Exercise 18, § 2

Let $X$ be a metric space such that, for each $x \in X$, there is an open ball with centre $x$ which, when considered as a subspace of $X$, has a countable base. Let $r_x$ be the least upper bound of the radii of balls with centre $x$ which have this property.

a) Show, using Zorn's lemma, that there is a maximal family $(B_\alpha)$ of open balls in $X$ which are pairwise disjoint and such that, if $x_\alpha$ is the centre of $B_\alpha$, the radius of $B_\alpha$ is $< r_{x_\alpha}$. Show that the union of the $B_x$ is dense in $X$, and hence that there is a dense subset $M$ of $X$ such that every open ball with centre at an arbitrary point $x \in X$ and radius $< r_x$ contains only a countable infinity of points of $M$ (note that such a ball can meet only a countable infinity of pairwise disjoint open sets, and use Proposition 12).

b) For each point $x \in M$, let $S_x$ denote the open ball with centre $x$ and radius $\frac{1}{3} r_x$. Show that the $S_x$ cover $X$, and that an arbitrary point of $X$ belongs only to a countable infinity of the $S_x$ [note that if $y \in S_x$, we have $d(x, y) \leq \frac{1}{2} r_y$].

c) Let $R$ be the following equivalence relation between points $x, y$ of $X$: there exists a sequence $(z_i)_{1 \leq i \leq n}$ of points of $M$ such that $x \in S_{z_1}$, $y \in S_{z_n}$ and $S_{z_i}$ meets $S_{z_{i+1}}$ for $1 \leq i \leq n - 1$. Show that the equivalence classes (mod $R$) are subspaces of $X$ which are both open and closed in $X$ and which have a countable base; in other words, $X$ is the topological sum of metric spaces with countable bases (Chapter I, § 2, no. 4).