indexed family

Indexed Family

An indexed family is a collection of objects (elements, numbers, sets, spaces, …) that are labeled by elements from another set, the index set:

Formally, and indexed family is a function which maps each index $i\in I$ to an element $x_i\in X$:

$$f:I\to Xi\mapsto f(i)=x_i$$

We write $(x_i{)}_{i\in I}$, or sometimes just $(x_i)$ if the index set is clear from context.

Duplicates are allowed, i.e. $x_i=x_j$ for $i\ne j$ is possible, i.e. different indices can point to the same element (not injective).

EXAMPLE

An ordered pair is a family indexed by the set $\{1,2\}$

sequences are indexed by natural numbers

A multiset is a family indexed by a set of sets.