Date: Mar 24, 2004 3:41 AM
Author: Robin Chapman
Subject: Re: Hex Win Proof?


Chan-Ho Suh wrote:

> In article <c3p0bk$2fl7ms$1@athena.ex.ac.uk>, Robin Chapman
> <rjc@ivorynospamtower.freeserve.co.uk> wrote:
>

>> Torben Ôø‡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 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_