Jordan–Schönflies theorem

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In mathematics, the Jordan–Schönflies theorem in geometric topology is a sharpening of the Jordan curve theorem in two dimensions.

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.