Exercise 29, § 8

a) Let $R$ be a Hausdorff equivalence relation on a non-empty topological space $X$, and let $\mathfrak{F}(X)$ be the space of non-empty closed subsets of $X$, with the topology induced by $\mathfrak{T}_\Theta$ (Exercise 12). Show that every $A \in \mathfrak{F}(X)$ which lies in the closure of $X/R$ [considered as a subset of $\mathfrak{F}(X)$] is contained in some equivalence class of $R$ (use reductio ad absurdum).

b) Suppose in addition that $X$ is regular and $R$ is open. Show that $X/R$ is then a closed subset of $\mathfrak{F}(X)$ with respect to the topology $\mathfrak{T}_\Theta$ (use Proposition 11 of § 7, no. 6).