# Index set

In mathematics, an **index set** is a set whose members label (or index) members of another set.^{[1]}^{[2]} 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[edit]

- An enumeration of a set S gives an index set , where
*f*:*J*→*S*is the particular enumeration of*S*. - Any countably infinite set can be (injectively) indexed by the set of natural numbers .
- For , the indicator function on
*r*is the function given by

The set of all such indicator functions, , is an uncountable set indexed by .

## Other uses[edit]

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.

^{[3]}

## See also[edit]

## References[edit]

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