Exercise 12, § 4

Let $X, Y$ be two Hausdorff spaces.

a) Suppose that there is a closed set $A$ in $X$ which is not a countable intersection of open sets, and that there is a countably infinite subset $B$ of $Y$ which is not closed. Let $b \in \overline{B} - B$, and consider the subsets $C = A \times B,\ D = (X - A) \times \{ b \}$ in $X \times Y$. Show that $C \cap \overline{D} = \overline{C} \cap D = \varnothing$, but that there is no pair $U, V$ of open subsets of $X \times Y$ such that $C \subset U,\ D \subset V$ and $U \cap V = \varnothing$.

b) If $X \times Y$ is completely normal then either (i) one of the spaces $X, Y$ is perfectly normal, or (ii) one of the spaces $X, Y$ has the property that every countably infinite subset is closed (cf. § 5, Exercise 16).

c) Let $X = [a, b]$ be an uncountable well-ordered set such that $[a, x[$ is countable for each $x < b$. Endow $X$ with the topology in which $\{x\}$ is open for each $x < b$ and the intervals $[x, b]$ ($x < b$) form a fundamental system of neighbourhoods of $b$. Show that $X \times X$ is completely normal but that $X$ is not perfectly normal.

d) Show that if $X$ is a compact space such that $X \times X \times X$ is completely normal, then $X$ is metrizable (use Theorem 1 of § 2, no. 4).