Exercise 15, § 4

Let $X$ be a compact space. Then the space $\mathfrak{F}(X)$ of non-empty closed subsets of $X$, with the topology induced by $\mathfrak{T}(\tilde{U})$ (Exercise 11), is compact (Chapter I, § 9, Exercise 13). Consider an ultrafilter $\Psi$ on $\mathfrak{F}(X)$, and suppose that for each entourage $V$ of the uniformity of $X$, and each $x \in \Psi'$, there exists $M \in \mathfrak{X}$ such that any two points of $M$ can be joined by a V-chain contained in $M$. Show that the limit point $A$ of $\Psi'$ in $\mathfrak{F}(X)$ is a connected subset of $X$ (use Proposition 6 of no. 4).