Jordan curve theorem
While the theorem seems intuitively obvious, it takes some ingenuity to prove it by elementary means.
"Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it."
More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.
A Jordan arc in the plane is the image of an injective continuous map of a closed and bounded interval
Theorem — Let X be an n-dimensional topological sphere in the (n+1)-dimensional Euclidean space Rn+1 (n > 0), i.e. the image of an injective continuous mapping of the n-sphere Sn into Rn+1.
It is first established that, more generally, if X is homeomorphic to the k-sphere, then the reduced integral homology groups of Y = Rn+1 \ X are as follows: This is proved by induction in k using the Mayer–Vietoris sequence.
When n = k, the zeroth reduced homology of Y has rank 1, which means that Y has 2 connected components (which are, moreover, path connected), and with a bit of extra work, one shows that their common boundary is X.
There is a strengthening of the Jordan curve theorem, called the Jordan–Schönflies theorem, which states that the interior and the exterior planar regions determined by a Jordan curve in R2 are homeomorphic to the interior and exterior of the unit disk.
In particular, for any point P in the interior region and a point A on the Jordan curve, there exists a Jordan arc connecting P with A and, with the exception of the endpoint A, completely lying in the interior region.
An alternative and equivalent formulation of the Jordan–Schönflies theorem asserts that any Jordan curve φ: S1 → R2, where S1 is viewed as the unit circle in the plane, can be extended to a homeomorphism ψ: R2 → R2 of the plane.
Unlike Lebesgue's and Brouwer's generalization of the Jordan curve theorem, this statement becomes false in higher dimensions: while the exterior of the unit ball in R3 is simply connected, because it retracts onto the unit sphere, the Alexander horned sphere is a subset of R3 homeomorphic to a sphere, but so twisted in space that the unbounded component of its complement in R3 is not simply connected, and hence not homeomorphic to the exterior of the unit ball.
[6] In reverse mathematics, and computer-formalized mathematics, the Jordan curve theorem is commonly proved by first converting it to an equivalent discrete version similar to the strong Hex theorem, then proving the discrete version.
[7] In image processing, a binary picture is a discrete square grid of 0 and 1, or equivalently, a compact subset of
: Both graph structures fail to satisfy the strong Hex theorem.
For this reason, when computing connected components in a binary image, the 6-neighbor square grid is generally used.
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.
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.
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.
[12]However, Thomas C. Hales wrote: Nearly every modern citation that I have found agrees that the first correct proof is due to Veblen...
[14]Earlier, Jordan's proof and another early proof by Charles Jean de la Vallée Poussin had already been critically analyzed and completed by Schoenflies (1924).
[15] 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, Ludwig Bieberbach, Luitzen Brouwer, Arnaud Denjoy, Friedrich Hartogs, Béla Kerékjártó, Alfred Pringsheim, and Arthur Moritz Schoenflies.
of the largest disk contained in the closed region bounded by the Jordan curve.
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 in reverse mathematics the Jordan curve theorem is equivalent to weak Kőnig's lemma over the system
In computational geometry, the Jordan curve theorem can be used for testing whether a point lies inside or outside a simple polygon.
Jordan curve theorem proof implies that the point is inside the polygon if and only if n is odd.
Adler, Daskalakis and Demaine[19] prove that a computational version of Jordan's theorem is PPAD-complete.
