deformation retract

Let $X$ and $Y$ be topological spaces such that $Y\\subset X$ .A deformation retract of $X$ onto $Y$ is a collection ofmappings $f_{t}:X\\rightarrow X$ , $t\\in[0,1]$ such that

1.
$f_{0}=id_{X}$ , the identity mapping on $X$ ,
2.
$f_{1}(X)\\subseteq Y$ ,
3.
$Y$ is a retract of $X$ via $f_{1}$ (that is, $f_{1}$ restricted to $Y$ is the identity on $Y$ )
4.
the mapping $X\\times I\\rightarrow X$ , $(x,t)\\mapsto f_{t}(x)$ is continuous, where the topology on $X\\times I$ is the product topology.

Of course, by condition 3, condition 2 can be improved: $f_{1}(X)=Y$ .

A deformation retract is called a strong deformation retract if condition 3 above is replaced by a stronger form: $Y$ is a retract of $X$ via $f_{t}$ for every $t\\in[0,1]$ .

Properties

•
Let $X$ and $Y$ be as in the above definition. Then a collection ofmappings $f_{t}:X\\rightarrow X$ , $t\\in[0,1]$ isa deformation retract (of $X$ onto $Y$ ) if and only if it is ahomotopy (http://planetmath.org/HomotopyOfMaps) rel Y $\\operatorname{id}_{X}$ and some retraction $r$ of $X$ onto $Y$ .

Examples

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If $x_{0}\\in\\mathbb{R}^{n}$ , then $f_{t}(x)=(1-t)x+tx_{0}$ , $x\\in\\mathbb{R}^{n}$ shows that $\\mathbb{R}^{n}$ deformation retracts onto $\\{x_{0}\\}$ .Since $\\{x_{0}\\}\\subset\\mathbb{R}^{n}$ ,it follows that deformation retract is not an equivalence relation.
•
The same map as in the previous example can be used to deformation retract any star-shaped set in $\\mathbb{R}^{n}$ onto a point.
•
we obtain adeformation retraction of $\\mathbb{R}^{n}\\backslash\\{0\\}$ onto the http://planetmath.org/node/186 $(n-1)$ -sphere $S^{n-1}$ by setting
$f_{t}(x)=(1-t)x+t\\displaystyle{\\frac{x}{||x||}},$
where $x\\in\\mathbb{R}^{n}\\backslash\\{0\\}$ , $n>0$ ,
•
The http://planetmath.org/node/3278Möbius strip deformation retracts onto the circle $S^{1}$ .
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The $2$ -torus with one point removed deformation retracts ontotwo copies of $S^{1}$ joined at one point. (The circles can bechosen to be longitudinal and latitudinal circles of the torus.)
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The characters E,F,H,K,L,M,N, and T all deformation retract onto the charachter I, while the letter Q deformation retracts onto the letter O.