Indexed family of sets

An indexed family of sets is a function $A\colon I\to\mathcal C$ where $I$ is an index set and $\mathcal C$ is a set whose elements are sets. One writes the family as $(A_i)_{i\in I}$ , where $A_i:=A(i)$ .

Remarks

Indexed families unify the notation for operations like union $\bigcup_{i\in I}A_i$ and intersection $\bigcap_{i\in I}A_i$ , which depend on an index set and a set assigned to each index.

Examples

For $I=\mathbb{N}$ , the assignment $n\mapsto\{n\}$ defines a family $(\{n\})_{n\in\mathbb{N}}$ .
For $I=\mathbb{R}$ , the assignment $x\mapsto (x,x+1)$ defines a family of intervals $( (x,x+1) )_{x\in\mathbb{R}}$ .