Exercise 14, § 9

The notations being those of Exercise 13, we endow $\mathfrak{F}(X)$ and $\mathscr{R}(X)$ with the topology induced by the topology $\mathfrak{T}_\Theta$ on $\mathfrak{P}_0(X)$ (I, p. 101, Exercise 12).

a) Now prove that if X is quasi-compact, the same is true of $\mathfrak{F}(X)$. (For every open set U in X, let $c(U)$ (resp. $m(U)$) be the set of closed subsets M of X such that $M \subset U$ (resp. $M \cap U \neq \varnothing$). The $c(U)$ and the $m(U)$ form a generator system of the topology of $\mathfrak{F}(X)$; use then Exercise 1 of I, p. 105).

b) Suppose that X is accessible; let X’ be the image of X by the mapping $x \mapsto \{x\}$ of X into $\mathfrak{F}(X)$. Let $\mathfrak{R}$ be a subspace of $\mathfrak{F}(X)$ containing X’; prove that if $\mathfrak{R}$ is quasi-compact, X is quasi-compact. (If $(U_\alpha)$ is an open covering of X, observe that, denoting by $\mathscr{V}_\alpha$ the set of the $M \in \mathfrak{F}(X)$ such that $M \cap U_\alpha \neq \varnothing$, $(\mathscr{V}_\alpha)$ is an open covering of $\mathfrak{F}(X)$.)

c) If X is locally compact, $\mathscr{R}(X)$ is open in $\mathfrak{F}(X)$ (use Proposition 10 of I, p. 65).

d) Suppose that X is accessible. Prove that, for $\mathscr{R}(X)$ to be separated (resp. regular, compact, locally compact), it is necessary and sufficient that X be so. (For the first two assertions, use Exercise 12 of I, p. 101 and Exercise 3 of I, p. 105; for the last two, use a), b) and Exercise 13.)