Exercise 24, § 1

Let $M$ be an uncountable set, let $X$ be the space $\mathbf{I}^M$ and let $A$ be the subspace $\{0\}^M$ of X.

a) Prove that the pair $(X,A)$ has the property of extension of stationary homotopies (Exercise 23).

b) Prove that the pair $(X,A)$ does not have the property of extension of homotopies. (Prove that the set A is not the intersection of a countable family of open sets.)

c) Prove that the pair $(X\times \mathbf{I},X\times \{0\} \cup A\times \mathbf{I})$ does not have the property of extension of stationary homotopies.