Remark 2

Let X be a separated topological space and let A be a subset of X such that the pair $(X,A)$ has the homotopy extension property. Let us denote by $i: A\rightarrow$ X the canonical injection. Let $j:$ Cyl($i$)$\rightarrow X\times \mathbf{I}$ be the canonical injection and let $r$ be a continuous retraction of the mapping $j$. The subspace $j$(Cyl($i$)) is equal to the set of pairs $(x, t)\in X\times \mathbf{I}$ such that $j(r(x, t)) = (x, t)$. Since the space $X\times \mathbf{I}$ is separated, the subset $j$(Cyl($i$)) is closed in $X\times \mathbf{I}$. The set A, equal to the set of $x\in X$ such that $(x,0)\in j$(Cyl($i$)), is then a closed subset of X.