Exercise 22, § 1

Let X and Y be topological spaces, let A be a subspace of X and let B be a subspace of Y.

(a) Suppose that the pairs $(X,A)$ and $(Y,B)$ are homeotopic. If the pair $(X,A)$ has the property of extension of homotopies, prove that the pair $(Y,B)$ has it analogously.

(b) Conversely suppose that the pairs $(X,A)$ and $(Y,B)$ have the property of extension of homotopies. Let $f: X\rightarrow Y$ be a homeotopy which induces, by passing to subspaces, a homeomorphism of A onto B. Prove that $f$ is a homeotopy of the pair $(X,A)$ onto the pair $(Y,B)$.