Definition:Indexing Set/Family of Sets

Definition
Let $S$ be a set and let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $I$ be an indexing set.

Let $\left \langle{S_i}\right \rangle_{i \mathop \in I}$ be a family of elements of $\mathcal P \left({S}\right)$ indexed by $I$.

Then is common to take for granted that all the elements of $\mathcal P \left({S}\right)$ are subsets of some set $S$.

Thus $\left \langle{S_i}\right \rangle_{i \mathop \in I}$ is referred to as an indexed family of sets.

Thus the phrase:
 * an (indexed) family $\left \langle {x_i} \right \rangle$ of subsets of $S$

is taken to mean:
 * a mapping $x$ from some indexing set $I$ into the power set $\mathcal P \left({S}\right)$ of $S$.

Also known as
Technically speaking, it is more correct to refer to $\left \langle{S_i}\right \rangle_{i \mathop \in I}$ as an indexed family $\left \langle {S_i} \right \rangle$ of subsets of $S$.

It is common to drop the word indexed and refer merely to a family of sets.