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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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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
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1 Answer

up vote 3 down vote accepted

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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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
    
I edited above. –  Sigur Aug 27 '12 at 23:55
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