Definition:Indexing Set/Family of Sets

Definition
Let $\mathcal S$ be a set of sets.

Let $I$ be an indexing set.

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

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

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

Also see

 * Definition:Indexed Family of Subsets, when each of $S_i$ is a subset of a set $S$.