Exercise 20, § 2

a) Give an example of a closed equivalence relation $R$ on a metrizable locally compact space $X$ which has a countable base such that $X/R$ is paracompact but such that there exists a point of $X/R$ which has no countable fundamental system of neighbourhoods [cf. Chapter I, § 10, Exercise 17 and Chapter IX, § 4, Exercise 24 a)].

b) Let $X$ be a metrizable space and let $R$ be a closed equivalence relation on $X$. Show that if every point of $X/R$ has a countable fundamental system of neighbourhoods, then each equivalence class (mod $R$) has a compact frontier in $X$. (Show that if the result were false there would exist a sequence in $X$, all of whose points were distinct, which had no cluster point, and whose image in $X/R$ was a sequence which converged to a point distinct from all the points of the sequence.) For each $z \in X/R$, let $C_z$ be the inverse image of $z$ in $X$, and let $F_z$ be the frontier of $C_z$ if this frontier is not empty; if $C_z$ is both open and closed, let less $F_z$ be any subset of $C_z$ consisting of a single point. Let $Y$ be the union of the sets $F_z$ as $z$ runs through $X/R$. Show that $Y/R_y$ is homeomorphic to $X/R$.