```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 ÃÂÃÂÃÂÃÂ¸ÃÂÃÂidius 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 notjust a matter of the intermediate value theorem.  If one path can bestraightened out, then one can apply the intermediate value theorem,but saying that you can straighten out a path is essentially thecontent of the Jordan curve theorem.  I'm very skeptical of this, because if you could show the paths mustintersect with the intermediate value theorem, then I think you have aproof of Jordan separation with just a little extra work.  So thiswould be a much simpler proof than I've ever seen of that.
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