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 $\left \langle {x_i} \right \rangle_{i \in I}$.

The mapping $x$ itself is called a family of elements of $S$ indexed by $I$.

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

Some sources call an indexed family an indexed collection.

The family $x$ is often seen with one of the following notations:


 * $\left \langle {x_i} \right \rangle_{i \in I}$


 * $\left({x_i}\right)_{i \in I}$


 * $\left\{{x_i}\right\}_{i \in I}$

There is little consistency in the literature.

The subscripted $i \in I$ is often left out, if it is obvious in the particular context.

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