I have a question that I've been wondering about the past day or so while trying to relate measure theory back to some general topology.
Is it ever possible for some family of (open) sets in $\mathbb{R}^2$ to be both a base for the usual topology on $\mathbb{R}^2$, as well as a semiring?
To avoid confusion, I mean a semiring of sets. So by semiring, I mean a collection of subsets of $\mathbb{R}^2$ which has $\emptyset$ as an element, is closed under finite intersections, and for any $A,B$ in the semiring, $A\setminus B=\bigcup_{i=1}^n C_i$ for disjoint $C_i$ in the semiring.