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<div>{{redirect|Retract|other meanings including concepts in group theory and category theory|Retraction (disambiguation)}}<br />
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.<br />
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== Definitions ==<br />
=== Retract ===<br />
Let ''X'' be a [[topological space]] and ''A'' a [[subspace (topology)|subspace]] of ''X''. Then a continuous map <br />
<br />
:<math>r:X \to A</math><br />
<br />
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 <br />
<br />
:<math>\iota : A \hookrightarrow X</math><br />
<br />
the [[Inclusion map|inclusion]], a retraction is a continuous map ''r'' such that <br />
<br />
:<math>r \circ \iota = id_A,</math><br />
<br />
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.<br />
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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.<br />
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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.<br />
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=== Neighborhood retract ===<br />
If there exists an [[open set]] ''U'' such that <br />
<br />
:<math>A \subset U \subset X</math><br />
<br />
and ''A'' is a retract of ''U'', then ''A'' is called a '''neighborhood retract''' of ''X''.<br />
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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.<br />
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=== Deformation retract and strong deformation retract===<br />
A continuous map <br />
<br />
:<math>F:X \times [0, 1] \to X \, </math><br />
<br />
is a ''deformation retraction'' of a space ''X'' onto a subspace ''A'' if, for every ''x'' in ''X'' and ''a'' in ''A'',<br />
<br />
:<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><br />
<br />
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]].<br />
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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).<br />
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''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. <br />
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If, in the definition of a deformation retraction, we add the requirement that <br />
<br />
:<math>F(a,t) = a\,</math><br />
<br />
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.)<br />
<br />
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 <br />
:<math>F(x,t)=\left((1-t)+{t\over \|x\|}\right) x.</math><br />
<br />
===Neighborhood deformation retract===<br />
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<br />
<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<br />
<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><br />
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==Properties==<br />
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. <br />
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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><br />
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==Notes==<br />
{{Reflist}}<br />
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==External links==<br />
* {{PlanetMath attribution|id=6255|title=Neighborhood retract}}<br />
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[[Category:Topology]]</div>86.27.227.84