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?
$a$ 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.