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.
Here is the theorem found on Kuratowski's book. Thanks for the reference, it is a very excellent book.
The Theorem II.4 cited above: