Exercise 29, § 8

a) Let R be a separated equivalence relation in a topological space X. In the space $\mathfrak{F}(X)$ of the non-empty closed subsets of X, endowed with the topology induced by $\mathfrak{T}_\Theta$ (I, p. 101, exerc. 12), now prove that every $A \in \mathfrak{F}(X)$ adhering to $X/R$ is contained in an equivalence class according to R (reason by contradiction).

b) Suppose in addition that X is regular and the relation R is open. Now prove that $X/R$ is then closed in $\mathfrak{F}(X)$ for the topology $\mathfrak{T}_\Theta$ (use prop. 11 of I, p. 51).

¶(30) Let X be a regular space. Now prove that, in order that X satisfy the condition (D_v) of I, p. 90, exerc. 7, it is necessary and sufficient that every locally finite family of non-empty open sets in X be countable. (To prove that the condition is necessary, consider a locally finite set $\mathfrak{U}$ of open sets in X. Endow X with a well-ordering; the set P of the $x \in X$ which are smallest elements (for this order) of an element of $\mathfrak{U}$ is equipotent to $\mathfrak{U}$; if for every $x \in P$, $V(x)$ is one of the sets of U having x as smallest element, the set $\mathscr{V}$ of the $V(x)$ for $x \in P$ is locally finite and equipotent to $\mathfrak{U}$, and for $y \in P$ and $y > x$, one has $x \notin V(y)$. Using the regularity of X and prop. 4 of I, p. 6, now prove that for every $x \in P$, there exists an open set $W(x)$ such that $x \in W(x) \subset V(x)$ and that, when x runs through P, the $\overline{W(x)}$ are pairwise without common point).