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Let $X$ be an arbitrary topological space and $A$ a subspace such that there exists a (strong) deformation retraction of $X$ to $A$.

Does it follow that $(X,A)$ has the homotopy extension property? If not, what are some nice counter-examples?

I suspect this might be a helpful fact to know when solving various exercises in algebraic topology. I can't think of an easy argument in any direction, so any thoughts would be welcome.

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This is almost true. What you need to additionally assume is that $A$ is a zero set, i.e. $A=\varphi^{-1}[\{0\}]$ for some continuous $\varphi\colon X\to [0,1]$. Since your $A$ is presumably closed, this will always be true in reasonable spaces (a strong separation axiom is needed which is true e.g. in metric spaces or locally finite CW complexes).

There is the following characterization: if $A\subseteq X$ is closed, then $(X,A)$ has the homotopy extension property iff $A$ is a zero set and some neighbourhood of $A$ strongly deformation retracts onto it. This is theorem 1 of this. It is also covered in May's A Concise Course

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    For posterity, this is the theorem on p. 45 of May’s$A$Concise Course.2018-11-11