Exercise 21, § 1

Let $X$ be a paracompact topological space, let $\mathscr{A}$ be a set of closed subsets of X such that $B\cap C\in \mathscr{A}$ for every pair $(B,C)$ of elements of $\mathscr{A}$. Let $A =\bigcup_{C\in\mathscr{A}}C$; suppose that, for every point $x$ of X, there exist a neighbourhood V of $x$ and a finite subset $\mathscr{A}_x$ of $\mathscr{A}$ such that $V\cap A = V\cap \bigcup_{C\in\mathscr{A}_x}C$. (a) Prove that A is a closed subset of X.

(b) Suppose that, for every $B\in \mathscr{A}$, the pair $(X,B)$ has the homotopy extension property. Prove that the pair $(X,A)$ has the same property.