# If $X$ is a connected subset of a connected space $M$ then the complement of a component of $M \setminus X$ is connected

I have an exercise found on a list but I didn't know how to proceed. Please, any tips?

Let $X$ be a connected subset of a connected metric space $M$. Show that for each connected component $C$ of $M\setminus X$ that $M\setminus C$ is connected.

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Interestingly, this was used to answer this question a few days ago. –  Dejan Govc Jun 12 '12 at 0:08
Ow. What a surprise! So, maybe there is the proof on that book. Thanks. –  Sigur Jun 12 '12 at 0:15
I have written the main ideas of proofs in that answer. They might be useful to you, the details probably won't be too hard to fill in. –  Dejan Govc Jun 12 '12 at 0:18

what is the Theorem II, 4 referred to in the proof? I don't see why $C \cup M$ must be connected... in fact, it seems that $M$ must lie in components of $\mathcal{X} - A$ other than $C$. –  Herng Yi Aug 27 '12 at 13:55