Exercise 32, § 4

a) In order that a regular space X be metrizable, it is necessary and sufficient that there exist a sequence $(\mathscr{B}_n)$ of locally finite families of open parts of X such that $\mathscr{B} = \bigcup_n \mathscr{B}_n$ be a basis of the topology of X (Nagata-Smirnov theorem). (To show that the condition is necessary, use the fact that X is paracompact, by considering for each n the covering of X formed by the open balls of radius $1/n$. To see that the condition is sufficient, first show that X is paracompact, using IX, p. 108, exerc. 29 a) and IX, p. 113, exerc. 25 c). For every couple of integers m, n and every $U \in \mathscr{B}_m$, let $U'$ be the union of the sets $V \in \mathscr{B}_n$ such that $\overline{V} \subset U$; show that $\overline{U} \subset U$. Let $f_U$ be a continuous mapping of X into $(0,1)$, equal to 0 in $X - U$ and to 1 in $U'$, and let

show that the distances $d_{mn}$ defining the topology of X.) In particular every regular space having a countable basis is metrizable.

b) Show that with the notations of a), one may in addition suppose that each of the families $\mathscr{B}_n$ is discrete (IX, p. 106, exerc. 24). (Use exerc. 27 a) by remarking (with the notations of this exercise) that in a metric space, each $W_v$ is the union of the open sets $W_{vm}$, where $W_{vm}$ is the set of points whose distance from $\mathbf{C} W_v$ is $> 1/m$. Deduce that in a metrizable space, the conditions (D_v) and (D_I) of I, p. 90, exerc. 7, are equivalent.)