Jordan curve theorem
In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an "inside" and an "outside". It was proved by Oswald Veblen in 1905. 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). Also, 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. The first to attempt a proof was Bernard Bolzano, followed by a number of other mathematicians including Camille Jordan, after whom the theorem is named. None could provide a correct proof, until Oswald Veblen finally did so in 1905. Several alternative proofs were found since then.
A rigorous 200,000-line formal proof of the Jordan curve theorem was produced in 2005 by an international team of mathematicians using the Mizar system.
Generalizations
There is a generalisation of the Jordan curve theorem to higher dimensions.
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 a generalisation of the Jordan curve theorem called the Jordan-Schönflies theorem which states that any Jordan curve in the plane 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.
References
- Oswald Veblen, Theory on plane curves in non-metrical analysis situs, Transactions of the American Mathematical Society 6 (1905), pp. 83–98.
- Ryuji Maehara, The Jordan curve theorem via the Brouwer fixed point theorem, American Mathematical Monthly 91 (1984), no. 10, pp. 641–643.