Definition:Indexing Set/Function

Definition
Let $I$ and $S$ be sets.

Let $x: I \to S$ be a mapping.

Let the domain $I$ of $x$ be the indexing set of $\left \langle {x_i} \right \rangle_{i \mathop \in I}$.

The mapping $x$ itself is called an indexing function for $S$.

Also known as
The mapping $x$ is often called a family of elements of $S$ indexed by $I$.

The object $\left \langle {x_i} \right \rangle_{i \mathop \in I}$ is often referred to as an $I$-indexed family.

Some sources call an indexed family an indexed collection.

Also see
If the the indexing set $I$ is finite or countable (and in particular if $I \subseteq \N$), then the family $\left \langle {x_i} \right \rangle_{i \mathop \in I}$ is often referred to as a sequence.