Schönflies Problem
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High Quality Content by WIKIPEDIA articles! In mathematics, the Schönflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schönflies. It states that not only does every simple closed curve in the plane 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 in the plane. An alternative statement is that if C subset mathbb R^2 is a simple closed curve, then there is a homeomorphism f : mathbb R^2 to mathbb R^2 su...
High Quality Content by WIKIPEDIA articles! In mathematics, the Schönflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schönflies. It states that not only does every simple closed curve in the plane 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 in the plane. An alternative statement is that if C subset mathbb R^2 is a simple closed curve, then there is a homeomorphism f : mathbb R^2 to mathbb R^2 such that f(C) is the unit circle in the plane. 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.
