How to denote a set of nested tuples?

Question

I want to define a set that elements that are nested tuples, like this one:

$$a =((1, 2), (1, 2, 3), (1, 2))$$

I want this so I can index elements and write $a_{2,3} = 3$.

My first idea was simply writing: $\mathbb{R}^2 × \mathbb{R}^3 × \mathbb{R}^2$. But this equals $\mathbb{R}^7$. So the tuples it contains couldn't have two indexes.

Is there a notation I can use for tuples of tuples?

Do you think $((1,2),(1,2,3),(1,2))=(1,2,1,2,3,1,2)$? If not, then why do you say that $\Bbb R^2\times\Bbb R^3\times\Bbb R^2$ "equals" $\Bbb R^7$? — anon, Jul 20 '14 at 07:28
@blue The wiki page on Tuples makes me think that the parentheses can be ignored. — Calmarius, Jul 20 '14 at 07:36
They can be, but do you see how my comment answers your question? — anon, Jul 20 '14 at 07:36

score 1 · Answer 1 · answered Jul 20 '14 at 07:58

$a$ is a vector of vectors, not necessarily from the same space. Thus $a_i$ has it's conventional meaning of the $i$th component. Then $a_{i,j}$ naturally has the meaning you want it to. You could also use $a_{ijk\dots}$. Also the two sets you mention are not equal, but isomorphic as many spaces eg. as real vector spaces. They are probably homeomorphic too.

How to denote a set of nested tuples?

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