Show that if $A$ is contractible and $i:A \to X$ is a cofibration, then there exists a retraction from $X$ onto $A$.

Question

Let $X$ be a Hausdorff space and $A \subset X$ a subspace. Show that if $A$ is contractible and $i:A \to X$ is a cofibration, then there exists a retraction from $X$ onto $A$.

If $A$ is contractible, then there exists a homotopy $H:A \times I \to A$ with $H(a,0)=\operatorname{id}_A(a)=a$ and $H(a,1)=c_{a_0}(a)=a_0$.

Since $i:A\to X$ is a cofibration there exists a retraction $$r:X \times I \to X \times \{0\} \cup A\times I.$$

I thought that I could somehow compose $r$ and $H$ to obtain a map $X\to A$, but I can't figure this out. Any help?

Answer 1

Let $h\colon A\times I\to A$ be a homotopy such that $h(a,0) = a_0$ and $h(a,1) = a$ for all $a\in A$. Extend this to a homotopy $H\colon X\times I\to A$ such that $H(x,0) = a_0$ for all $x\in X$. Then $H(-,1)\colon X\to A$ is the desired retraction, since it extends $h(-,1) = \mathrm{id}_A$.