deformation retract is transitive
Proposition.
Let $Z\mathrm{\subset}Y\mathrm{\subset}X$ be nested topological spaces^{}. If there exist a deformation retraction (http://planetmath.org/DeformationRetraction) of $X$ onto $Y$ and a deformation retraction of $Y$ onto $Z$, then there also exists a deformation retraction of $X$ onto $Z$. In other words, “being a deformation retract^{} of” is a transitive relation.
Proof.
Since $Y$ is a deformation retract of $X$, there is a homotopy^{} $F:I\times X\to X$ between ${\mathrm{id}}_{X}$ and a retract^{} $r:X\to Y$ of $X$ onto $Y$. Similarly, there is a homotopy $G:I\times Y\to Y$ between ${\mathrm{id}}_{Y}$ and a retract $s:Y\to Z$ of $Y$ onto $Z$.
First notice that since both $r$ and $s$ fix $Z$, the map $sr:X\to Z$ is a retraction.
Now define a map $\stackrel{~}{G}:I\times X\to X$ by $\stackrel{~}{G}=iG({\mathrm{id}}_{I}\times r)$, where $i:Y\hookrightarrow X$ is inclusion. Observe that

•
$\stackrel{~}{G}(0,x)=r(x)$ for any $x\in X$;

•
$\stackrel{~}{G}(1,x)=sr(x)$ for any $x\in X$; and

•
$\stackrel{~}{G}(t,a)=a$ for any $a\in Z$.
Hence $\stackrel{~}{G}$ is a homotopy between the retractions $r$ and $sr$.
Finally we must glue together the homotopies (http://planetmath.org/GluingTogentherContinuousFunctions) $F$ and $\stackrel{~}{G}$ to get a homotopy between ${\mathrm{id}}_{X}$ and $sr$. To do this, define a function $H:I\times X\to X$ by
$$H(t,x)=\{\begin{array}{cc}F(2t,x),\hfill & 0\le t\le \frac{1}{2}\hfill \\ \stackrel{~}{G}(2t1,x),\hfill & \frac{1}{2}\le t\le 1.\hfill \end{array}$$ 
Since $F(1,x)=\stackrel{~}{G}(0,x)=r(x)$, the gluing yieds a continuous map^{}. By construction,

•
$H(0,x)=x$ for all $x\in X$;

•
$H(1,x)=sr(x)$ for all $x\in X$; and

•
$H(t,a)=a$ for any $a\in Z$.
Hence $H$ is a homotopy between the identity map on $X$ and a retraction of $X$ onto $Z$. We conclude that $H$ is a deformation retraction of $X$ onto $Z$. ∎
Title  deformation retract is transitive 

Canonical name  DeformationRetractIsTransitive 
Date of creation  20130322 15:43:59 
Last modified on  20130322 15:43:59 
Owner  mps (409) 
Last modified by  mps (409) 
Numerical id  4 
Author  mps (409) 
Entry type  Result 
Classification  msc 55Q05 