The nature of number

[CliveStaples]

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.

But, to my understanding, nothing gets mapped to the pole--but rather to every neighborhood around the 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.

I don't think I understand what you mean by "lying across this hole". Do you mean points on either side of the hole?

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?).[/quote]

Hmm. That's an interesting line of attack; I hadn't thought of that. The result would immediately follow, since compactness is preserved by homeomorphisms.

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.

Which doesn't make sense; the ring lies on the torus, whereas the hole does not lie on the new shape.

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...

I think this is the easiest proof for me to follow.