Exercise 34, § 4

a) Let X be a metrizable space, R a closed equivalence relation in X, such that every equivalence class according to R is compact. Then X/R is metrizable (To apply the Nagata-Smirnov theorem (IX, p. 114, exerc. 32), consider a covering $\mathscr{R} = (U_\lambda)_{\lambda \in L}$ of X formed of open sets saturated for R; show that there exists a sequence $(\mathscr{S}_n)$ having the following properties: 1° each $\mathscr{S}_n$ is a locally finite family of open sets in X saturated for R; 2° $\mathscr{S} = \bigcup_n \mathscr{S}_n$ is an open covering of X finer than $\mathscr{R}$.

For this, provide L with a well ordered set structure; let d be a distance defining the topology of X; for every $\lambda \in L$ and every integer $n > 0$, let $W_{n,\lambda}$ be the largest open set saturated for R contained in the set of the $x \in X$ such that $d(x, X - U_\lambda) > 2^{-n}$, $F_{n,\lambda}$ the saturated for R of the set of the $x \in X$ such that $d(x, X - U_\lambda) \geq 2^{-n}$, $G_{n,\lambda}$ the set of the $x \in W_{n,\lambda}$ such that $x \notin F_{n+1,\mu}$ for $\mu < \lambda$, $V_{n,\lambda}$ the largest open set saturated contained in the set of the $y \in X$ such that $d(x, G_{n,\lambda}) < 2^{-n-3}$. Show that the set $\mathscr{S}_n$ of the $V_{n,\lambda}$ for $\lambda \in L$ answers the question.)

b) Extend the result of a) to the case where one supposes the relation R closed and the space X/R such that every point admits a countable fundamental system of neighbourhoods (cf. IX, p. 96, exerc. 23 b).

c) Let X be a topological space, $(F_\alpha)$ a locally finite closed covering of X. Show that if each of the subspaces $F_\alpha$ is metrizable, X is metrizable (consider X as quotient space of the topological sum of the F).

d) Show that a paracompact space of which every point has a metrizable neighbourhood is metrizable (apply c)). (Cf. I, p. 114, exerc. 22 b) and IX, p. 113, exerc. 14 b).)