Jordan Curve Theorem - Proof and Generalizations

Proof and Generalizations

The Jordan curve theorem was independently generalized to higher dimensions by H. Lebesgue and L.E.J. Brouwer in 1911, resulting in the Jordan–Brouwer separation theorem.

Let X be a topological sphere in the (n+1)-dimensional Euclidean space Rn+1, i.e. the image of an injective continuous mapping of the n-sphere Sn into Rn+1. Then the complement Y of X in Rn+1 consists of exactly two connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior). The set X is their common boundary.

The proof uses homology theory. It is first established that, more generally, if X is homeomorphic to the k-sphere, then the reduced integral homology groups of Y = Rn+1 \ X are as follows:

This is proved by induction in k using the Mayer–Vietoris sequence. When n = k, the zeroth reduced homology of Y has rank 1, which means that Y has 2 connected components (which are, moreover, path connected), and with a bit of extra work, one shows that their common boundary is X. A further generalization was found by J. W. Alexander, who established the Alexander duality between the reduced homology of a compact subset X of Rn+1 and the reduced cohomology of its complement. If X is an n-dimensional compact connected submanifold of Rn+1 (or Sn+1) without boundary, its complement has 2 connected components.

There is a strengthening of the Jordan curve theorem, called the Jordan–Schönflies theorem, which states that the interior and the exterior planar regions determined by a Jordan curve in R2 are homeomorphic to the interior and exterior of the unit disk. In particular, for any point P in the interior region and a point A on the Jordan curve, there exists a Jordan arc connecting P with A and, with the exception of the endpoint A, completely lying in the interior region. An alternative and equivalent formulation of the Jordan–Schönflies theorem asserts that any Jordan curve φ: S1 → R2, where S1 is viewed as the unit circle in the plane, can be extended to a homeomorphism ψ: R2 → R2 of the plane. Unlike Lebesgues' and Brouwer's generalization of the Jordan curve theorem, this statement becomes false in higher dimensions: while the exterior of the unit ball in R3 is simply connected, because it retracts onto the unit sphere, the Alexander horned sphere is a subset of R3 homeomorphic to a sphere, but so twisted in space that the unbounded component of its complement in R3 is not simply connected, and hence not homeomorphic to the exterior of the unit ball.