Proposition   Let $Z\subset Y\subset X$ be nested topological spaces. If there exist a deformation retraction 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 $\id_X$ and a retract $r:X\to Y$ of $X$ onto $Y$ Similarly, there is a homotopy $G:I\times Y\to Y$ between $\id_Y$ 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 $\widetilde{G}:I\times X\to X$ by $\widetilde{G}=iG(\id_I\times r)$ where $i:Y\hookrightarrow X$ is inclusion. Observe that

• $\widetilde{G}(0,x)=r(x)$ for any $x\in X$
• $\widetilde{G}(1,x)=sr(x)$ for any $x\in X$ and
• $\widetilde{G}(t,a)=a$ for any $a\in Z$

Hence $\widetilde{G}$ is a homotopy between the retractions $r$ and $sr$

Finally we must glue together the homotopies $F$ and $\widetilde{G}$ to get a homotopy between $\id_X$ and $sr$ To do this, define a function $H:I\times X\to X$ by $$ H(t,x)=\begin{cases} F(2t,x), & 0\le t\le\frac{1}{2} \\ \widetilde{G}(2t-1, x), & \frac{1}{2}\le t\le 1. \end{cases} $$ Since $F(1,x)=\widetilde{G}(0,x)=r(x)$ 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 $Z$