Proposition 17

Every Hausdorff quotient space of a compact metrizable space is compact and metrizable.

Equivalently, if $f$ is a continuous mapping of a compact metrizable space $X$ into a Hausdorff space $X'$, then $f(X)$ is a metrizable subspace of $X'$ (Chapter I, § 9, no. 4, Theorem 2, Corollary 4).

Let $X$ be a compact metrizable space, and let $R$ be an equivalence relation on $X$ such that $X/R$ is Hausdorff. Then $X/R$ is compact (Chapter I, § 9, no. 4, Theorem 2), hence by Proposition 16 it is enough to show that the topology of $X/R$ has a countable base. To do this, we use the facts that $R$ is closed (Chapter I, § 10, no. 4, Proposition 8) and that the classes mod $R$ are compact. Let $\varphi$ be the canonical mapping of $X$ onto $X/R$, and let $(U_n)$ be a countable base of the topology of $X$. Let $z$ be any point of $X/R$ and let $V$ be a neighbourhood of $z$ in $X/R$; then $\overline{\varphi}^{-1}(V)$ is a neighbourhood in $X$ of the compact set $\overline{\varphi}^{-1}(z)$. If $x$ is any point of $\overline{\varphi}^{-1}(z)$, there is a set $U_n$ containing $x$ and contained in $\overline{\varphi}^{-1}(V)$, and therefore by the Borel-Lebesgue axiom there is a finite open covering $(U_{n_k})_{1 \leq k \leq r}$ of $\overline{\varphi}^{-1}(z)$ such that, if $W$ denotes $\bigcup_k U_{n_k}$, $W$ is a neighbourhood of $\overline{\varphi}^{-1}(z)$ contained in $\overline{\varphi}^{-1}(V)$. Since $R$ is closed, it follows that $\varphi(W)$ is a neighbourhood of $z$ in $X/R$, contained in $V$ (Chapter I, § 5, no. 4, Proposition 10). Let $\mathscr{B}$ denote the set of interiors of sets of the form $\varphi(W)$, where $W$ runs through the set $\mathscr{F}$ of all finite unions of sets of the form $U_n$; we have then shown that $\mathscr{B}$ is a base of the topology of $X/R$, and since $\mathfrak{F}$ is countable, so is $\mathscr{B}$.