Date: Mar 23, 2004 10:08 PM
Author: Chan-Ho Suh
Subject: Re: Hex Win Proof?
In article <firstname.lastname@example.org>, Robin Chapman
> 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.