\begin{definition}[Definition:Indexing Set/Family of Subsets]
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 $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$ indexed by $I$.
Then $\family {S_i}_{i \mathop \in I}$ is referred to as an '''indexed family of subsets (of $S$ by $I$)'''.
\end{definition}