Possible Duplicate:
Bourbaki exercise on connected sets
Let $X \subset M$ be both connected spaces. If $C$ is a connected component of $M-X$, then $M-C$ is connected.
My try: Let´s suppose that $A,B$ form a separation of $M-C$ i.e $ M - C = A \cup B $ and also holds that $ \overline A \cap B = A \cap \overline B = \varnothing $ . Clearly $ M = C \cup A \cup B $ is a disjoint union. And since $C$ it´s a connected component, $C$ it's closed. If I prove that $ A \cup B $ it's closed, then I'm ready. How can I prove that no limit points of $A$ or of $B$, lies in $C$? Please help me!
And clearly I can suppose that X lies in A without loss of generality. But also considering that I don't know how to do it.