(July 16, 2012 at 2:37 pm)jonb Wrote:(July 16, 2012 at 2:14 pm)CliveStaples Wrote: 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).
Is the same true for a sphere enclosing a sphere? At what point does the surface of the object become the object?
I'm not sure what you mean by "a sphere enclosing a sphere." Suppose that S1 is a sphere of radius r1, and S2 a sphere of radius r2. If r2>r1, and S1 and S2 share the same center, then S2 encloses S1. This doesn't change S1's topology or S2's topology.
Or do you mean the surface enclosing the region 'between' the spheres?
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