Exercise 24, § 4

a) Let $X$ be a metrizable space and let $R$ be a closed equivalence relation on $X$, all of whose equivalence classes are compact. Show that $X/R$ is metrizable. [Apply the Nagata-Smirnov theorem (Exercise 22) by proving the analogue of Lemma 3 of no. 5 for open coverings of $X$ formed of sets which are saturated with respect to $R$; let $F_{n\alpha}$ denote the largest saturated open set contained in the set of all $x \in X$ such that $d(x, X - U_\alpha) > 2^{-n}$, let $F'_{n\alpha}$ denote the saturation of the set of all $x \in X$ such that $d(x, X - U_\alpha) \geq 2^{-n}$ and let $G_{n\alpha}$ denote the set of all $x \in F_{n\alpha}$ such that $x \notin F'_{n+1, \beta}$ for all $\beta < \alpha$.]

b) Extend the result of a) to the case in which the relation $R$ is closed and every point of $X/R$ has a countable fundamental system of neighbourhoods [use Exercise 20 b) of § 2].

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

d) A paracompact space, in which every point has a metrizable neighbourhood, is metrizable [apply c); cf. Chapter I, § 9, Exercise 12 b), also Chapter IX, § 4, Exercise 25 d), § 2, Exercise 15 and § 5, Exercise 15].