Remark 1

Let X be a topological space and let A be a subset of X such that the pair $(X,A)$ possesses the homotopy extension property. Let Y be a topological space, let $f: X\rightarrow Y$ be a continuous mapping and let $\sigma : A\times \mathbf{I}\rightarrow Y$ be a homotopy whose origin is the mapping $f|A$. The mapping $\overline{\sigma}: A\times \mathbf{I}\rightarrow Y$ defined by $(a, t)\mapsto \sigma (a,1-t)$ is a homotopy with term $f|A$; let $\tau : X\times \mathbf{I}\rightarrow Y$ be a homotopy with term $f$ which extends $\overline{\sigma}$. The mapping $\overline{\tau}: X\times \mathbf{I}\rightarrow Y$ given by $(x, t)\mapsto \tau (x,1-t)$ is then a homotopy which extends $\sigma$ and whose origin is the mapping $f$.

Let X be a topological space, let A be a subspace of X and let $i: A\rightarrow X$ be the canonical injection. Let $\alpha_i: A\times \mathbf{I}\rightarrow$ Cyl($i$) and $\beta_i: X\rightarrow$ Cyl($i$) denote the canonical mappings. Let $j:$ Cyl($i$)$\rightarrow X\times \mathbf{I}$ be the unique continuous mapping such that $j(\alpha_i(a, s)) = (i(a), s)$ and $j(\beta_i(x)) = (x,1)$ for $a\in A,s\in \mathbf{I}$ and $x\in X$. It is injective; its image is the subspace $(A\times \mathbf{I})\cup (X\times \{1\})$ of $X\times \mathbf{I}$. The mapping $j$ is closed if A is closed in X. It is not always strict (III, p. 325, exerc. 17).