Indexset

An index set is a set used to label members of a collection of objects in mathematics. Given a set I, an indexed family or family indexed by I is a collection {A_i}_{i∈I} where each i ∈ I is associated with an object A_i in some common type (for example sets, groups, vector spaces, topological spaces). The standard notation (A_i)_{i∈I} emphasizes the dependence on i, while I serves as the parameter domain that organizes the collection. The index set is not itself one of the objects; it simply provides a labeling scheme. Two indexings can define the same family if there is a bijection φ: I → J such that A_i = B_{φ(i)} for all i.

Examples include sequences, where I = N and A_n denotes the nth term; families of subsets A_i ⊆ X indexed by I; or families of vector spaces (V_i)_{i∈I}. The product of the family, denoted ∏_{i∈I} X_i, consists of all functions f: I → ⋃ X_i with f(i) ∈ X_i; when every X_i equals the same X, this is the Cartesian product X^I. A related notion is a net: if I is a directed set (a preorder with a common upper bound property), then a family (x_i)_{i∈I} is a net in a space, used to generalize sequences in topological spaces.

In category theory, an indexed family can be viewed as a functor from the discrete category on I to a target category C, illustrating how index sets formalize parameterized families.

Overview: the index set provides a framework for organizing and referencing a collection of objects, influencing notation, operations like products, and convergence concepts in various areas of mathematics.