Exercise 3, § 6

a) Every separated scattered space is completely regular. The locally compact non-normal spaces defined in IX, p. 103, exerc. 12 and IX, p. 113, exerc. 14, are scattered. b) One says that a topological space $X$ is strongly scattered if for every closed subset $A$ of $X$ and every neighbourhood $U$ of $A$, there exists an open and closed neighbourhood of $A$ contained in $U$. Every separated strongly scattered space is normal. Every compact totally disconnected space is strongly scattered. In order that a normal space $X$ be strongly scattered, it is necessary and sufficient that its Stone-Čech compactification $\beta X$ (IX, p. 9) be totally disconnected (use exerc. 23 b) of IX, p. 106). c) Let $X$ be a separated strongly scattered space, $(U_n)$ a growing sequence of non-empty open sets in $X$. Now prove that there exists a sequence $(G_n)$ of sets both open and closed in $X$, pairwise without point in common, such that $G_n \subset U_n$ for all $n$, and $\bigcup_n G_n = \bigcup_n U_n$ (define the $G_n$ by induction). If moreover $X$ is perfectly normal (IX, p. 103, exerc. 11), every open set in $X$ is the union of a countable family of sets both open and closed, pairwise disjoint. d) Let $X$ be a normal space, union of a sequence $(A_n)$ of strongly scattered subspaces. Now prove that $X$ is strongly scattered. (Let $B, C$ be two closed sets in $X$ such that $B \cap C = \varnothing$. Define by induction two growing sequences $(G_n), (H_n)$ of open sets in $X$ such that $\overline{G}_n \cap \overline{H}_n = \varnothing, A_n \subset G_n \cup H_n$ and $B \cap A_n \subset G_n, C \cap A_n \subset H_n$).