Remark 1

Let X be a topological space and let A be a subspace of X; let $i$ denote the canonical injection of A into X. The canonical retraction $r_i$ of the cylinder Cyl($i$) onto its base maps the subspace $\alpha_i(A\times \{0\})$ into A. Let $\rho :$ Cone$(i)\rightarrow X/A$ be the continuous mapping deduced from it by passing to the quotient spaces. Suppose that the pair $(X,A)$ has the homotopy extension property. By III, p. 243, Theorem 1, the mapping $r_i$ defines a homotopy equivalence of the pair (Cyl($i$)$, \alpha_i(A\times \{0\})$) with the pair $(X,A)$. By Remark 4 of III, p. 253, the mapping $\rho$ is then a homotopy equivalence of the pair (Cone$(i), s$) with the pair $(X, s_{X/A})$. In particular it is a homotopy equivalence.