Exercise 20, § 4

a) A regular space $X$ is paracompact if, for each open covering $\mathfrak{R}$ of $X$, there is a sequence $(\mathscr{S}_n)$ such that $\mathscr{S} = \bigcup_n \mathscr{S}_n$ is an open covering of $X$ which refines $\mathfrak{R}$ and such that each $\mathscr{S}_n$ is a locally finite family (cf. Lemmas 4, 5 and 6).

b) Deduce from a) that in a paracompact space every countable union of closed sets is a paracompact subspace (cf. $§ 5$, Exercise 15).

c) In a regular space $X$, let $\mathfrak{F}$ be a locally finite family of closed sets each of which is a paracompact subspace of $X$. Show that the union of the sets of $\mathfrak{F}$ is a paracompact subspace of $X$ (use Lemma 6 of no. 5; cf. $§ 5$, Exercise 15).

d) Show that if $X$ is a paracompact space and $Y$ is a regular space which is a countable union of compact sets, then $X \times Y$ is paracompact [embed $Y$ in its Stone-\v{C}ech compactification and use b); cf. $§ 5$, Exercise 16].