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 may be indexed or labeled by means of the elements of a set , then is an index set. The indexing consists of a surjective function from onto , and the indexed collection is typically called an indexed family, often written as .
Examples An enumeration of a set gives an index set , where is the particular enumeration of . Any countably infinite set can be (injectively) indexed by the set of natural numbers . For , the indicator function on is the function given by
The set of all such indicator 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 Friendly-index set
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