Countability over indexed families I'm strugling with countability over indexed sets compared to ordinary sets. Basically we say that any set $A$ is countable if there is a bijective function $f$ such that $f:\mathbb{N}\longrightarrow A$. Here my problem is that I don't know if there is a similar standard definition when we talk about families of the form$\{a_i\mid i\in I\}$. Essentially what I'm looking for is something like: 
"A family of sets is countable if and only if..."
Intuitively I'd say that a family of sets $\{a_i\mid i\in I\}$ is countable if there is bijection between $I$ and $\mathbb{N}$, but here I don't know if this is a consequence of the definition or this can be admited as such. 
Any comments are appreciated. 
Thanks. 
 A: The answer depends on one's definition of "family of sets". There are basically two possibilities:


*

*A family of sets $(a_i)_{i \in I}$ is a mapping $f:I \to A$;

*A family of sets $\{a_i\}_{i \in I}$ is the image of a mapping $f: I \to A$.


(Remark: Both notations (round vs. curly braces) are seen for both concepts; I distinguish them notationally for clarity.)
The difference is in whether we consider $a_i$ to uniquely determine $i$, when a set occurs multiple times in the family.

In the first case, it's easy: set-theoretically, we have $f = \{(i, a_i): i \in I\}$, and it's clearly bijective with $I$ (projection to first coordinate). So in this case, $(a_i)_{i \in I}$ is countable precisely when $I$ is.
In the second case, it's more difficult. Countability of $I$ means that $\{a_i\}_{i \in I}$ is finite or countable. But even if $I$ is uncountable, it may be that $\{a_i\}_{i \in I}$ is countable.

So in either case, because of our definition of "family of sets" as a set, we can apply the definition of countability for sets. But in only one of these cases this can be translated to a definition in terms of $I$.
