Exercise 15, § 4

Let $X, Y$ be two separated spaces. (a) Suppose that there exists in $X$ a closed set $A$ which is not an intersection of open sets, and in $Y$ a countable infinite non-closed set $B$. Let $b \in \overline{B} - B$. Consider in $X \times Y$ the sets $C = A \times B, D = (X - A) \times \{b\}$. Now prove that $C \cap \overline{D} = \overline{C} \cap D = \varnothing$, but that there do not exist pairs $(U, V)$ of open sets in $X \times Y$ such that $C \subset U, D \subset V$ and $U \cap V = \varnothing$. (b) In order that $X \times Y$ be completely normal, it is necessary that one of the spaces $X, Y$ be perfectly normal or that in one of the spaces $X, Y$ every countable infinite set be closed (cf. IX, p. 113, exerc. 15). (c) Let $X = [a, b)$ be an uncountable well ordered set such that $\{a, x\}$ is countable for every $x < b$. Equip $X$ with the topology for which $\{x\}$ is open for every $x < b$, and the intervals $]x, b]$ form a fundamental system of neighbourhoods of $b$ (for $x < b$). Now prove that $X \times X$ is completely normal but that $X$ is not perfectly normal. (d) Now prove that if $X$ is a compact space such that $X \times X \times X$ is completely normal, $X$ is metrizable (use the th. 1 of IX, p. 15).