Definition:Indexing Set/Notation

Definition
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 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.