# 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 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 (Aj)jJ.

## Examples

• An enumeration of a set S gives an index set , where f : JS is the particular enumeration of S.
• Any countably infinite set can be indexed by the set of natural numbers .
• For , the indicator function on r is the function given by The set of all the functions is an uncountable set indexed by .

## Other uses

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

## See also

1. Weisstein, Eric. "Index Set". Wolfram MathWorld. Wolfram Research. Retrieved 30 December 2013.
2. Munkres, James R. (2000). Topology. 2. Upper Saddle River: Prentice Hall.
3. Goldreich, Oded (2001). Foundations of Cryptography: Volume 1, Basic Tools. Cambridge University Press. ISBN 0-521-79172-3.