Jordan Curve Theorem and its Generalizations

The Jordan curve theorem is deceptively simple:

Jordan Curve Theorem Any continuous simple closed curve in the plane, separates the plane into two disjoint regions, the inside and the outside.

For a long time this result was considered so obvious that no one bothered to state the theorem, let alone prove it. The result was first stated as a theorem in Camille Jordan's famous textbook, "Cours d'Analyze de l'École Polytechnique" in 1887, and hence bears his name. Jordan found that proving this theorem is by no means easy, and in fact the proof he gave in his textbook is completely wrong.

The theorem is indeed obvious for smooth curves (hint: standard elementary calculus texts show how to compute the outward normal vector to such a curve) and not too difficult to extend to piecewise smooth curves. However this approach completely breaks down for nowhere smooth simple closed curves like the Koch snowflake shown here:

(You might also take a look at the Java version of the snowflake to appreciate the difficulties that arise when one tries to distinguish between inside and outside points near the curve.)

The first correct proof of the Jordan curve theorem was given by Oswald Veblen in 1905. However his proof left open the question of whether the inside and outside of all such curves were homeomorphic to the inside and outside of the standard circle in the plane (ie. the unit complex numbers). This strong form of the Jordan curve theorem was proved by A. Schönflies in 1906. (His proof contained some errors which were fixed by L E J Brouwer in 1909.)

Jordan-Schönflies Curve Theorem For any simple closed curve in the plane, there is a homeomorphism of the plane which takes that curve into the standard circle.

Brouwer then considered higher dimensional analogs of this question. In 1912 he showed:

Jordan-Brouwer Separation Theorem Any imbedding of the n-1 dimensional sphere into n-dimensional Euclidean space, separates the Euclidean space into two disjoint regions.

Brouwer was unable to prove the analog of the Jordan-Schönflies theorem, that the inside and outside of such an imbedded sphere are homeomorphic to the inside and outside of the standard sphere in Euclidean space (ie. the sphere of unit vectors). In 1921 J. W. Alexander announced that he had a proof of this result. However before he published his paper, he discovered a mistake in his proof. Then in 1924 he discovered a counterexample to the analog of the Jordan-Schönflies theorem in 3-dimensions, what came to be known as "Alexander's horned sphere":

The outside of the horned sphere is not simply connected, unlike the outside of the standard sphere: a loop around one of the horns of the horned sphere can't be continuously deformed to a point without passing through the horned sphere.

In 1948 R. H. Fox and E. Artin gave simpler examples of nonstandard imbeddings of spheres in 3-dimensional Euclidean space:

as well as arcs:

You might also like to check out the following discussion of an analog of the Jordan curve theorem for closed surfaces.