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 {Aj}jJ.


  • 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 such indicator functions,   , is an uncountable set indexed by  .

Other usesEdit

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.[3]

See alsoEdit


  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.