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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.