つづき

Proofs of the Jordan–Schoenflies theorem
For smooth or polygonal curves, the Jordan curve theorem can be proved in a straightforward way.

Polygonal curve

Continuous curve

Smooth curve
Proofs in the smooth case depend on finding a diffeomorphism between the interior/exterior of the curve and the closed unit disk (or its complement in the extended plane). This can be solved for example by using the smooth Riemann mapping theorem, for which a number of direct methods are available, for example through the Dirichlet problem on the curve or Bergman kernels.[10]

Generalizations

https://en.wikipedia.org/wiki/Arthur_Moritz_Schoenflies
Arthur Moritz Schoenflies (German: [ˈʃøːnfliːs]; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology.
The Schoenflies problem is to prove that an
(n-1)-sphere in Euclidean n-space bounds a topological ball, however embedded.
This question is much more subtle than it initially appears.
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