History and Further Proofs
The statement of the Jordan curve theorem may seem obvious at first, but it is a rather difficult theorem to prove. Bernard Bolzano was the first to formulate a precise conjecture, observing that it was not a self-evident statement, but that it required a proof. It is easy to establish this result for polygonal lines, but the problem came in generalizing it to all kinds of badly behaved curves, which include nowhere differentiable curves, such as the Koch snowflake and other fractal curves, or even a Jordan curve of positive area constructed by Osgood (1903).
The first proof of this theorem was given by Camille Jordan in his lectures on real analysis, and was published in his book Cours d'analyse de l'École Polytechnique. There is some controversy about whether Jordan's proof was complete: the majority of commenters on it have claimed that the first complete proof was given later by Oswald Veblen, who said the following 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, Thomas C. Hales wrote:
| “ | 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 pointed out that the special case of simple polygons is not only an easy exercise, but was not really used by Jordan anyway, and quoted Michael 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. | ” |
Jordan's proof and another early proof by de la Vallée-Poussin were later critically analyzed and completed by Shoenflies (1924).
Due to the importance of the Jordan curve theorem in low-dimensional topology and complex analysis, it received much attention from prominent mathematicians of the first half of the 20th century. Various proofs of the theorem and its generalizations were constructed by J. W. Alexander, Louis Antoine, Bieberbach, Luitzen Brouwer, Denjoy, Hartogs, Kerékjártó, Alfred Pringsheim, and Schoenflies.
Some new elementary proofs of the Jordan curve theorem, as well as simplifications of the earlier proofs, continue to be carried out.
A short elementary proof of the Jordan curve theorem was presented by A. F. Filippov in 1950.
- A proof using the Brouwer fixed point theorem by Maehara (1984).
- A proof using non-standard analysis by Narens (1971).
- A proof using constructive mathematics by Gordon O. Berg, W. Julian, and R. Mines et al. (1975).
- A proof using non-planarity of the complete bipartite graph K3,3 was given by Thomassen (1992).
- A simplification of the proof by Helge Tverberg.
The first formal proof of the Jordan curve theorem was created by Hales (2007a) in the HOL Light system, in January 2005, and contained about 60,000 lines. Another rigorous 6,500-line formal proof 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.
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