An explicative definition of what is meant by $\{A_i\}_{i\in I}$?

Question

What does $\{A_i\}_{i\in I}$ mean exactly?

I know it's an index, but what exactly is that?

Answer 1

It denotes a collection $\mathscr A$ of objects, and it means there is surjection $f: I\rightarrow \mathscr A$. One often has that $f$ is injective, so that objects $f(i) = A_i\in \mathscr A$ and $f(j)= A_j\in\mathscr A$ corresponding to distinct indices $i\in I$ and $j\in I$ are themselves distinct, but this need not be the case.

It simply means that there are sufficiently many members of $I$ to enumerate the members of $\mathscr A$.

Answer 2

To supplement MPW's answer and the comments, the other important detail about this notation is absolutely nothing is implied about the cardinality (size) of the index set $I$. It could be a finite collection, countably infinite collection, uncountably infinite....we use the term "an arbitrary collection" to indicate any particular size.