Difference between revisions of "Deformation retract"
From Encyclopedia of Mathematics
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| − | A subset $A\subset X$ with the following property: There exists a homotopy of the identity mapping of $X$ to some mapping $X\to A$ under which all points of the set $A$ remain fixed. If, under the homotopy, the points from $X\setminus A$ remain in $X\setminus A$, $A$ is known as a strong deformation retract. A deformation retract of a space $X$ has the same [[homotopy type]] as does $X$. See also [[Retract]]; [[Retract of a topological space]]. | + | A subset $A\subset X$ with the following property: There exists a [[homotopy]] of the identity mapping of $X \to X$ to some mapping $X\to A$ under which all points of the set $A$ remain fixed. If, under the homotopy, the points from $X\setminus A$ remain in $X\setminus A$, $A$ is known as a strong deformation retract. A deformation retract of a space $X$ has the same [[homotopy type]] as does $X$. See also [[Retract]]; [[Retract of a topological space]]. |
[[Category:Algebraic topology]] | [[Category:Algebraic topology]] | ||
Latest revision as of 19:45, 5 February 2016
of a topological space $X$
A subset $A\subset X$ with the following property: There exists a homotopy of the identity mapping of $X \to X$ to some mapping $X\to A$ under which all points of the set $A$ remain fixed. If, under the homotopy, the points from $X\setminus A$ remain in $X\setminus A$, $A$ is known as a strong deformation retract. A deformation retract of a space $X$ has the same homotopy type as does $X$. See also Retract; Retract of a topological space.
How to Cite This Entry:
Deformation retract. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Deformation_retract&oldid=37658
Deformation retract. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Deformation_retract&oldid=37658
This article was adapted from an original article by E.G. Sklyarenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article