class

a class is a collection of sets with a shared property

let $\text{[math]}$ be a nonempty set and let $\text{[math]}$ (power set of $\text{[math]}$ ) be a class of subsets of $\text{[math]}$ .

a class of sets $\text{[math]}$ is called

$\text{[math]}$ -closed (closed under intersections) or a $\text{[math]}$ -system if $\text{[math]}$ whenever $\text{[math]}$ ,
$\text{[math]}$ -closed (closed under countable intersections) if $\text{[math]}$ for any choice of countably many sets $\text{[math]}$ ,
$\text{[math]}$ -closed (closed under unions) if $\text{[math]}$ whenever $\text{[math]}$ ,
$\text{[math]}$ -closed (closed under countable unions) if $\text{[math]}$ for any choice of countably many sets $\text{[math]}$ ,
$\text{[math]}$ -closed (closed under differences) if $\text{[math]}$ whenever $\text{[math]}$ , and
closed under complements if $\text{[math]}$ for any set $\text{[math]}$ .

[cite:@klenke_prob_2020 definition 1.1]