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

Then $\left \langle{S_i}\right \rangle_{i \in I}$ is called an indexed family $\left \langle {S_i} \right \rangle$ of subsets of $S$.

Also known as
It is common to drop the word indexed and refer merely to a family of subsets.

It is also common to take for granted that all the elements of $\mathcal P \left({S}\right)$ are subsets of some set $S$, and merely refer to a family of sets.