Example 3

Let $i$ be the canonical injection of A into B and let us take for the space $B'$ the cylinder of the mapping $i$; let $\alpha_i: A\times \mathbf{I}\rightarrow$ Cyl($i$) and $\beta_i: B\rightarrow$ Cyl($i$) be the canonical mappings and $r_i:$ Cyl($i$)$\rightarrow B$ the canonical retraction of the cylinder Cyl($i$) onto its base. Let $A_0$ be the subspace $\alpha_i(A\times \{0\})$ of Cyl($i$) and let $f_0: A_0\rightarrow X$ be the mapping $f\circ r_i$. For $a\in A$, we have

Let $\eta : X\cup_{f_0}$ Cyl($i$)$\rightarrow X\cup_fB$ be the unique continuous mapping such that $\eta \circ \alpha_{f_0}=\alpha_f$ and $\eta \circ \beta_{f_0}=\beta_f\circ r_i$. Suppose that the pair $(B,A)$ has the homotopy extension property. Then the mapping $r_i$ defines a homotopy equivalence of the pair (Cyl($i$)$,A_0$) with the pair $(B,A)$ (III, p. 243, th. 1). By Corollary 1, the mapping $\eta$ then defines a homotopy equivalence of the pair $(X\cup_{f'}B', \alpha_{f'}(X))$ with the pair $(X\cup_fB, \alpha_f(X))$. In particular, $\eta$ is a homotopy equivalence.