Nice proof of the Jordan curve theorem? - MathOverflow

Nice proof of the Jordan curve theorem?

2009-12-11T02:28:16Z

As a student, I was taught that the Jordan curve theorem is a great example of an intuitively clear statement which has no simple proof.

What is the simplest known proof today?

Is there an intuitive reason why a very simple proof is not possible?

Answer by fuzzytron for Nice proof of the Jordan curve theorem?

2009-12-11T02:50:40Z

There's a short proof (less than three pages) that uses Brouwer's fixed point theorem, available here:

The Jordan Curve Theorem via the Brouwer Fixed Point Theorem

The goal of the proof is to take Moise's "intuitive" proof and make it simpler/shorter. Not sure whether you'd consider it "nice," though.

Answer by Kevin Lin for Nice proof of the Jordan curve theorem?

2009-12-11T03:06:20Z

It depends on what you mean by "simple". If you know homology, the proof is not very hard (less than 1 page), see for example, section 2.B ("Classical Applications") of Hatcher's book "Algebraic Topology".

Answer by ivane for Nice proof of the Jordan curve theorem?

2009-12-11T05:01:00Z

You should compare with: "Geometric Topology in Dimensions 2 and 3", Moise, Edwin E. (1977). Springer-Verlag and tell

Answer by Konrad Swanepoel for Nice proof of the Jordan curve theorem?

2009-12-11T14:21:56Z

Carsten Thomassen's proof is relatively simple:

Carsten Thomassen, The Jordan-Schönflies theorem and the classification of surfaces. Amer. Math. Monthly 99 (1992), no. 2, 116-130.

By the way, the Jordan Curve Theorem has a formal proof (one that can be checked by a computer): Thomas C. Hales, The Jordan curve theorem, formally and informally. Amer. Math. Monthly 114 (2007), no. 10, 882-894.

Hales bases the formal proof on Thomassen's.

The following is a survey on the older papers on the subject:

H. Guggenheimer, The Jordan curve theorem and an unpublished manuscript by Max Dehn. Archive for History of Exact Sciences 17 (1977), 193-200.

Answer by Ady for Nice proof of the Jordan curve theorem?

2009-12-11T16:26:00Z

Several proofs are here:

http://www.maths.ed.ac.uk/~aar/jordan/index.htm

Among them, Tverberg's (1980) could (and should) be mentioned.

But, after reading (and reading)

http://www.math.sunysb.edu/~bishop/classes/math401.F09/HalesDefense.pdf ,

I really like Jordan's proof itself.

Answer by Ronnie Brown for Nice proof of the Jordan curve theorem?

2012-01-08T10:25:58Z

There is a proof of the Jordan Curve Theorem in my book "Topology and groupoids" which also derives results on the Phragmen-Brouwer Property. Also published as

`Groupoids, the Phragmen-Brouwer property and the Jordan curve theorem', J. Homotopy and Related Structures 1 (2006) 175-183.

The van Kampen Theorem for the fundamental groupoid on a set of base points is used to prove that if $X$ is pathconnected and the union of open path connected sets $U,V$ whose intersection has $n$ path components, then the fundamental group of $X$ contains the free group on $n-1$ generators as a retract.

May 30: The question asks why there is not a simple proof. Perhaps the following Figure 9.10 from the above book will explain why a proof is not expected to be so so easy; how do you decide whether a point in the middle is inside or outside?

Answer by nemesiso for Nice proof of the Jordan curve theorem?

2012-05-30T10:13:38Z

A nice and simple proof using mod 2 intersection theory is given in the book Differential Topology by Guillemin,Pollack.

Answer by Vladimir Kanovei for Nice proof of the Jordan curve theorem?

2013-05-11T19:35:27Z

An elementary proof by means of nonstandard analysis (by reduction to the case of polygons) and elementary combinatorics is given in Kanovei & Reeken, A nonstandard proof of the Jordan curve theorem, RAE 1999, 24, 161--170