Exercise 26, § 5

Let $X$ be a locally compact metrizable space, let $d$ be a metric compatible with the topology of $X$, and let $R$ be a closed equivalence relation on $X$, all of whose equivalence classes are compact. Then $X/R$ is locally compact (Chapter I, $\S$ 10, Proposition 9) and metrizable ($\S$ 4, Exercise 24).

a) For each pair of points $u,\ v$ of $X/R$, set

where $\varphi : X \to X/R$ is the canonical mapping, and for each $u \in X/R$ let Show that $\lambda$ is upper semi-continuous on $X/R$ and that, for each $\alpha > 0$, the set of points $u \in X/R$ such that $\lambda(u) \geq \alpha$ is nowhere dense. [Show that otherwise there would exist a compact subset $K$ of $X$ and an infinite sequence $(x_n)$ of points of $K$ such that $d(x_m,\ x_n) > \frac{1}{2} \alpha$ for $m \neq n$.]

b) Deduce from a) that there is a dense subset $M$ of $X/R$ which is a countable intersection of open sets in $X/R$ and is such that for each $x \in \overline{\varphi}(M)$ the image under $\varphi$ of every neighbourhood of $x$ in $X$ is a neighbourhood of $\varphi(x)$ in $X/R$ (use Theorem 1).