## Complement of a simply connected domain [closed]

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)!

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 The question you ask does not quite match the title of your post. Moreover, please see mathoverflow.net/howtoask and mathoverflow.net/faq#whatnot – Yemon Choi Nov 26 2011 at 19:47 Looks like homework AND does not make any sense. – Igor Rivin Nov 26 2011 at 19:48 I hope that, the question is now clear. – Marijan Nov 26 2011 at 20:04 You still don't say why you want to know the answer, or give evidence that you've done some special cases or tried to use existing results – Yemon Choi Nov 26 2011 at 21:13