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Assuming the Jordan Curve Theorem, we can consider the 2 connected components of the complement of the simple closed curve C in the Riemann sphere. I am trying to establish the Jordan-Schoenflies theorem via Caratheodory's mapping theorem. Is there a basic way of establishing that the connected components are simply connected so that we can get a conformal mapping from the unit disk to the component and hence use Caratheodory's theorem?

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  • $\begingroup$ See Thm. 5.7 in Marshall's Complex Analysis. I also asked a similar question on MO partially inspired by your question mathoverflow.net/questions/447853/… $\endgroup$
    – D.R.
    May 30, 2023 at 6:43

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The Jordan Curve Theorem says that any simple closed curve $C$ in the two-dimensional sphere $S^2$ separates $S^2$ into two connected regions (that is, $S^2 \setminus C$ has exactly two nonempty connected components).

The Jordan-Schoenflies Theorem strengthens this by stating that these two regions are homeomorphic to an open disk.

Based only on the Jordan Curve Theorem, is there a basic way of establishing the connected components are simply connected? It depends on your understanding of "basic", but it seems to me that the simple connectedness of the regions is not a straightforward corollary of Jordan Curve. In fact, you have to strengthen the proof.

Look at higher dimensions, for example an embedded copy $C$ of $S^2$ in the three-dimensional sphere $S^3$. Then $S^3 \setminus C$ also has exactly two nonempty connected components, but they are not necessarily simply connected. An example for this phenomenom is the Alexander horned sphere. See https://en.wikipedia.org/wiki/Alexander_horned_sphere.

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    $\begingroup$ Thanks for the answer. I am aware of the example of the Alexander horned sphere and the extension to higher dimensions via Jordan-Brouwer Theorem. What I wanted to know is how to prove the Schoenflies Theorem via Caratheodory's mapping theorem. The proof as outlined here (en.wikipedia.org/wiki/Schoenflies_problem#cite_ref-6) is starightforward but only if one can apply Riemann mapping which needs the components to be simply connected. Hence my question. $\endgroup$
    – shc
    Nov 22, 2018 at 22:52
  • $\begingroup$ The horned sphere is an example that the components are not necessarily simply connected. This is a special feature in dimension two, and it is not trivial. The proof goes far deeper than just invoking Jordan Curve. That is, there is a gap between Jordan curve and the applicability of Caratheodory. $\endgroup$
    – Paul Frost
    Nov 22, 2018 at 23:01
  • $\begingroup$ Certainly. However I am only interested in the dimension 2 case. If a proof without invoklng Schoenflies can be given then the RMT/Caratheodory proof will be complete. Else there seems to be some circular reasoning here, atleast in my head. $\endgroup$
    – shc
    Nov 22, 2018 at 23:04
  • $\begingroup$ As I said: To fill the gap you have to go back to the origin and prove more than Jordan Curve. Simple connectedness is not an addition you will get almost for free. $\endgroup$
    – Paul Frost
    Nov 22, 2018 at 23:12
  • $\begingroup$ @shc If you want, I shall delete my answer. This gives you a better chance that somebody else has a look to you question (if a question has an answer, then this is visible at first glance in the list of questions, and many people will not have a closer look in this case). $\endgroup$
    – Paul Frost
    Nov 23, 2018 at 8:20

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