Why are these sets disjoint? (Munkres' Topology) There's a step in a proof in Munkres' Topology that doesn't make sense to me.
Theorem: Let A be a connected subspace in X. If $A \subset B \subset \bar{A}$, then B is also connected.
Proof: Let A be connected, and $A \subset B \subset \bar{A}$. Suppose that $B = C \cup D $ is a separation of B. A must thus lie entirely in C or D. Suppose  $A \subset C$. Then, $\bar{A} \subset \bar{C}$; Since $\bar{C}$ and D are disjoint, B cannot intersect D.
Here he has lost me. Why must $\bar{C}$ and $D$ be disjoint? Sure, C and D are disjoint, but why must the closure of C and D be disjoint as well? 

$\overline C$ and $D$ are disjoint by Lemma 23.1 on p.148 of Munkres.
