Homotopy extension property

Remarks

If $A$ is closed in $X$, then $(X, A)$ has the homotopy extension property if and only if $X \times \set{0} \cup A \times I$ is a retract of $X \times I$. Example: $(D^n, S^{n-1})$ has it, because the radial projection from the point $(0, \ldots, 0, 2)$ of $\R^{n+1}$ gives a retraction of $D^n \times I$ onto $D^n \times \set{0} \cup S^{n-1} \times I$. Non-example: let $X = \set{0} \cup \set{1/n : n \ge 1}$ and $A = \set{0}$. A retraction $r$ of $X \times I$ onto $X \times \set{0} \cup A \times I$ would fix $(0, 1)$, and would send each connected set $\set{1/n} \times I$ to $(1/n, 0)$. Then $r(1/n, 1) = (1/n, 0) \to (0, 0) \neq r(0, 1)$, so $r$ is not continuous and the pair fails the property. Every CW pair has the homotopy extension property.