Date: Mar 23, 2004 10:08 PM
Author: Chan-Ho Suh
Subject: Re: Hex Win Proof?

In article <c3p0bk$2fl7ms$1@athena.ex.ac.uk>, Robin Chapman

<rjc@ivorynospamtower.freeserve.co.uk> 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.