Proposition 7

In a metrizable space, every closed set is the intersection of a countable family of open sets, and every open set is the union of a countable family of closed sets.

Let $d$ be a metric compatible with the topology of a metrizable space $X$. If $A$ is a closed subset of $X$, it is the intersection of the open sets $V_{1/n}(A)$ [the set of all $x \in X$ such that $d(x, A) < 1/n$; cf. Proposition 2]. The second part of the proposition follows by taking complements.