Exercise 25, § 4

(a) Let X be a regular space such that, for every open covering $\mathscr{R}$ of X, there exists a locally finite covering $\mathscr{A}$ (not necessarily open) of X which is finer than $\mathscr{R}$. Then, for every open covering $\mathscr{R}$ of X, there exists a closed locally finite covering $\mathscr{F}$ of X which is finer than $\mathscr{R}$. (For every $x \in X$, let $V_x$ be an open neighbourhood of $x$ such that $\overline{V}_x$ is contained in a set of $\mathscr{R}$. Consider a locally finite covering $\mathscr{B}$ finer than the covering formed by the $V_x$, then the covering $\mathscr{F}$ formed by the closures of the sets of $\mathscr{B}$.) (b) Let X be a separated space such that, for every open covering $\mathscr{R}$ of X, there exists a closed locally finite covering $\mathscr{F}$, finer than $\mathscr{R}$. Prove that X is paracompact. (Let $\mathscr{R}$ be an open covering of X, $\mathscr{A}$ a locally finite covering of X, finer than $\mathscr{R}$; for every $x$, let $W_x$ be an open neighbourhood of $x$ meeting only a finite number of sets of $\mathscr{A}$. Let $\mathscr{F}$ be a closed locally finite covering finer than the covering formed by the $W_x$. For every $A \in \mathscr{A}$, let $U_A$ be a set of $\mathscr{R}$ containing A, and let $C_A$ be the union of the sets $F \in \mathscr{F}$ such that $A \cap F = \varnothing$. Prove that when A ranges over $\mathscr{A}$, the sets $U_A \cap (X - C_A)$ form an open locally finite covering finer than $\mathscr{R}$.) (c) Deduce from a) and b) that if X is a regular space such that, for every open covering $\mathscr{R}$ of X, there exists a locally finite covering (not necessarily open) finer than $\mathscr{R}$, X is paracompact.