Properties of classes of sets closed under set-subtraction

Theorem statement

If $\mathcal{A}$ is a class of subsets of $\Omega$ such that

• $$\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]$$ - that is closed under set-subtraction (or $\backslash$-closed)

Then we have:

1. $\mathcal{A}$ is $\cap$-closed
2. $\mathcal{A}$ is $\sigma$-$\cup$-closed$\implies$ $\mathcal{A}$ is $\sigma$-$\cap$-closed
3. Any countable union of sets in $\mathcal{A}$ can be expressed as a countable disjoint union of sets in $\mathcal{A}$

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Proof:

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