Whether the unbounded connected component of the complement of the closure of a bounded simply connected domain in the extended complex plane $\overline C$ is a Jordan domain (in $\overline C= S^2\subset R^3$)?
Explanation: If $G\subset \overline C$ is a bounded simply connected domain, then $H:=$ unbounded connected component of Interior$(\overline C\setminus G)$ is a symply connected domain in $\overline C$. My question is whether $H$ is a Jordan domain in $\overline C$?
It is very natural question! I am trying to find and define the smallest Jordan domain $G'$ containing a simply connected bounded domain $G$ in the complex plane (it will be equal to the complement of $H$, provided the answer to the question is affirmative)!