Let $\mathcal{U} = \{U_i\}_{i\in I} $ be a collection of open sets with the property that the set $\bigcap_{i\in J} U_i $ is open for all subsets $J$ of $I$.
Is there a name for such collections of open sets?
Both locally finite collections and point-finite collections have this property, but these notions are too strong (just think of infinite discrete spaces).