For finite $\sigma$-algebras, to check closure of countable unions, does it suffice to show that the union of two events is also an event?
I have a finite, supposedly $\sigma$-algebra $\Sigma$ and I want to show if it is closed under countable unions. My question is if it is true that if $A,B\in\Sigma$ and $A\cup B\in\Sigma$ it means that $\Sigma$ is closed under countable unions.
I think it is because in the context of finite $\sigma$-algebras i think closure of countable unions is equivalent to the closure of finite unions, and finite unions are basically unions of two sets, repeatedly.
But I don't know how to prove it, so I cannot believe myself yet.