Jordan curve theorem

The Jordan curve theorem is a result in the mathematical branch of topology.

Statement

Each closed Jordan curve in the Euclidean plane decomposed into two disjoint these areas, their common border is the Jordan curve and whose union is the whole plane with the Jordan curve. Exactly one of the two areas is limited.

History

This sentence seems so obvious that generations of mathematicians have used it without explicitly formulate it, let alone to prove it. The evidence is, however, extremely difficult and time-consuming. A first - yet incorrect - proof attempt was published in 1887 by Camille Jordan in the third volume of his work Cours d'analyze de l' Ecole Polytechnique. The first correct proof of the Jordan curve theorem was provided in 1905 by Oswald Veblen. The Jordan curve theorem is now about in geographic information systems application in the point -in- polygon test by Jordan.

Generalization

Jordan - Brouwer separation theorem

The Jordan curve theorem was generalized by Luitzen Brouwer to the so-called Jordan - Brouwer separation theorem. This theorem states that the complement of a compact -dimensional submanifold of exactly two connected components has. One of the two has the property that its conclusion is a compact manifold with boundary whose boundary is precisely the mentioned submanifold. The proof of this theorem is usually done with the mapping degree or with the help of algebraic topology.

Set of Schoenflies

Another generalization is the set of Schoenflies, after each homeomorphism between the unit circle and a Jordan curve in the plane can be continued to the whole plane. Here, the generalization does not apply to higher dimensions.