Proposition 3.19

Let $M$ be any topological space that is a Baire space. For subsets of $M$ ordered by inclusion, call a collection a quasipoint if it is nonempty, excludes $\varnothing$, contains for every two of its members a member contained in their intersection, and is maximal under inclusion among collections with these properties. Let $C$ be any quasipoint consisting of Borel subsets of $M$, with maximality taken among collections of Borel subsets. Then the following are equivalent: there exists a quasipoint $B$ consisting of open subsets of $M$, with maximality taken among collections of open subsets, such that for every $U\in B$ there is an $A\in C$ whose underlying subset equals $U$; and, for every open subset $U\subseteq M$, there is no $A\in C$ whose underlying subset equals $\overline U\setminus U$. No nonemptiness assumption on $M$ is imposed; if $M$ is empty, no such $C$ exists.