Exercise 2, § 6

a) Let $X$ be a metrizable space. Show that the following properties are equivalent:

$\alpha)$ There exists a metric $d$ compatible with the topology of $X$, with respect to which $X$ is an ultrametric space ($§ 2$, Exercise 4).

$\beta)$ $X$ is strongly zero-dimensional.

$\gamma)$ There exists a sequence $(\mathscr{B}_n)$ of locally finite families of open and closed subsets of $X$ such that $\mathscr{B} = \bigcup_n \mathscr{B}_n$ is a base for the topology of $X$.

[To show that $\alpha)$ implies $\beta)$, note that if $F$ is a closed set in an ultrametric space $X$, then the set of all $x \in X$ such that $d(x, F) = \rho > 0$ is both open and closed in $X$. To show that $\beta)$ implies $\gamma)$, use Lemma 3 of $§ 4$, no. 5 and Exercise 1 c) of $§ 6$. Finally, to establish that $\gamma)$ implies $\alpha)$, reduce to the case where $\mathscr{B}_n = (U_{n, \lambda})_{\lambda \in L}$, and if $f_{n, \lambda}$ is the characteristic function of $U_{n, \lambda}$, consider the mapping $x \mapsto f(x) = f_{n, \lambda}(x)$ of $X$ into the product group $G^{N \times L}$, where $G = \mathbf{Z}/2\mathbf{Z}$ (identified with the set $\{0, 1\}$); for each $n \in \mathbf{N}$, let $G_n$ be the subgroup of $G$ consisting of the elements whose $(k, \lambda)$ coordinates are zero for all $\lambda \in L$ and all $k < n$; show that if $G$ is endowed with the group topology for which $(G_n)$ is a fundamental system of neighbourhoods of 0, then $f$ is a homeomorphism of $X$ onto a subspace of $G$; and complete the proof by noting that the topology of $G$ can be defined by an invariant metric with respect to which $G$ is an ultrametric space.]

b) Deduce from a) that every zero-dimensional Hausdorff space $X$ which has a countable base is a strongly zero-dimensional metrizable space. (Show by using $§ 2$, no. 8, Proposition 13 that there exists a countable base formed of sets which are both open and closed.) Furthermore, $X$ is then homeomorphic to a subspace of Cantor's triadic set $K$ (cf. $§ 2$, Exercise 12 (*).)

(*) It is not known whether every zero-dimensional metrizable space is strongly zero-dimensional.