If A is a connected subset of a disconnected set X s.t. X=MUN , M,N nonempty closed disjoint sets, how do we show, A is either contained in M or in N? I can start a proof, but then, I am kind of stuck. I would go by contradiction and say A intersection M is non empty and A intersection N is non empty. Hence A would be the union of 2 non empty disjoint sets. But since A is connected, A intersection M and A intersection N cannot be both open. So without loss of generality, say A intersection M is not open. Hence X\(A intersection M) is not closed. Then I get stuck. Any help?