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.
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.