View source for Jordan theorem

A plane simple closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j0543701.png" /> decomposes the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j0543702.png" /> into two connected components and is their common boundary. Established by C. Jordan [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j0543703.png" />) is bounded; it is characterized by the fact that the order of every point in it with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j0543704.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j0543705.png" />; the other (the outside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j0543706.png" />) is unbounded, and the orders of its points with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j0543707.png" /> are zero. For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j0543708.png" /> of the bounded component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j0543709.png" /> and every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437010.png" />, there exists a simple arc with ends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437012.png" /> and all points of which, except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437013.png" />, are contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437014.png" /> (Schoenflies' theorem).

The Jordan (curve) theorem can be generalized according to the dimension: Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437015.png" />-dimensional submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437016.png" /> homeomorphic to a sphere decomposes the space into two components and is their common boundary; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437017.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437018.png" />-dimensional theorem is sometimes called the Jordan–Brouwer theorem or Jordan–Brouwer separation theorem.

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



====Comments====
Jordan's theorem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437019.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437020.png" />) is often called the Jordan curve theorem; it was first proved rigorously by O. Veblen [[#References|[a4]]].

C. Kuratowski strengthened Schoenflies' theorem by showing that there is in fact a [[Homeomorphism|homeomorphism]] from the closed unit disc that maps the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437021.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437022.png" /> and the interior onto the inside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437023.png" />; the Riemann mapping theorem (cf. [[Riemann theorem|Riemann theorem]]) then actually gives a homeomorphism that is analytic in the interior.

The analogue of this strengthening is false in higher dimensions, as is shown by the famous  "horned sphere of Alexanderhorned sphere"  of J.W. Alexander (see [[#References|[a1]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bing,   "The geometric topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054370/j05437024.png" />-manifolds" , Amer. Math. Soc.  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Dugundji,   "Topology" , Allyn &amp; Bacon  (1966)  (Theorem 8.4)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. van Mill,   "Infinite-dimensional topology, prerequisites and introduction" , North-Holland  (1988)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  O.Veblen,   "Theory of plane curves in non-metrical Analysis Situs"  ''Trans. Amer. Math. Soc.'' , '''6'''  (1905)  pp. 83–98</TD></TR></table>