Jordan–Schönflies theorem 713320 224730608 2008-07-10T04:23:59Z 190.21.231.202 In [[mathematics]], the '''Jordan–Schönflies theorem''', or simply the '''Schönflies theorem''', of [[geometric topology]] is a sharpening of the [[Jordan curve theorem]]. ==Formulation== It states that not only does every [[simple closed curve]] in the [[plane (mathematics)|plane]] separate the plane into two regions, one (the "inside") [[bounded set|bounded]] and the other (the "outside") unbounded; but also that these two regions are [[homeomorphic]] to the inside and outside of a standard [[circle]]. Putting this more precisely: The plane can be mapped onto itself through a [[continuous function (topology)|continuous]] [[bijection]] whose [[inverse]] is also continuous (the definition of a [[homeomorphism]]) to make the simple closed curve become a circle; the inside and outside of the curve become those of the circle. Such a theorem is only valid in two dimensions. In three dimensions there are [[counterexample]]s 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. ==Generalizations== There does exist a higher-dimensional generalization due to [[Morton Brown]] and independently [[Barry Mazur]] with [[Marston Morse]], which is also called the '''Schönflies theorem'''. It states that, if an (''n''&nbsp;&minus;&nbsp;1)-dimensional [[sphere]] ''S'' is embedded into the ''n''-dimensional sphere ''S<sup>n</sup>'' in a [[locally flat]] way (that is, the embedding extends to that of a thickened sphere), then the pair (''S<sup>n</sup>'', ''S'') is homeomorphic to the pair (''S<sup>n</sup>'', ''S''<sup>''n''&minus;1</sup>), where ''S''<sup>''n''&minus;1</sup> is the equator of the ''n''-sphere. Brown and Mazur received the [[Veblen Prize]] for their contributions. The Schonflies problem can be posed in categories other than the topologically locally flat category, i.e. does a smoothly (piecewise-linearly) embedded ''n-1''-sphere in the ''n''-sphere bound a smooth (piecewise-linear) ''n''-ball? For ''n'' = 4, the problem is still open for both categories. See [[Mazur manifold]]. ==References== *Brown, Morton (1960), A proof of the generalized Schoenflies theorem. ''Bull. Amer. Math. Soc.'', vol. 66, pp. 74&ndash;76. {{MathSciNet| id = 0117695}} *Mazur, Barry, On embeddings of spheres., ''Bull. Amer. Math. Soc.'' 65 1959 59--65. {{MathSciNet| id = MR0117693}} *Morse, Marston, A reduction of the Schoenflies extension problem., ''Bull. Amer. Math. Soc.'' 66 1960 113--115. {{MathSciNet| id = 0117694}} [[Category:Geometric topology]] [[Category:Homeomorphisms]] [[Category:Mathematical theorems]] [[de:Satz von Schönflies]] [[es:Teorema de Jordan-Schönflies]]