indexed family

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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:

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

which maps each index $i\in I$ to an element $x_i\in X$.
We denote it as $(x_i{)}_{i\in I}$

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 $M:U\to {\mathbb{N}}_0$ mapping each element from $U$ to its multiplicity in the multiset.