Definition:Indexing Set/Note on Terminology

Definition
Let $I$ and $S$ be sets.

Let $x: I \to S$ be a family of sets indexed by $I$.
 * (This terminology is not absolutely established, but it is one of the standard choices among related slight variants; in the sequel it and it alone will be used.) An unacceptable but generally accepted way of communicating the notation and indicating the emphasis is to speak of a family $\left\{{x_i}\right\}$ in $X$, or of a family $\left\{{x_i}\right\}$ of whatever the elements of $X$ may be; when necessary, the index set $I$ is indicated by some such parenthetical expression as $\left({i \in I}\right)$. Thus, for instance, the phrase "a family $\left\{{A_i}\right\}$ of subsets of $X$" is usually understood to refer to a function $A$, from some set $I$ of indices, into $\mathcal P \left({X}\right)$.
 * $\S 9$: Families