Exercise 4, § 6

a) Let $X$ be a metrizable space. Prove that the following properties are equivalent: $\alpha$ there exists a distance compatible with the topology of $X$ and for which $X$ is an ultrametric space (IX, p. 91, exer. 4); $\beta$ $X$ is strongly scattered (exer. 3). $\gamma$ there exists a sequence $(\mathscr{B}_n)$ of locally finite families of subsets which are both open and closed in $X$ such that $\mathscr{B} = \bigcup_n \mathscr{B}_n$ is a basis for the topology of $X$.

To prove that $\alpha$ implies $\beta$), remark that if $F$ is a closed set in an ultrametric space $X$, the set of $x \in X$ such that $d(x, F) = \rho > 0$ is both open and closed. To prove that $\beta$ implies $\gamma$), use exer. 3 c) of IX, p. 118. 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 $\mathbf{G}^{N \times L}$, where $\mathbf{G} = \mathbf{Z}/2\mathbf{Z}$ (identified with the set $\{0, 1\}$) ; for every $n \in \mathbf{N}$, let $G_n$ be the subgroup of $G$ formed by the elements whose projections of index $(k, \lambda)$ are zero for every $\lambda \in L$ and every $k < n$; prove that if $G$ is provided with the group topology for which $(G_n)$ is a fundamental system of neighbourhoods of $0$, $f$ is a homeomorphism of $X$ onto a subspace of $G$; conclude by remarking that the topology of $G$ can be defined by an invariant distance for which $G$ is an ultrametric space.

b) Deduce from a) that every separated scattered space X which admits a countable basis is a strongly scattered metrizable space (prove that there exists a countable basis formed of sets which are both open and closed, using Prop. 13 of IX, p. 19). Moreover, X is then homeomorphic to a subspace of the triadic set of Cantor K (cf. IX, p. 92, exer. 11).¹