9
$\begingroup$

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.

  • 5
    Interestingly, this was used to answer this [question](http://math.stackexchange.com/questions/153376/antisymmetry-among-cut-points/156333#156333) a few days ago.2012-06-12
  • 0
    Ow. What a surprise! So, maybe there is the proof on that book. Thanks.2012-06-12
  • 0
    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

1 Answers 1

6

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

  • 0
    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 $C$.2012-08-27
  • 0
    I edited above.2012-08-27