Exercise 29, § 4

a) Let $(\mathfrak{S}_n)$ (for $n \geq 1$) be a sequence of locally finite families formed of open subsets of a topological space $X$. Suppose that $\mathfrak{S} = \bigcup_n \mathfrak{S}_n$ is a covering of $X$.

There then exists a locally finite covering $\mathscr{B}$ (not necessarily open) of $X$ which is finer than $\mathfrak{S}$. (Let $U_n$ be the union of the open sets of all the families $\mathfrak{S}_k$ for $k \leq n$; now prove that if $A_n = U_n - U_{n-1}$ (with $U_0 = \varnothing$) the set $\mathscr{B}$ of the parts $V \cap A_n$, where $V \in \mathfrak{S}_n$ and $n$ runs through the set of integers $\geq 1$, answers the question.)

b) Let $X$ be a topological space, $(A_i)_{i \in I}$ a family of subsets of $X$; one says that $(A_i)$ is locally finite at a point $x \in X$ if there exists a neighbourhood of $x$ meeting $A_i$ for only a finite number of indices $i \in I$. One says that a subspace $Y$ of $X$ is relatively paracompact in $X$ if, for every set $\mathscr{F}$ of open subsets of $X$, forming a covering of $Y$, there exists a set $\mathfrak{S}$ of open subsets of $X$, forming a covering of $Y$ finer than $\mathscr{F}$, and locally finite at every point of $Y$. If $X$ is separated, $Y$ is then a paracompact space (cf. IX, p. 113, exerc. 14). In a paracompact space, every closed subspace is relatively paracompact. In a separated space, every compact subspace is relatively paracompact.

c) Let $Y$ be a closed subspace of a normal space $X$; now prove that if $Y$ is relatively paracompact in $X$, then, for every set $\mathscr{F}$ of open subsets of $X$, forming a covering of $Y$, there exists a locally finite set $\mathfrak{S}$ of open subsets of $X$, finer than $\mathscr{F}$ and forming a covering of $Y$. (One may suppose that $\mathscr{F}$ is locally finite at every point of Y. For every $y \in Y$, let $U_y$ be a neighbourhood of $y$ meeting only a finite number of sets of $\mathscr{F}$. Let V be the union of the $U_y$ for $y \in Y$, W a neighbourhood of Y such that $\overline{W} \subset V$; consider the intersections of W and the sets of $\mathscr{F}$.

d) In a normal space X, now prove that a countable union of closed subspaces relatively paracompact in X is a relatively paracompact subspace in X (use a) and c) and IX, p. 107, exerc. 25 c)). In particular, in a paracompact space X, every countable union of closed subspaces is a relatively paracompact subspace in X (and fortiori paracompact) (cf. IX, p. 113, exerc. 14).

e) Now prove that every product of a paracompact space X and a completely regular space Y, a countable union of compact sets, is paracompact (embed Y in $\beta Y$ and use d)) (cf. IX, p. 113, exerc. 14).