Exercise 25, § 1

(a) Let $X$ and $Y$ be topological spaces, let $A$ be a closed subspace of $X$ and let $B$ be a closed subspace of $Y$. Let $f: X\rightarrow Y$ be a continuous mapping; it is assumed that $f$ induces, by passing to the subspaces, a homeomorphism of $A$ onto $B$. If $f$ has a left inverse up to homotopy and if $(Y,B)$ has the property of extension of stationary homotopies (exer. 23), the same is true of $(X,A)$.

(b) If the pairs $(X,A)$ and $(Y,B)$ are homotopic, then $(X,A)$ has the property of extension of stationary homotopies if and only if the same is true of $(Y,B)$.

(c) Let C be the set of points of the plane of the form $(t/n, t)$, for $t\in \mathbf{I}$ and $n\in \mathbf{N}^*$, let X be its closure and let $A =\{0\} \times \mathbf{I}$. Prove that the pair $(X,A)$ does not have the property of extension of stationary homotopies.