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.