I saw the following surprising statement in Wikipedia:
When $D\subseteq\Bbb C$ is a simply connected compact set, then its complement $E=D^c$ is a simply connected domain in the Riemann sphere that contains $\infty$, ...
This property sounds like something specific to $S^2$. Is it true that the complement of a compact simply connected subset of $S^n$ is simply connected? Is this true if $D$ is not necessarily compact/closed? How do you prove this?
A related question (which may need to be moved elsewhere): is there a subset $D\subseteq S^2$ such that there are no nontrivial paths in either $D$ or $D^c$?