Definition:Indexing Set/Family

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

Let $x: I \to S$ be an indexing function for $S$.

The codomain $S$ consisting of the terms $\left \langle {x_i} \right \rangle_{i \mathop \in I}$, along with the indexing function $x$ itself, is called a family of elements of $S$ indexed by $I$.

Also defined as
The usage here follows.

It is more usual to find the term family of elements of $S$ indexed by $I$ to refer to the mapping $x$ itself.

However, it feels more intuitively clear to refer to the mapping itself as a function and the set which is the codomain as a family.

Hence the usage on.

Also known as
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 $\left \langle {x_i} \right \rangle_{i \in I}$ is often referred to as a sequence.