Indexmenge

Indexmenge, in mathematics, is the German term for index set. An index set I is a set whose elements label members of a family of objects {A_i} or similar, with i ranging over I. The index set itself is just a set and need not carry extra structure, though in many contexts it has additional features such as an order or a directed structure.

In a typical situation, one writes a family of objects indexed by I as {A_i | i ∈ I}. The index set serves only to label the members; the actual values of the A_i are determined by the indices. Examples include a finite collection {a_1, a_2, a_3} with index set {1, 2, 3}, a matrix with entries a_{ij} indexed by I for rows and J for columns, or a sequence (x_n) where I is the set of natural numbers N.

The index set is central to operations such as summation or product over a family: ∑_{i∈I} a_i or ∏_{i∈I} a_i. In analysis and topology, indexed families are used to define nets, where convergence is described with respect to a directed index set I. A directed set provides a notion of eventual behavior needed for limits that extend beyond sequences.

Index sets can be finite, countable, or uncountable, and they may be ordered or unordered depending on the context. When multiple indexing dimensions are involved, one uses product index sets like I × J.

See also: indexing, indexed family, net, directed set.