# Jordan theorem

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A plane simple closed curve decomposes the plane into two connected components and is their common boundary. Established by C. Jordan [1]. Together with the similar assertion: A simple arc does not decompose the plane, this is the oldest theorem in set-theoretic topology.

Of the two components, one (the inside of ) is bounded; it is characterized by the fact that the order of every point in it with respect to is ; the other (the outside of ) is unbounded, and the orders of its points with respect to are zero. For any point of the bounded component and every point , there exists a simple arc with ends and and all points of which, except , are contained in (Schoenflies' theorem).

The Jordan (curve) theorem can be generalized according to the dimension: Every -dimensional submanifold of homeomorphic to a sphere decomposes the space into two components and is their common boundary; for this was proved by e dimension','../l/l057830.htm','Lebesgue function','../l/l057840.htm','Lebesgue inequality','../l/l057850.htm','Lebesgue integral','../l/l057860.htm','Lebesgue measure','../l/l057870.htm','Lebesgue summation method','../l/l057940.htm','Lebesgue theorem','../l/l057950.htm','Measure','../m/m063240.htm','Metric space','../m/m063680.htm','Metric theory of functions','../m/m063700.htm','Orthogonal series','../o/o070370.htm','Perron method','../p/p072370.htm','Potential theory','../p/p074140.htm','Regular boundary point','../r/r080680.htm','Singular integral','../s/s085570.htm','Suslin theorem','../s/s091480.htm','Urysohn–Brouwer lemma','../u/u095860.htm','Vitali variation','../v/v096790.htm')" style="background-color:yellow;">H. Lebesgue, and in the general case by L.E.J. Brouwer, whence the -dimensional theorem is sometimes called the Jordan–Brouwer theorem or Jordan–Brouwer separation theorem.

#### References

 [1] C. Jordan, "Cours d'analyse" , 1 , Gauthier-Villars (1893) [2] Ch.J. de la Valleé-Poussin, "Cours d'analyse infinitésimales" , 2 , Libraire Univ. Louvain (1925) [3] P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian) [4] J.A. Dieudonné, "Foundations of modern analysis" , Acad. Press (1961) (Translated from French) [5] W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948) [6] A.F. Filippov, "An elementary proof of Jordan's theorem" Uspekhi Mat. Nauk , 5 : 5 (1950) pp. 173–176 (In Russian)