Jordan-Schoenflies Theorem

The Jordan-Schoenflies Theorem is an extension of the Jordan Curve Theorem, which states that every simple closed curve in the plane divides the plane into an “inside” and “outside” region. The Jordan-Schoenflies Theorem goes further by asserting that not only does the curve separate the plane, but the inside region is homeomorphic to a disk. While the Jordan Curve Theorem is considered obviously true for all dimensions, the Jordan-Schoenflies Theorem is only true in two dimensions, and failed quite famously in three dimensions due to the existence of wild embeddings.

Formally if there is a there is a simple closed curve:

$$C\subset {\mathbb{R}}^2$$

Then there exists a homeomorphism:

$$f:{\mathbb{R}}^2\to {\mathbb{R}}^2$$