Homotopy extension property of a pushout - converse statement

algebraic-topology

It turns out to be false.

Take $X$ and $A \subset X$ such that $(X,A)$ doesn't have HEP for $W$. Take $Y=\{*\}$. Then pair $(Z,Y)$ is cofibration, because for any given $\{*\} \times I \rightarrow W$ we can obtain $Z \times I \rightarrow W$ just by mapping $Z$ to $\{*\}$. So it gives a counterexample.

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