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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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    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.2012-06-12

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Here is the theorem found on Kuratowski's book. Thanks for the reference, it is a very excellent book. print screen


The Theorem II.4 cited above:

Theorem II.4

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    I edited above.2012-08-27