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In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem. ==Original formulation== 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 is a simple closed curve, then there is a homeomorphism such that is the unit circle in the plane. Elementary proofs can be found in , , and . If the curve is smooth then the homeomorphism can be chosen to be a diffeomorphism. This can also be deduced by solving the Dirichlet problem on the curve (extending results of Kneser, Rado and Choquet); or by showing that the Riemann mapping for the interior of the curve extends smoothly to the boundary, which can be proved either using the Dirichlet problem or Bergman kernels.〔See: * * * * * 〕 Such a theorem is valid only 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Schoenflies problem」の詳細全文を読む スポンサード リンク
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