Exercise 14, § 4

Let $X$ be a compact space, and let $\mathfrak{B}$ be a filter base on $X$ consisting of connected closed sets. Show that the (non-empty) intersection of the sets of $\mathfrak{B}$ is closed and connected (argue by reductio ad absurdum, using Proposition 4 of no. 3 and Chapter I, § 9, no. 1, Theorem 1).