| Jordan Curve Theorem |
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| CATEGORIES ABOUT JORDAN CURVE THEOREM | |
| topology | |
| mathematical theorems | |
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.
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
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