Let $\Lambda$ and $S$ be sets such that there exists a surjection $f \colon \Lambda \to S$ Then $\Lambda$ is an indexing set for $S$ Also, $S$ is indexed by $\Lambda$

In such situations, the elements of $S$ could be referenced by using the indexing set $\Lambda$ such as $f(\lambda)$ for some $\lambda \in \Lambda$ 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 $\Lambda$ such as $s_{\lambda}$ for some $\lambda \in \Lambda$ If, however, the surjection from $\Lambda$ to $S$ were called $s$ this notation would be quite similar 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 $f_i \colon B \to X$ defined by $f_i(j)=f(i,j)$ is a partial function;
3. For every $j\in B$ the map $f_j \colon A \to X$ defined by $f_j(i)=f(i,j)$ is a partial function.