Let X = compact connected set in $R^2$. Let $X^c$ be its complement.
Am I right to say this:
The number of connected components of $X^c$ roughly refers to the number of "holes" in $X$. So, I can make $X^c$ contain any arbitrary number of connected components by starting with a closed and bounded set, then deleting the same number of open sets from its interior and defining the resulting set as $X$?