TITLE: If $X$ is a connected subset of a connected space $M$ then the complement of a component of $M \setminus X$ is connected
QUESTION [11 upvotes]: 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.

REPLY [7 votes]: 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: