Question

Hi All

I have a question.

Imagine a bundle of stickes ||||........ , the first stick is at
position zero , the second at position 1, the third at position 2 , ad
infanitum. This bundle can be easily bijected to Peano's
natural numbers beginning from zero N={0,1,2,3,.........}, so it has
cardinality equal to
Aleph-0.

Now image a two dimensional stick bundle as below
..
..
..
|||||.......
|||||........
|||||........

This also can be bijected to N={0,1,2,3,} , this can be made easily in
a zigzag manner as below:
..
.. .
16 . .
15 . . .
7 14 . .
68 13 . .
259 12 . .
13410 11......

So even this two dimensional bundle of stickes has cardinality equal to
Aleph-0 since it can
be bijected to N= {0,1,2,3,.......}


In a similar manner a three dimentional bundle can be imagined to be
bijected to N and thus
has also cardinality equal to Aleph-0.

In reality any n-dimensional bundle of stickes can be easily bijected
to N and thus has
Aleph-0 cardinality.

My question is what if we had an Aleph-0 dimensional bundle of stickes
, then can that
be bijected to N, and how can that be imagined.

Zuhair

.