Re: thickened sphere

T.Foliakis wrote:
Is the product of a two-sphere S^2 with the unit interval I the same
thing as a "thickened sphere"? I believe a thickened sphere should be
something like the set of points in R^3 having distance between 1-e and
1+e from the origin. doesn't the product structure of S^2xI demand that
[...]

The unit interval [0,1] has the same topology as the interval [10,11].
It seems you were thinking of a number t in [0,1] as if it
parameterized fibers, going away from or else towards the origin.

Suppose we give x, y, z as names for cartesian
coordinate-variables in R^3. Then it's maybe better to
think of t in [0,1] as a w-coordinate argument, where now
we are working in R^4 and the w-axis is perpendicular to
the x, y and z axes. That way S^2 x [0,1] is
the union of shells S^2 x t, where t is varying along the w-axis.

In the same way, S^1 x [0,1] with S^1 having x and y coordinates,
and [0,1] having a z coordinate, gives a union of circles
whose plane is perpendicular to the z-axis: a finite
height cylindrical surface.

If this works, S^2 x [0,1] might be equivalent to a dimension 3
"hyper-cylinder" in R^4...

David Bernier





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