From: Brandsma Subject: Re: Jordan Curve theorem. Date: Thu, 27 May 1999 14:16:09 +0200 Newsgroups: sci.math Keywords: Where to find proofs of the Jordan Curve Theorem? Carl Christian Kjelgaard Mikkelsen wrote: > Greetings ! > > I have been fighting the Jordan curve theorem on and off for some time > now. The theorem says that a closed (f(a)=f(b)) continuous curve f > : [a,b] -> R2, with no double points, that is f : (a,b) -> R2 is > injective, will divide the plane into two connected components, one > which is unbounded and another which is bounded. Does any one know a > simple proof of this theorem ? I would like a reference, please. > I do not think there is such a thing as a really easy proof, but the most elementary I know is the one you can find in Jan van Mill's book "infinite-dimensional topology", the proof there only uses the Brouwer fixed point theorem for dimension 2 (of which an elementary proof can also be found in the chapter Plane Topology). Henno Brandsma > Yours sincerly > > Carl Christian K. Mikkelsen > Aarhus University > Denmark ============================================================================== From: jmccarty@sun1307.ssd.usa.alcatel.com (Mike McCarty) Subject: Re: Jordan Curve theorem. Date: 27 May 1999 18:43:12 GMT Newsgroups: sci.math Churchill give a pretty simple proof in his "Complex Analysis" (or whatever the name of his book on the theory of functions of a complex variable). In article <374D2421.7CF6FD13@daimi.au.dk>, Carl Christian Kjelgaard Mikkelsen wrote: [original post requoted -- djr] ---- char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);} This message made from 100% recycled bits. I don't speak for Alcatel <- They make me say that. ============================================================================== From: Simba Karkhanis Subject: Re: Jordan Curve theorem. Date: Mon, 31 May 1999 11:04:58 -0400 Newsgroups: sci.math I learned complex analysis from Ahlfor's book Complex Analysis. As a supplement, my prof gave out a one page proof due to Estermann on the Jordan curve theorem. I can type it up if interested. "Benjamin P. Carter" wrote: > Ahlfors's graduate text on complex analysis does not include the Jordan > Curve theorem, because to do so would require a nontrivial amount of > topology, and Ahlfors states that the Jordan Curve theorem is never needed > for complex analysis. > > Books on algebraic topology typically prove the generalized Jordan Curve > theorem: if you remove from the n-sphere S^n a subset C homeomorphic to > S^(n-1), what remains has two components whose common boundary is C. > The proof depends on homology theory. > -- > Ben Carter ============================================================================== From: ray@nmsu.edu (R. Mines) Subject: Re: Jordan Curve theorem. Date: 01 Jun 1999 14:53:36 -0600 Newsgroups: sci.math You can find a simple proof of the Jordan Curve Theorem in The constructive Jordan curve theorem, {\sl Rocky Mountain J. Math.,} {\bf 2} (1975), 225--236 (G.~Berg, W.~Julian, R.~Mines and F.~Richman). It is constructive in that the proof shows how to determine if a point in in the bounded or unbounded regions, and it show how to contruct a path joing two points that are in the same component of the complement. -- Ray Mines ray@nmsu.edu Mathematical Sciences New Mexico State University Las Cruces, New Mexico 88003