equivalence]].
 
A retract need not be a deformation retract. For instance, having a single point as a deformation retract would imply a space is path connected (in fact, it would imply contractibility of the space).
 
''Note:'' An equivalent definition of deformation retraction is the following. A continuous map ''r'': ''X'' → ''A'' is a deformation retraction if it is a retraction and its composition with the inclusion is homotopic to the identity map on ''X''. In this formulation, a deformation retraction carries with it a homotopy between the identity map on ''X'' and itself.
 
If, in the definition of a deformation retraction, we add the requirement that
 
:<math>F(a,t) = a\,</math>
 
for all ''t'' in [0, 1], ''F'' is called a '''strong deformation retraction'''. In other words, a strong deformation retraction leaves points in ''A'' fixed throughout the homotopy. (Some authors, such as [[Allen Hatcher]], take this as the definition of deformation retraction.)
 
As an example, the [[n-sphere|''n''-sphere]] ''S<sup>n</sup>'' is a strong deformation retract of '''R'''<sup>''n''+1</sup>\{0}; as strong deformation retraction one can choose the map
:<math>F(x,t)=\left((1-t)+{t\over \|x\|}\right) x.</math>
 
===Neighborhood deformation retract===
A closed subspace ''A'' is a '''neighborhood deformation retract''' of ''X'' if there exists a continuous map <math>u:X \rightarrow I</math> (where <math>I=[0,1]</math>) such that <math>A = u^{-1} (0)</math> and a homotopy
<math>H:X\times I\rightarrow X</math> such that <math>H(x,0) = x</math> for all <math>x \in X</math>, <math>H(a,t) = a</math> for all
<math>(a,t) \in A\times I</math>, and <math>h(x,1) \in A</math> for all <math>x \in u^{-1} [ 0 , 1)</math>.<ref name='steenrod'>{{cite journal | journal= Michigan Math. J. | last1=Steenrod | first1=N. E. | title=A convenient category of topological spaces | volume=14 | issue=2 | year=1967 | pages=133–152}}</ref>
 
==Properties==
Deformation retraction is a particular case of homotopy equivalence. In fact, two spaces are [[homotopy equivalent]] [[if and only if]] they are both deformation retracts of a single larger space.  
 
Any topological space which deformation retracts to a point is [[contractible space|contractible]] and vice versa. However, there exist contractible spaces which do not strongly deformation retract to a point.<ref name='hatcher'>{{Citation | last1=Hatcher | first1=Allen | title=Algebraic topology | url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html | publisher=[[Cambridge University Press]] | isbn=978-0-521-79540-1 | year=2002}}</ref>
 
==Notes==
{{Reflist}}
 
==External links==
* {{PlanetMath attribution|id=6255|title=Neighborhood retract}}
 
[[Category:Topology]]

Latest revision as of 14:48, 28 August 2014

from His Holiness to the Lord of the universe

In East Timor shrine strong clouds can only be regarded as common,オークリー サングラス 登山, is the 'main quasi Universe,ゴルフ サングラス オークリー,' a sacred place ...... because of the limited universe withstand East Timor,オークリー レディース サングラス, in the case of the presence of so many of the Lord of the universe,オークリー サングラス ジョウボーン, the universe they are a group of quasi- Lord simply can not break through,オークリーサングラス画像, not the supply of holy origin of the universe to break.

from His Holiness to the Lord of the universe,オークリーサングラス画像, the divine power consumption is extremely alarming. No source supply is impossible.

His Holiness ...... Treasure of the Sierra

from such an embarrassing primary identity quasi universe, all of a sudden become the most dazzling East Timor holy status of high ...... even more than any master universe,オークリー サングラス 人気, the universe strongest! East Timor ancestor shrine even met him personally, and he even said: 'rhino Wong Anatomy Board also proved your potential ...... even if you never get this heritage, and will also get my holy best cultivation. You do not need to try to pressure ......! '

'If the performance is good enough for you! unlimited potential,オークリー サングラス ジョウボーン!' 相关的主题文章：

even immediately salute. Other Peng work of the Lord

A point, continue to climb up, the ultimate goal is - gold, wood, water, fire,オークリーサングラス カスタム, earth, wind, lightning, light,オークリー 自転車 サングラス, time, space, this ten basic rules, one to the plenary, two to do are integrated into 'Chaos Theory'. 'Feng Luo His words, revealing the presence of all color shocked, shocked even the ax founders stand up.

even across founders Theme ax,スポーツサングラス オークリー, a white robe figure appeared out of thin air on another Theme.

'You say Chaos Theory?' Luo Feng stared at the man in white robes,オークリー サングラス フロッグスキン, with eyes ever light.

'original ancestors!' ax founders surprised.

'teacher,登山 サングラス オークリー.'

chaos, darkness,オークリー サングラス 調整, and the thrill of climbing three of them surprised, even immediately salute. Other Peng work of the Lord, the Lord of the East Green,オークリーサングラス画像, virtual gold Lord, Lord Dragon, the shortage of primary discriminator is excited,サングラスケース オークリー, are even opening shouted: '! Original ancestors.'

Luo Feng also amazed watching this white robe man ...... he is passing 相关的主题文章：

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