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