# Jordan theorem

A plane simple closed curve $\Gamma$ decomposes the plane $\mathbf R^2$ 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 $\Gamma$) is bounded; it is characterized by the fact that the order of every point in it with respect to $\Gamma$ is $\pm1$; the other (the outside of $\Gamma$) is unbounded, and the orders of its points with respect to $\Gamma$ are zero. For any point $x$ of the bounded component $A$ and every point $x_0\in\Gamma$, there exists a simple arc with ends $x_0$ and $x$ and all points of which, except $x_0$, are contained in $A$ (Schoenflies' theorem).

The Jordan (curve) theorem can be generalized according to the dimension: Every $(N-1)$-dimensional submanifold of $\mathbf R^N$ homeomorphic to a sphere decomposes the space into two components and is their common boundary; for $N=3$ 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 $N$-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)

Jordan's theorem in $\mathbf R^2$ (or $\mathbf C$) is often called the Jordan curve theorem; it was first proved rigorously by O. Veblen [a4].
C. Kuratowski strengthened Schoenflies' theorem by showing that there is in fact a homeomorphism from the closed unit disc that maps the boundary $S^1$ onto $\Gamma$ and the interior onto the inside of $\Gamma$; the Riemann mapping theorem (cf. Riemann theorem) then actually gives a homeomorphism that is analytic in the interior.
 [a1] R.H. Bing, "The geometric topology of $3$-manifolds" , Amer. Math. Soc. (1983) [a2] J. Dugundji, "Topology" , Allyn & Bacon (1966) (Theorem 8.4) [a3] J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1988) [a4] O.Veblen, "Theory of plane curves in non-metrical Analysis Situs" Trans. Amer. Math. Soc. , 6 (1905) pp. 83–98