Pg 269, 7.2.5 lemma Let $i:A \rightarrow X$ be a cofibration. Let $H:X \times \mathbb{I} \rightarrow Y$ be a homotopy $f \simeq g$, and let $\mathcal{G}$ be a homotopy rel end maps $G=H(i \times 1)$ to a homotopy $G' : u \simeq v$. Then $H$ is homotopic rel end maps to a homotopy $H'$ such that $H'(i \times 1)=G'$.
I am struggling to understand the proof :
- What is the purpose of this step.
$W$ is a retract of $Z=X \times \mathbb{I} \times \mathbb{I}$.
EDIT: From comment, the only thing we need is any map on $W$ extends to a map on $Z$. This is proven with out using the fact $W$ is a retract.
- Why this argument works:
$T \cong X \times \mathbb{I} \times 0$, implies $T$ and $A \times \mathbb{I} \times \mathbb{I}$ have the gluing property?
I couldn't construct the diagram chase from gluing property with $X \times \mathbb{I} \times 0$. In fact, I thought $T$ and $A \times \mathbb{I} \times \mathbb{I}$ satisfy gluing property simply being closed subspaces of their union.