Exercise 22, § 2

Let $X$ be a Hausdorff topological space with a countable base $(U_n)$, and let $R$ be a Hausdorff equivalence relation on $X$ such that every point of $X/R$ has a countable fundamental system of neighbourhoods. Show that the topology of $X/R$ has a countable base. [Let $\varphi : X \to X/R$ be the canonical mapping, and show that the interiors of finite unions of sets $\varphi(U_n)$ form a base of the topology of $X/R$: if $V$ is a neighbourhood of a point $z \in X/R$, and $(W_k)$ is a sequence of sets belonging to the base $(U_n)$, contained in $\overline{\varphi^{-1}(V)}$ and covering $\overline{\varphi^{-1}(z)}$, show that there is a finite number of indices $k$ such that the union of the $\overline{\varphi^{-1}(W_k)}$ is a neighbourhood of $z$. Argue by contradiction: if this statement were false we could construct a sequence $(\gamma_n)$ of distinct points of $X/R$, tending to $z$ and such that $\gamma_n$ did not belong to the union of the $\varphi(W_k)$ for $k \leq n$; then show that the union of the $\overline{\varphi^{-1}(\gamma_n)}$ would be closed in $X$.]

Show that the same conclusion is valid if $R$ is assumed to be closed and every equivalence class mod $R$ is compact (similar method).