Exercise 27, § 4

a) Let X be a normal space, $(U_\lambda)_{\lambda \in L}$ a locally finite open covering of X. Now prove that there exists a countable open covering $(V_n)_{n \in \mathbf{N}}$ of X, where, for each $n \in \mathbf{N}$, $V_n$ is the union of a family $(W_v)_{v \in K_n}$ of pairwise disjoint open sets forming a covering of $V_n$ finer than $(U_\lambda)_{\lambda \in L}$. (Let $(f_\lambda)_{\lambda \in L}$ be a continuous partition of unity subordinate to $(U_\lambda)_{\lambda \in L}$. For every finite subset J of L, let $W_j$ be the set of the $x \in X$ such that $f_\lambda(x) < f_\mu(x)$ for all the pairs $(\lambda, \mu)$ such that $\lambda \in L - J$, $\mu \in J$. Take for $K_n$ the set of subsets of L having n elements.) b) Let X be a topological space, $\mathscr{O}$ the set of open sets of X. One says that a subset $\mathscr{U}$ of $\mathscr{O}$ is quasi-full if it satisfies the following conditions: $\alpha)$ $\mathscr{U}$ is a covering of X; $\beta)$ every open subset of a set belonging to $\mathscr{U}$ belongs to $\mathscr{U}$; $\gamma)$ every finite union of sets of $\mathscr{U}$ belongs to $\mathscr{U}$; $\delta)$ every union of a family of sets belonging to $\mathscr{U}$ and pairwise disjoint, belongs to $\mathscr{U}$. Now prove that in order that the space X be paracompact, it is necessary and sufficient that every quasi-full subset $\mathscr{U}$ of $\mathscr{O}$ be equal to $\mathscr{O}$. (To see that the condition is necessary, use a) to construct a countable locally finite open covering $(V_n)_{n \in \mathbf{N}}$ of X such that $V_n \in \mathscr{U}$ for every $n \in \mathbf{N}$. Then, with the aid of a continuous partition of unity $(f_n)$ subordinate to

(V_n) and of the function $f(x) = \sum_{n > 0} n f_n(x)$, construct an open covering $(W_n)$ of $X$ by open sets of $\mathscr{U}$ such that $\overline{W}_n \subset W_{n+1}$ for every $n \geq 0$. To see that the condition is sufficient, given an open covering $\mathscr{R}$ of $X$, consider the set $\mathscr{U}$ of the open sets of X such that for every set $A \in \mathscr{U}$, the intersections with $A$ of the sets of $\mathscr{R}$ constitute a covering of $A$ possessing a finer and locally finite open covering.)