It is my understanding that if I write to write a collection of sets as $\{E_i\}$, I am implying that the set is countable, and what I realy mean is $E_1, E_2, E_3, \ldots$. However, if I write $\{E_{\alpha} \}_{\alpha \in I}$, there is a possibility that the collection is arbitrary.
I am trying to understand why this notation works. If I set $I = \mathbb{N}$, then the two notations are equivalent. In theory, there could be a strictly increasing function that maps $i$ to $f(i) \in I$, but I am not sure of what exactly the "contents" of $I$ are. Could there, for example, be an index $0.1$? Could $I = [0,1]$? If the collection is arbitrary, is $I$ itself an uncountable set?
Generally function $f=(F,A,B)$ is defined by triple, where $A$, $B$ are sets, $F$ is functional graph and domain $pr_1F=A$. Sometimes functional graph is called family, the domain is called index set and the range $pr_2F=B$ is called set of elements of family. Indicial notation $f_x$ is used to denote the value of $f$ at the element $x$.