indekset

An indekset, often called an index set, is a set used to label or index a family of objects. It provides a uniform way to refer to each member of the family by associating to every index i in the set I a corresponding object A_i. The index set itself does not impose any particular structure on the family unless I carries additional structure (for example, being ordered).

Notation and concepts

A family indexed by I is written as {A_i}_{i∈I} or (A_i)_{i∈I}, where each i maps to A_i. The index set I can be any set, and the objects A_i can be numbers, vectors, topologies, functions, spaces, or other mathematical objects. If I has an order, the family can be treated as a sequence; if I is merely a set without order, the family is an unordered collection.

Typical constructions

Many constructions rely on the index set. For example, the product over I of a family {A_i} is written ∏_{i∈I} A_i and consists of all choices of an element from each A_i. Direct sums, limits, or colimits indexed by I are defined analogously. In analysis and probability, an indexed family {X_i}_{i∈I} may represent random variables or functional elements parameterized by I.

Examples

- A_i = real numbers, yielding a family of real values indexed by I.

- A_i = vector spaces, giving a family of spaces whose product ∏_{i∈I} A_i is a common construction.

- X_i for i in I as random variables on a shared probability space.

Remarks

The index set I is not the same as the objects A_i; different index sets can index isomorphic or identical families. The concept is fundamental across areas of mathematics for organizing parameterized families and for defining constructions that depend on a parameter.