indexing set
Let $\mathrm{\Lambda}$ and $S$ be sets such that there exists a surjection $f:\mathrm{\Lambda}\to S$. Then $\mathrm{\Lambda}$ is an indexing set for $S$. Also, $S$ is indexed by $\mathrm{\Lambda}$.
In such situations, the elements of $S$ could be referenced by using the indexing set $\mathrm{\Lambda}$, such as $f(\lambda )$ for some $\lambda \in \mathrm{\Lambda}$. On the other hand, quite often, indexing sets are used without explicitly defining a surjective function. When this occurs, the elements of $S$ are referenced by using subscripts (also called indices) which are elements of $\mathrm{\Lambda}$, such as ${s}_{\lambda}$ for some $\lambda \in \mathrm{\Lambda}$. If, however, the surjection from $\mathrm{\Lambda}$ to $S$ were called $s$, this notation would be quite to the function notation: $s(\lambda )={s}_{\lambda}$.
Indexing sets are quite useful for describing sequences, nets, summations, products^{}, unions, and intersections^{}.
Multiple indices are possible. For example, consider the set $X=\{{x}_{aa},{x}_{ab},{x}_{ac},{x}_{bb},{x}_{bc},{x}_{cc}\}$. Some people would consider the indexing set for $X$ to be $\{aa,ab,ac,bb,bc,cc\}$. Others would consider the indexing set to be $\{a,b,c\}\times \{a,b,c\}$. (The double indices can be considered as ordered pairs.) Thus, in the case of multiple indices, it need not be the case that the underlying function $f$ be a surjection. On the other hand, $f$ must be a partial surjection. For example, if a set $X$ is indexed by $A\times B$, the following must hold:

1.
For every $x\in X$, there exist $i\in A$ and $j\in B$ such that $f(i,j)=x$;

2.
For every $i\in A$, the map ${f}_{i}:B\to X$ defined by ${f}_{i}(j)=f(i,j)$ is a partial function^{};

3.
For every $j\in B$, the map ${f}_{j}:A\to X$ defined by ${f}_{j}(i)=f(i,j)$ is a partial function.
Title  indexing set 
Canonical name  IndexingSet 
Date of creation  20130322 16:07:51 
Last modified on  20130322 16:07:51 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  9 
Author  Wkbj79 (1863) 
Entry type  Definition 
Classification  msc 03E99 
Synonym  index set^{} 
Defines  subscript 
Defines  index 
Defines  indices 
Defines  indexed by 
Defines  double indices 
Defines  multiple indices 