Exercise 15, § 11

Let $X$ be a connected, locally connected space. a) Let $M, N$ be two disjoint non-empty closed subsets of $X$. Show that $C(M \cup N)$ has a component whose closure meets both $M$ and $N$. [If $C$ is a component of $C(M \cup N)$ whose closure does not meet $M$, observe that $C$ is both open and closed in $CN$]. b) If $A$ is a closed subset of $X$ distinct from $X$, deduce from a) that every component of $A$ meets the closure of $CA$.