> In article <email@example.com>, Robin Chapman > <firstname.lastname@example.org> wrote: > >> Torben ÃÂÃÂÃÂÃÂ¸ÃÂÃÂidius Mogensen wrote: >> >> > >> > Indeed, if we want to prove it to a mathematician who does not already >> > accept the intermediate value theorem (of which the intersection >> > property is a simple consequence). >> >> Is it? >> >> Now I presume this "intersection property" can be paraphrased as >> "a path with endpoints at two opposite vertices of a square with >> all other points in the interior of the square must meet a path >> with endpoints at the other two vertices of the square with >> all other points in the interior of there square". >> >> That's a simple consequence of the intermediate value theorem, is it? >> >> I must be stupid, since the only way I can see to prove that is >> using the Jordan Curve Theorem. :-( > > I don't think you're being stupid, unless I'm being stupid also :-) > > I can't see a way to prove this without Jordan separation. It's not > just a matter of the intermediate value theorem. If one path can be > straightened out, then one can apply the intermediate value theorem, > but saying that you can straighten out a path is essentially the > content of the Jordan curve theorem.
More than that --- it's almost the Schoenflies theorem. On the other hand, if one is dealing with a path on a lattice, like we are doing here, then one can do the straightening stepwise and end us with a nice "theta" shape which we can apply the IVT to.
> I'm very skeptical of this, because if you could show the paths must > intersect with the intermediate value theorem, then I think you have a > proof of Jordan separation with just a little extra work. So this > would be a much simpler proof than I've ever seen of that.
-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html "Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9" Francis Wheen, _How Mumbo-Jumbo Conquered the World_