Index set

In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set $A$ may be indexed or labeled by means of the elements of a set $J$, then $J$ is an index set. The indexing consists of a surjective function from $J$ onto $A$, and the indexed collection is typically called an indexed family, often written as ${A_{j}}_{j∈J}|undefined$.

Examples

 * An enumeration of a set $S$ gives an index set $$J \sub \N$$, where $f : J → S$ is the particular enumeration of $S$.
 * Any countably infinite set can be (injectively) indexed by the set of natural numbers $$\N$$.
 * For $$r \in \R$$, the indicator function on $r$ is the function $$\mathbf{1}_r\colon \R \to \{0,1\}$$ given by $$\mathbf{1}_r (x) := \begin{cases} 0, & \mbox{if } x \ne r \\ 1, & \mbox{if } x = r. \end{cases} $$

The set of all such indicator functions, $$\{ \mathbf{1}_r \}_{r\in\R}$$, is an uncountable set indexed by $$\mathbb{R}$$.

Other uses
In computational complexity theory and cryptography, an index set is a set for which there exists an algorithm $I$ that can sample the set efficiently; e.g., on input $1^{n}$, $I$ can efficiently select a poly(n)-bit long element from the set.