习题 2, § 6

a) 令 $X$ 为可度量的空间。证明以下性质是等价的：

$\alpha)$ There exists a metric $d$ compatible with 拓扑 of $X$, with respect to which $X$ is an ultrametric 空间 ($§ 2$, 习题 4).

$\beta)$ $X$ 是强零维的。

$\gamma)$ There exists 序列 $(\mathscr{B}_n)$ of locally finite families of open and closed 子集 of $X$ such that $\mathscr{B} = \bigcup_n \mathscr{B}_n$ is 基 for 拓扑 of $X$.

[To show that $\alpha)$ implies $\beta)$, 注 that if $F$ is a closed 集合 in an ultrametric 空间 $X$, then 集合 of all $x \in X$ such that $d(x, F) = \rho > 0$ is both open and closed in $X$。要证明$\beta)$蕴含$\gamma)$，请使用$§ 4$的引理 3、no. 5和$§ 6$的习题 1 c)。 Finally, to establish that $\gamma)$ implies $\alpha)$, 约化 to the case where $\mathscr{B}_n = (U_{n, \lambda})_{\lambda \in L}$, and if $f_{n, \lambda}$ is the characteristic 函数 of $U_{n, \lambda}$, consider 映射 $x \mapsto f(x) = f_{n, \lambda}(x)$ of $X$ into 乘积群 $G^{N \times L}$，其中 $G = \mathbf{Z}/2\mathbf{Z}$（与集合 $\{0, 1\}$ 标识）； for each $n \in \mathbf{N}$, let $G_n$ be 子群 of $G$ 由……组成 of 元素 whose $(k, \lambda)$ coordinates are 零 for all $\lambda \in L$ and all $k < n$; show that if $G$ is endowed with 群拓扑 for which $(G_n)$ is a fundamental 系统 of 邻域 of 0, then $f$ is a homeomorphism of $X$ onto 子空间 of $G$； and complete 证明 by noting that 拓扑 of $G$ can be defined by an invariant metric with respect to which $G$ is an ultrametric 空间.]

b) Deduce from a) that every zero-dimensional Hausdorff 空间 $X$ which has a countable 基 is a strongly zero-dimensional metrizable 空间. (Show by using $§ 2$, no. 8, 命题 13 that there exists a countable 基 formed of 集合 which are both open and closed.) Furthermore, $X$ is then homeomorphic to 子空间 of Cantor's triadic 集合 $K$（cf.$§ 2$，习题12（*）。）

(*) 尚不清楚是否每个零维可度量空间都是强零维的。