Exercise 16, § 4

Let $X$ be a connected compact space and let $A, B$ be two disjoint non-empty closed subsets of $X$. Show that there is a component of $C(A \cup B)$ whose closure meets both $A$ and $B$. (Let $U$ be an entourage of the uniformity of $X$ such that $\hat{U}(A) \cap \hat{U}(B) = \varnothing$. Show first that the set $\mathfrak{M}_U$ of components of $C(U(A) \cup U(B))$ which meet both $U(A)$ and $\overline{U(B)}$ is not empty, by proving that for each symmetric entourage $W \subset U$ there exist $x \in U(A)$ and $y \in U(B)$ and a $W$-chain which joins $x$ to $y$ and all of whose points other than $x$ and $y$ belong to $C(U(A) \cup U(B))$, and then applying Exercise 15. For each entourage $V \subset U$, show that every $K \in \mathfrak{M}_V$ contains a set $H \in \mathfrak{M}_U$, and that the set $\mathfrak{M}_{U,V}$ of sets $H \in \mathfrak{M}_U$ which are contained in a set $K \in \mathfrak{M}_V$ is closed in $\mathfrak{F}(X)$; and conclude that as $V$ runs through the set of entourages contained in $U$, the intersection of the sets $\mathfrak{M}_{U,V}$ is not empty).