Jordan curve theorem

Illustration of the Jordan curve theorem. The Jordan curve (drawn in black) divides the plane into an "inside" region (light blue) and an "outside" region (pink).

In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane (also known as a Jordan curve) divides the plane into an "inside" and an "outside" region, and any path connecting a point of one region to a point of the other intersects that loop somewhere. The first to give a proof was Camille Jordan (1887). It has generally been thought that his proof was flawed and that Veblen (1905) gave the first rigorous proof, but this view has been disputed by Hales (2007b).

Theorem and proofs

The precise mathematical statement is as follows.

Let c be a simple closed curve (i.e. a Jordan curve) in the plane R². Then the complement of the image of c consists of two distinct connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior). The image of c is the boundary of each component.

The statement of the Jordan curve theorem seems obvious, but it was a very difficult theorem to prove. It was easy to establish the result for simple curves such as polygonal lines but the problem came in generalising it for all kind of curves which included nowhere differentiable curves such as the Koch snowflake. The first to formulate a version of the theorem, observing that it was not a self-evident statement but one requiring proof, was Bernard Bolzano.

The first proof was given by Camille Jordan (1887) in his book series Cours d'analyse de l'École Polytechnique, after whom the theorem is named. There is some controversy about whether Jordan's proof is complete: the majority of authors have claimed that the first complete proof was given by Oswald Veblen (1905), who said about Jordan's proof: "His proof, however, is unsatisfactory to many mathematicians. It assumes the theorem without proof in the important special case of a simple polygon, and of the argument from that point on, one must admit at least that all details are not given". However Hales (2007b) said "Nearly every modern citation that I have found agrees that the first correct proof is due to Veblen...In view of the heavy criticism of Jordan’s proof, I was surprised when I sat down to read his proof to find nothing objectionable about it. Since then, I have contacted a number of the authors who have criticized Jordan, and each case the author has admitted to having no direct knowledge of an error in Jordan’s proof." Hales also points out that the special case of simple polygons is not only an easy exercise, but is not really used by Jordan anyway, and quotes Reeken as saying "Jordan’s proof is essentially correct... Jordan’s proof does not present the details in a satisfactory way. But the idea is right and with some polishing the proof would be impeccable".

Several simpler proofs have been found, such as one using the Brouwer fixed point theorem by Maehara (1984), a proof using non-standard mathematics by Narens (1971), and a proof using constructive mathematics by Gordon O. Berg, W. Julian, and R. Mines et al. (1975). The first formal proof was found by Hales (2007a) in the HOL Light system, in January 2005 and has about 60,000 lines. Another rigorous 6,500-line formal proof of the Jordan curve theorem was produced in 2005 by an international team of mathematicians using the Mizar system. Both the Mizar and the HOL Light proof rely on libraries of previously proved theorems, so these two sizes are not comparable. Nobuyuki Sakamoto and Keita Yokoyama (2007) showed that the Jordan curve theorem is equivalent in proof-theoretic strength to the weak König's lemma.

Generalizations

There is a generalisation of the Jordan curve theorem to higher dimensions (sometimes called the Jordan-Brouwer Separation Theorem).

Let X be a continuous, injective mapping of the sphere Sⁿ into Rⁿ⁺¹. Then the complement of the image of X consists of two distinct connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior). The image of X is their common boundary.

There is another generalisation of the Jordan curve theorem, called the Jordan–Schönflies theorem, which states that any Jordan curve in the plane, viewed as a mapping of the circle S¹ into the plane R², can be extended to a homeomorphism of the plane. This is a much stronger statement than the Jordan curve theorem. This generalisation is false in higher dimensions, and a famous counterexample is Alexander's horned sphere. The unbounded component of the complement of Alexander's horned sphere is not simply connected, and so the mapping of Alexander's horned sphere cannot be extended to all of R³.

Another generalization of the Jordan curve theorem states that if M is any compact, connected boundaryless n-dimensional sub-manifold of Rⁿ⁺¹, then M separates Rⁿ⁺¹ into two regions: one compact, the other non-compact.

References

• Berg, Gordon O.; Julian, W.; Mines, R.; Richman, Fred (1975), "The constructive Jordan curve theorem", The Rocky Mountain Journal of Mathematics 5: 225–236, MR0410701, ISSN 0035-7596 
• Hales, Thomas C. (2007a), "The Jordan curve theorem, formally and informally", The American Mathematical Monthly 114 (10): 882–894, MR2363054, ISSN 0002-9890 
• Hales, Thomas (2007b), "Jordan's proof of the Jordan Curve theorem", Studies in logic, grammer and rhetoric 10 (23), http://mizar.org/trybulec65/4.pdf 
• Jordan, Camille (1887), Cours d'analyse, pp. 587-594, http://www.maths.ed.ac.uk/~aar/jordan/jordan.pdf 
• Maehara, Ryuji (1984), "The Jordan Curve Theorem Via the Brouwer Fixed Point Theorem", The American Mathematical Monthly (Mathematical Association of America) 91 (10): 641–643, MR0769530, ISSN 0002-9890, http://www.jstor.org/stable/2323369 
• Narens, Louis (1971), "A nonstandard proof of the Jordan curve theorem", Pacific Journal of Mathematics 36: 219–229, MR0276940, ISSN 0030-8730, http://projecteuclid.org/euclid.pjm/1102971282 
• Sakamoto, Nobuyuki; Yokoyama, Keita (2007), "The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic", Archive for Mathematical Logic 46 (5): 465–480, doi:10.1007/s00153-007-0050-6, MR2321588, ISSN 0933-5846 
• Veblen, Oswald (1905), "Theory on Plane Curves in Non-Metrical Analysis Situs", Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 6 (1): 83–98, ISSN 0002-9947, http://www.jstor.org/stable/1986378 

External links

• The full 6,500 line formal proof of Jordan's curve theorem in Mizar.
• Historical material relating to the Jordan curve theorem
• A simple proof of Jordan curve theorem (PDF)