Schoenflies problem

The 1908 proved by Arthur Moritz Schoenflies set of Schoenflies forms an essential link between the topology and the combinatorial problem of coloring maps (four- color theorem ). Clearly he states: Malt is a closed curve (without crossovers ) to a rubber blanket, then you can forgive the cloth so that the curve of a circle.

Set

It is a closed Jordan curve and denote the unit circle. Then each homeomorphism can be continued to a homeomorphism.

Higher dimensions

The immediate generalization of Schoenflies to higher dimensions is not valid, since in three dimensions Alexander sphere (see, and web link ) provides a counterexample.

In contrast, Morton Brown has generalized the theorem as follows: If one - dimensional sphere locally flat embedded in a - dimensional sphere, so the couple is homeomorphic to, the equator of the sphere is. (This embedding is locally flat, if there is an embedding, which matches up with. )

Conclusion

The set of Schoenflies attracts immediately the Jordan curve theorem by itself: the two disjoint regions, in which is decomposed, are straight ( the restricted area ) and ( the infinite domain ).