Index set

Not to be confused with indexed sets, or index sets in computability theory.

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.

Examples

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

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

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 1ⁿ, I can efficiently select a poly(n)-bit long element from the set.

See also

• Friendly-index set