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?(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Connected sets

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