Proposition 16

A compact space $X$ is metrizable if and only if its topology has a countable base.

The condition is necessary. By Proposition 14, for each integer $n \geq 1$ there is a finite subset $A_n$ of $X$ such that the distance of $A_n$ from every point of $X$ is $\leq 1/n$; the countable set $A = \bigcup_n A_n$ is therefore dense in $X$, and the result follows from Proposition 12 of no. 8.

The condition is sufficient. Let $(U_n)$ be a countable base of the topology of $X$. Every neighbourhood of a point of the diagonal $\Delta$ of $X \times X$ therefore contains a neighbourhood of the form $U_n \times U_n$; applying the Borel-Lebesgue axiom to the compact subset $\Delta$ of $X \times X$, it follows that every neighbourhood of $\Delta$ contains a finite union of sets of the form $U_n \setminus U_n$, which is a neighbourhood of $\Delta$. Hence the neighbourhoods of $\Delta$ which are finite unions of sets of the form $U_n \times U_n$ form a fundamental system of entourages of the uniformity of $X$ (Chapter II, § 4, no. 1, Theorem 1), and the result therefore follows from Theorem 1 of no. 4.