The nature of number

[CliveStaples]

Thanks for your help, but before we move into other dimensional use’s of this form. I think it would be nice to examine a possibility if this graph stands.



We seem to have a range of zeros or a stretched zero which is not a point, but that has length.
And although the value does not change we can see a definite progression along the line.

Is this a problem for theories such as those of Grigori Perelman. As a flow around an object will not necessarily be able to categorise what the shape is as, a distance of zero could contain enough differences to produce an inconsistent result..

My thinking is this way; I have a tours. I press the middle until it is almost connected, but is not actually connected. The central hole exists, but it occupies Zero space. If I categorise the shape by the flow over it, it will fulfil all the criteria of being a sphere, however if I categorise the same shape by the internal side of this zero space skin, it is in all effect a torus. We can do the same the other way round.
Now how could you examine a shape to find holes of zero size and get any meaningful result? So is it not so that Grigori Perelman’s theorem is only proof to a given resolution and not an absolute?

The torus would be topologically distinguishable from a sphere--any loop enclosing the hole could not be shrunk to a point while remaining on the torus, whereas every loop on a sphere can be shrunk to a point.

If they're topologically distinguishable, then they will be geometrically distinguishable (since geometric manifolds incorporate topological manifolds).