Exercise 1, § 6

a) Every zero-dimensional space is completely regular. The non-normal locally compact spaces defined in § 4, Exercise 26 b) and § 5, Exercise 15 are zero-dimensional.

b) A topological space $X$ is said to be strongly zero-dimensional if for each closed subset $A$ of $X$ and each neighbourhood $U$ of $A$, there is an open and closed neighbourhood of $A$ contained in $U$. Every strongly zero-dimensional space is normal. A normal space is strongly zero-dimensional if and only if its Stone-Čech compactification ($§ 1$, Exercise 7) is totally disconnected [use Exercise 17 c) of § 4].

c) Let $X$ be a strongly zero-dimensional Hausdorff space and let $(U_n)$ be an increasing sequence of non-empty open sets in $X$. Show that there exists a sequence $(G_n)$ of mutually disjoint sets which are both open and closed in $X$, such that $G_n \subset U_n$ for each $n$, and $\bigcup_n G_n = \bigcup_n U_n$ (define the $G_n$ by induction). If, moreover, $X$ is perfectly normal ($§ 4$, Exercise 8), every open set in $X$ is the union of a countable family of mutually disjoint sets which are both open and closed in $X$.

d) Let $X$ be a normal space which is the union of a sequence $(A_n)$ of strongly zero-dimensional subspaces. Show that $X$ is strongly zero-dimensional. [Let $B, C$ be two disjoint closed subsets of $X$; define by induction two increasing sequences $(G_n), (H_n)$ of open sets in $X$ such that $\overline{G}_n \cap \overline{H}_n \neq \varnothing,\ A_n \subset G_n \cup H_n,\ B \cap A_n \subset G_n,\ C \cap A_n \subset H_n.$]