Indexed family of sets

An indexed family of sets is a function

from an index set $I$ to the class of sets. It is typically denoted by $\{A_i\}_{i\in I}$.

Indexed families let one take unions/intersections over arbitrary index sets (including infinite ones) and are pervasive in analysis (e.g., sequences of sets correspond to the case $I=\mathbb{N}$).

Examples:

• A sequence of sets $(A_n)_{n\in\mathbb{N}}$ is an indexed family with $I=\mathbb{N}$.
• In $\mathbb{R}$, $A_n := (-1/n,1/n)$ for $n\in\mathbb{N}$ defines an indexed family of open intervals.
• If $I=\{1,2,3\}$ and $A_1=\{0\}$, $A_2=\{1,2\}$, $A_3=\varnothing$, then $\{A_i\}_{i\in I}$ is a finite indexed family.