Definition:Indexing Set/Family of Subsets

Definition
Let $S$ be a set.

Let $I$ be an indexing set.

For each $i \in I$, let $S_i$ be a corresponding subset of $S$.

Let $\left \langle{S_i}\right \rangle_{i \mathop \in I}$ be a family of subsets of $S$ indexed by $I$.

Then $\left \langle{S_i}\right \rangle_{i \mathop \in I}$ is referred to as an indexed family of subsets (of $S$ by $I$).

Also defined as
As the set of subsets of $S$ is the power set $\mathcal P \left({S}\right)$ of $S$, $\left \langle{S_i}\right \rangle_{i \mathop \in I}$ can also be defined as a mapping from some indexing set $I$ into the power set $\mathcal P \left({S}\right)$ of $S$.

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