sigma–ring of sets

A $\sigma$ -ring of sets is a nonempty collection $\mathcal{S}$ of sets such that

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if $A\in\mathcal{S}$ and $B\in\mathcal{S}$ then $A-B\in\mathcal{S}$ and
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if $A_{i}\in\mathcal{S}$ for $i=1,2\ldots,$ then $\cup_{i=1}^{\infty}A_{i}\in\mathcal{S}$

\cap_{i=1}^{\infty}A_{i}=A-\cup_{i=1}^{\infty}(A-A_{i})

where $A=\cup_{i=1}^{\infty}A_{i}$ .

$\sigma$ -rings are used in measure theory.