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'' − 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''−1</sup>), where ''S''<sup>''n''−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. 74–76. {{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]]