Exercise 16, § 2

Let $X$ and $Y$ be two countably compact spaces. Show that the product $X \times Y$ is countably compact in each of the following two cases: (i) one of $X, Y$ is compact; (ii) one of $X, Y$ is such that each point has a countable fundamental system of neighbourhoods. [In case (i), supposing that $Y$ is compact, consider an increasing sequence $(G_n)$ of open sets whose union is $X \times Y$; for each $n$, let $H_n$ be the set of all $x \in X$ such that $\{ x \} \times Y \subset G_n$; show that $H_n$ is open and that $X = \bigcup_n H_n$.]