1) Does a unit square contain the same amount of points as a unit line?
(We are talking about real numbers here.)

2) If yes, that means there has to be a 1-to-1 mapping between those
points. Give a function that expresses such a mapping.

3) If the answer to the first question is yes, then it follows that
the amount of points inside a unit cube is also the same as the amount
of points on a unit line. The same for a four-dimensional hypercube,
and so on. Can you give a generic function that gives a 1-to-1 mapping
between a unit line and an n-dimensional unit cube?

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?

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".)

-- 
                                                          - Warp

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