The nature of number

[CliveStaples]

1. This mapping of the torus i) fails to be differentiable ii) fails to be an embedding (even in the weakest topological sense) iii) does not seem to be well-defined (holes of zero width are o.k. if we're just deleting a point from a plane, but have you thought about what it means for a cylinder to have zero width?)

i) Proof? It seems like you're actually claiming that every mapping of the torus with these particular endpoints fails to be differentiable (since no particular map was given).

ii) Proof? This result would actually imply (i), since differentiable functions are necessarily continuous.

iii) I suspect there's actually a bijection from the surface he described (a 'torus' with a hole consisting of one point) to the unit torus, which would mean that the mapping is well-defined (in the sense of being a function). Or did you mean that the resulting object actually fails to be a torus?

4. A reprise of #1: Topological properties aren't invariant under the stuff you're doing to torii/spheres here, so you and Perelman are playing different games in yet another sense..

This is basically just another claim that any mapping with the endpoints he gave must fail to be homeomorphic. Can you prove this?