Remark

Let X be a topological space and let A be a subset of X.

1) If the canonical mapping of X onto $X/A$ is a homotopy equivalence, there exists a homotopy $\sigma : X\times \mathbf{I}\rightarrow X$ whose initial mapping is Id$_X$ and which is constant on $A\times \{1\}$. But it may happen that there exists no such homotopy satisfying moreover the condition $\sigma (A\times \mathbf{I})\subset A$ (III, p. 322, exerc. 4).

2) There may exist a homotopy joining the identity mapping of X to a mapping of X into X which is constant on A, without the spaces X and $X/A$ being homotopy equivalent (III, p. 325, exerc. 14).

3) Suppose that A is contractible. If the pair $(X,A)$ has the homotopy extension property, the canonical mapping of X onto $X/A$ is a homotopy equivalence. Let in fact $\sigma : A\times \mathbf{I}\rightarrow A$ be a homotopy whose initial mapping is the identity mapping of A and whose terminal mapping is a constant mapping. There then exists a homotopy $\sigma '$ with initial mapping Id$_X$ which extends $\sigma$. The assertion then follows from proposition 10.

4) Let X and Y be topological spaces, let A be a nonempty subspace of X and let B be a subspace of Y. Let $f: X\rightarrow Y$ be a continuous mapping such that $f(A)\subset B$. Let $\varphi : X/A\rightarrow Y/B$ denote the continuous mapping deduced from $f$ by passing to quotients. If $f$ defines a homotopy equivalence of the pair $(X,A)$ onto the pair $(Y,B)$, the mapping $\varphi$ is a homotopy equivalence of the pair $(X/A, s_{X/A})$ onto the pair $(Y/B, s_{Y/B})$. Let P be a topological space reduced to a point, and let $\alpha$ and $\beta$ be the constant mappings of A and B into P. Observe that the mapping $\varphi$ is identified with the mapping of $P\cup_{\alpha}X$ into $P\cup_{\beta}Y$ deduced from $f$ and the identity mapping of P. The assertion then follows from corollary 1 of III, p. 249.