Definition:Indexing Set/Function

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

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

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 element $i \in I$ under $x$.

When used in this context, the mapping $x$ is referred to as an indexing function for $S$.

Also defined as
Some treatments make the added stipulation that $x$ needs to be surjective for this definition to be valid. This presupposes the condition that the whole of $S$ is thus indexed.

However, the definition as given on does not impose that condition. If such a condition is required for a specific construct, then surjectivity will be imposed as appropriate.

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.