Exercise 17, § 4

Let $X$ be a connected locally compact space. Let $K$ be any non-empty compact subset of $X$ distinct from $X$. Show that every component of $K$ meets the frontier of $K$ in $X$ (use the Corollary to Proposition 6 of no. 4). Deduce that if $A$ is any non-empty relatively compact open subset of $X$ distinct from $X$, then the closure of every component of $A$ meets $CA$.