Let $E$ be a set, and let $S$ be any set of subsets of $E$, such that $S$ contains the empty set. Can you identify the smallest set of subsets $T$ of $E$ such that $T$ contains all elements of $S$, and $T$ is stable by both finite union and by any intersections ? If possible write the generic form of an element of $T$ by using elements of $S$, with unions and intersections.