Jordan curve theorem

In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an "inside" and an "outside". The precise mathematical statement is as follows.

Let c be a simple closed curve (i.e. a Jordan curve) in the plane R². Then the complement of the image of c consists of two distinct connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior).

The statement of the Jordan curve theorem seems obvious, but it is a very difficult theorem to prove, and an incorrect proof was originally given by Jordan.

There is a generalisation of the Jordan curve theorem to higher dimensions.

Let X be a continuous, injective mapping of the sphere Sⁿ into Rⁿ⁺¹. Then the complement of the image of X consists of two distinct connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior).

There is a generalisation of the Jordan curve theorem called the Jordan-Schonflies theorem which states that any Jordan curve in the plane can be extended to a homeomorphism of the plane. This is a much stronger statement than the Jordan curve theorem. This generalisation is false in higher dimensions, and a famous counterexample is Alexander's horned sphere. The unbounded component of the complement of Alexander's horned sphere is not simply connected, and so Alexander's horned sphere cannot be extended to all of R³.