Proof that interior of a simple closed curve in plane is simply connected

Question

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?

Answer 1

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.