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Suppose $X$ and $Y$ are CW-complexes and $A\subset X$ and $B\subset Y$ have properties such that the product topologies of $X\times B$ and $A\times Y$ are CW-complexes, such as when $A$ and $B$ are locally compact. So complexes $X\times B \cup A\times Y$ have the homotopy extension property in $(X\times Y)_c$, but they're also spaces that are subtopologies of $X\times Y$, so must they have the HEP in $X\times Y$?

In particular I'd like to show it for the case of $(X\times X, X\times \{e\} \cup \{e\}\times X)$ where $e$ is a $0$-cell in $X$. The case $(X\times X, X\times \{e\})$ is easy enough from extending the retract for the pair $(X,e)$, but beyond that I'm pretty stuck.

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