# indexing set

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 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. 1.

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

2. 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. 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.

 Title indexing set Canonical name IndexingSet Date of creation 2013-03-22 16:07:51 Last modified on 2013-03-22 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