Exercise 20, § 1

Let $X$ be a topological space and let $A$, $B$ and $C$ be closed subspaces of $X$ such that $A = B\cap C$.

a) If the pairs $(X,B)$, $(X,C)$ and $(X,A)$ have the homotopy extension property, prove that the pair $(X,B\cup C)$ has the same property.

b) Suppose that the pairs $(X,A)$ and $(X,B\cup C)$ have the homotopy extension property. Prove that the pairs $(X,B)$ and $(X,C)$ have the same property.

c) Suppose that $X$ is a normal space and that $A\subset \mathring{B}\cup \mathring{C}$. If the pairs $(X,B)$ and $(X,C)$ have the homotopy extension property, prove that the pair $(X,A)$ has the same property.