Exercise 12, § 1

Let $X$ and $Y$ be topological spaces; one says that the homotopy extension property holds for continuous mappings with target $X$ and source $Y$ if, for every continuous mapping $F: Y \rightarrow X$, every closed subspace $B$ of $Y$ and every homotopy $\sigma : B\times \mathbf{I}\rightarrow X$ of origin $F|B$, there exists a homotopy $\widetilde{\sigma}: Y\times \mathbf{I}\rightarrow X$ of origin $F$ which extends $\sigma$.

(a) Let $X$ be a topological space which is an absolute neighbourhood retract. Prove that the homotopy extension property holds for continuous mappings with target $X$ and source a metrizable space of countable type.

(b) Prove the equivalence of the following two properties:

(i) The space $X$ is an absolute neighbourhood retract;

(ii) Every point of $X$ possesses a neighbourhood V such that for every metrizable topological space of countable type $Y$, every closed subspace $B$ of $Y$, every continuous mapping of $B$ into V extends to a continuous mapping of $Y$ into $X$.

(c) Suppose that $X$ is semi-locally contractible (cf. Exercise 9) and that the homotopy extension property holds for continuous mappings with target $X$. Prove that $X$ is an absolute neighbourhood retract. [^1]

(d) Prove that the space $X =\mathbf{Q}$ is not an absolute neighbourhood retract but that the homotopy extension property holds for continuous mappings with target $X$.

[^1]: K. Borsuk constructed a semi-locally contractible space which is not an absolute neighbourhood retract, see “Sur un espace localement contractile qui n’est pas un rétracte absolu de voisinage”, Fund. Math. 35, (1948), p. 175–180.