A $\sigma$ algebra at most of numerable cardinality is finite. Let $\mathcal{A}$ a $\sigma$ algebra at most of numerable cardinality (i.e $|\mathcal{A}|\leq \aleph_0$). Show that $\mathcal{A}$ is finite. I try to build a disjoint subcollection of $\mathcal A$ to try this exercise but I can not think what this subcollection can be. Thanks for your help!

Suppose that $\mathcal A$ is a $\sigma$-algebra of subsets of the set $X$. Suppose further that $\mathcal B$ is a countably infinite subcollection of $\mathcal A$. For each $x\in X$, consider the sets $$E_x=\bigcap\\{E\in\mathcal B:x\in E\\}.$$ To show that $\\{E_x:x\in X\\}$ is a infinite disjoint subcollection, look at the relation $\sim$ on $X$ given by $x\sim y$ iff $y\in E_x$.