Exercise 12, § 8

Let $X$ be a topological space which has at least two points. a) Show that the topologies $\mathfrak{T}_\Omega$ and $\mathfrak{T}_\Phi$ on $\mathfrak{P}_0(X)$ (§ 2, Exercise 7) are not topologies of an accessible space. b) Let $\mathfrak{T}_\Theta$ denote the least upper bound of the topologies $\mathfrak{T}_\Omega$ and $\mathfrak{T}_\Phi$. Show that the set $\mathfrak{F}(X)$ of non-empty closed subsets of $X$, with the topology induced by $\mathfrak{T}_\Theta$, is a Kolmogoroff space, and that if $X$ is accessible, so is $\mathfrak{F}(X)$. c) Supposing that $X$ is accessible, show that the space $\mathfrak{F}(X)$ (with the topology induced by $\mathfrak{T}_\Theta$) is Hausdorff if and only if $X$ is regular.