In article <firstname.lastname@example.org>, Robin Chapman wrote: >> 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'd be suspicious of any use of well-known curve theorems without going over their proofs and making sure they apply to paths on the Hex board, because a path on the Hex board can, without intersecting itself, close off a region of the board.