Indexed family

In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers, is a collection of real numbers, where a given function selects one real number for each integer (possibly the same) as indexing.

More formally, an indexed family is a mathematical function together with its domain ${\displaystyle I}$ and image ${\displaystyle X}$ (that is, indexed families and mathematical functions are technically identical, just points of view are different). Often the elements of the set ${\displaystyle X}$ are referred to as making up the family. In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set ${\displaystyle I}$ is called the index set of the family, and ${\displaystyle X}$ is the indexed set.

Sequences are one type of families indexed by natural numbers. In general, the index set ${\displaystyle I}$ is not restricted to be countable. For example, one could consider an uncountable family of subsets of the natural numbers indexed by the real numbers.