I am trying to solve Exercise 0.26 from Hatcher's Algebraic Topology in a manner different from the hint in the book. It says that if $(X,A)$ satisfies the homotopy extension property, then $X \times \{0\} \cup A \times I$ is a deformation retract of $X \times I$. The hint is to use a corollary from the book.
I would like to solve this by directly using the homotopy extension property. In the end I need a map $X \times I \times I \to X \times I$ that is the identity on $X \times I \times \{0\}$, is the identity on $(X \times \{0\} \cup A \times I) \times I$, and is a retraction to $X \times \{0\} \cup A \times I$ on $X \times I \times \{1\}$. I can show that $(X \times I, X \times \{0\} \cup A \times I)$ has the homotopy extension property. So, what I need are maps $f_1: X \times I \to X \times I$ and $f_2 :(X \times \{0\} \cup A \times I) \times I \to X \times I$. Then homotopy extension kicks in to give me a map $H:X \times I \times I \to X \times I$. It seems that $f_2$ should just be projection, so that $H$ keeps $X \times \{0\} \cup A \times I$ fixed during the deformation retract. But, putting $f_1 := \operatorname{Id}$ doesn't work. Which map should it be? Thanks.
