Jordan–Schönflies theorem

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In mathematics, the Jordan–Schönflies theorem, or simply the Schönflies theorem, in geometric topology is a 2-dimensional sharpening of the Jordan curve theorem.

It states that not only does every simple closed curve separate the plane into two regions, one (the "inside") bounded and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle. That is, the plane can be stretched and squeezed through a continuous bijection whose inverse is also continuous (the definition of a homeomorphism) to make the simple closed curve become a circle.

Such a theorem is only valid in two dimensions. In three dimensions there are counterexamples such as Alexander's horned sphere. Although they separate space into two regions, those regions are so twisted and knotted that they are not homeomorphic to the inside and outside of a normal sphere.

There does exist a higher-dimensional generalization due to Morton Brown, which is also called the Schönflies theorem. It states that, if an (n − 1)-dimensional sphere S is embedded into an n-dimensional sphere Sⁿ in a locally flat way (that is, the embedding is nice in any small neighborhood), then the pair (Sⁿ, S) is homeomorphic to the pair (Sⁿ, Sⁿ⁻¹), where Sⁿ⁻¹ is the equator of the n-sphere.