Let $\mathcal{A}$ be an algebra of set (in a space $X$), such that any subcollection of disjoint sets in $\mathcal{A}$ is finite. Prove that $\mathcal{A}$ is finite.
I already found a boring brute force proof of this fact. But the finiteness property "smell" like a compactness argument of some sort - specifically, we seems to be lifting a local finiteness property to a global one. Hence I am looking for a slick purely topological proof of this fact. But for the life of me I can't figure out which topological structure to put on it to make the argument fall out. Anyone have any clues?
Thank you.