> > the Moebius theorem seems to precede JCT by about 50 years. > >
Proginoskes wrote 17 Feb 2006: > > The usual proof of Mobius's proof is through the use of Euler's > > formula > > for polyhedra: > > v - e + f = 2 > > In the "map" with 5 mutual adjacencies, f = 5, e = C(5,2) = 10, and v <= 2/3 * 10. (Thus v <= 6.) Substituting into this equation, we get > > 2 = v - e + f <= 6 - 10 + 5 = 1. > > Of course, Euler's polyhedron formula may have been proved using the JCT. > > --- Christopher Heckman >
So, if the Moebius theorem of 4 maximum mutual adjacencies uses the Euler formula for a proof means, then we can see the Jordan Curve theorem is a corollary or lesser statement of the Moebius theorem. For if there were a path that did not cross the closed loop, the path would be establishing a 5th mutual adjacency.
I have always felt that a simple statement in math, begging for a proof, has a simple easy quick proof, if one is willing to look hard enough. And that when people come up with long boring "case studies" as proofs, they are not mathematicians but fakesters who like to grovel in case studies.
So now I have two proofs in mathematics that require their more general statement proven first and then the corollary gives the other proof, Beal with FLT and now Moebius with Jordan Curve Theorem.
Any other path to a proof of Jordan Curve Theorem is just grovelling in the mud of case studies.