The nature of number

[Categories+Sheaves]

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?

i) If we're skewing a torus by making the space in the center look like an arbitrarily thin cylinder... (yes yes this becomes a sphere with a line connecting two poles. This was my interpretation of what he said and I ran with it) you can't define a pushforward of the tangent space (of the torus being mapped into 3-space, when you treat it as a manifold) for any of the points that get mapped to a pole.
ii) Failure to be injective. Or at the very least: if he's insisting that the hole have "width zero", that doesn't bode well for the hausdorff-ness/distinctness of the points lying 'across' this hole/gap.
iii) Under that interpretation of jonb's construction (hrm, this looks like the particular shape he was actually talking about... but you still get the shape I was talking about by mapping the torus and then putting it through [some charitable attempt at] the inverse mapping of the sphere) what you end up with is neither a torus nor a sphere, since that 'compressed' shape won't be compact (surely the one-point hole is a limit point?).

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

Either:
a) there is some 'inner ring' that gets mapped to the point that's supposed to be a hole, and this fails to be injective.
b) If the torus' "hole" is genuinely a point: That compactness thing from earlier.

Although the first part of the 'playing different games' comment was probably too overstated/harsh...