John VanSickle <evi### [at] kosherhotmailcom> wrote:
> > 4) So the next question is: Does a countably-infinite-dimensional unit
> > cube contain the same amount of points as a unit line? If yes, can you
> > give a 1-to-1 mapping between them?

> Yes, as you can see from above.  The unit line, unit square, unit cube, 
> unit hypercube, etc., all have the same countability.

You demonstrated that an n-dimensional unit cube has the same amount of
points as a unit line. However, that's not what the question is asking.

> > 5) And the logical extreme: Does an uncountably-infinite-dimensional
> > unit cube contain the same amount of points as a unit line? Explain why.
> > (Also explain how the number of dimensions can be uncountably infinite.
> > That seems to defy the definition of "dimension".)

> The hypercube of infinite dimension may have a higher countability than 
> any finite-dimensioned cube, but it is certainly no lower in countability.

I'm not sure that answer makes sense...

-- 
                                                          - Warp

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