\begin{definition}[Definition:Indexing Set/Notation]
The family of elements $x$ of $S$ indexed by $I$ is often seen with one of the following notations:
:$\family {x_i}_{i \mathop \in I}$
:$\paren {x_i}_{i \mathop \in I}$
:$\set {x_i}_{i \mathop \in I}$
There is little consistency in the literature, but $\paren {x_i}_{i \mathop \in I}$ is perhaps most common.
The preferred notation on {{ProofWiki}} is $\family {x_i}_{i \mathop \in I}$.
The subscripted $i \in I$ is often left out, if it is obvious in the particular context.
Note the use of $x_i$ to denote the image of the index $i$ under the indexing function $x$.
As $x$ is actually a mapping, one would expect the conventional notation $\map x i$.
However, this is generally not used, and $x_i$ is used instead.
Category:Definitions/Indexed Families
\end{definition}