Definition:Indexing Set/Family

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

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

Let $x_i$ denote the image of an element $i \in I$ of the domain $I$ of $x$.

Let $\family {x_i}_{i \mathop \in I}$ denote the set of the images of all the elements $i \in I$ under $x$.

The image $\Img x$, consisting of the terms $\family {x_i}_{i \mathop \in I}$, along with the indexing function $x$ itself, is called a family of elements of $S$ indexed by $I$.

Also known as
The object $\family {x_i}_{i \mathop \in I}$ 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 $\family {x_i}_{i \mathop \in I}$ is often referred to as a sequence.