Inmathematics, anindex set is aset whose members label (or index) members of another set.^[1][2] For instance, if the elements of a setA may beindexed orlabeled by means of the elements of a setJ, thenJ is an index set. The indexing consists of asurjective function fromJ ontoA, and the indexed collection is typically called anindexed family, often written as{A_j}_j∈J.

• Anenumeration of a setS gives an index set${\displaystyle J\subset \mathbb{N}}$, wheref :J →S is the particular enumeration ofS.
• Anycountably infinite set can be (injectively) indexed by the set ofnatural numbers${\displaystyle \mathbb{N}}$.
• For${\displaystyle r\in \mathbb{R}}$, theindicator function onr is the function${\displaystyle {\mathbf{1}}_r:\mathbb{R}\to \{0,1\}}$ given by

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

Incomputational complexity theory andcryptography, an index set is a set for which there exists an algorithmI that can sample the set efficiently; e.g., on input1ⁿ,I can efficiently select a poly(n)-bit long element from the set.^[3]

• Friendly-index set

• Mathematical notation
• Basic concepts in set theory

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