I have looked in Halmos and Engelking, which would be the natural places to look, and could not find anything related to this. I am trying to understand how the complete Boolean algebra of regular opens looks like. My problem has to do with how one computes regular closure of intersections of arbitrary sets.
Specifically, if $(U_{n\in \omega})$ is a family of arbitrary sets, and I know that $\bigcap_{n\in \omega}U_{n}$ is regular open (i.e., $\bigcap_{n\in \omega}U_{n}=int(cl(\bigcap_{n\in \omega}U_{n}))$, do we then have that $$\bigcap_{n\in \omega}U_{n}=int(\bigcap_{n\in\omega}int(cl(U_{n})))$$ always holds?