Exercise 13, § 9

With the notation of Exercise 13, endow $\mathfrak{F}(X)$ and $\mathfrak{K}(X)$ with the topology induced by the topology $\mathfrak{T}_\Theta$ on $\mathfrak{B}_0(X)$ (§ 8, Exercise 12).

a) Show that if $X$ is quasi-compact, then so is $\mathfrak{F}(X)$. (For each open set $U$ in $X$, let $c(U)$ [resp. $m(U)$] be the set of all closed subsets $M$ of $X$ such that $M \subset U$ (resp. $M \cap U \neq \varnothing$); the $c(U)$ and $m(U)$ form a subbase of $\mathfrak{T}_\Theta$. Then use Exercise 1).

b) Suppose that $X$ is accessible, and let $X'$ be the image of $X$ under the mapping $x \mapsto \{x\}$ of $X$ into $\mathfrak{F}(X)$. Let $\mathfrak{H}$ be a subspace of $\mathfrak{F}(X)$ which contains $X'$. Show that if $\mathfrak{H}$ is quasi-compact then so is $X$. [If $(U_\alpha)$ is an open covering of $X$, remark that if $\mathfrak{B}_\alpha$ denotes the set of all $M \in \mathfrak{F}(X)$ such that $M \cap U_\alpha \neq \varnothing$, then $\{ \mathfrak{B}_\alpha \}$ is an open covering of $\mathfrak{F}(X)$].

c) Show that if $X$ is locally compact, then $\mathfrak{K}(X)$ is open in $\mathfrak{F}(X)$ (use Proposition 10 of § 7).

d) Suppose that $X$ is accessible. Show that $\mathfrak{K}(X)$ is Hausdorff (resp. regular, compact, locally compact) if and only if $X$ is. [For the first two assertions, use Exercise 12 of § 8 and Exercise 3 of § 9; for the last two assertions, use a) and b) and Exercise 12].