\begin{definition}[Definition:Indexing Set/Indexed Set]
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 $x$.
The image of $x$, that is, $x \sqbrk I$ or $\Img x$, is called an '''indexed set'''.
That is, it is the '''set indexed by $I$'''
\end{definition}