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An index set is just the domain $I$ of some function $f:I\to X$. It's just a notational distinction between a function domain and an index set - when we think if it as an index set, we write $f_i$ rather than $f(i)$.
Both the Wikipedia and Wolfram links you provide indicate that the function $f$ should be $1-1$ and onto, but I don't actually think that is necessary. For example, if we have a sequence $a_1,\dots,a_n,\dots$ then the index set is $\mathbb N$ whether or not the $a_i$ are distinct.