Proposition 3.9

Let $M$ be a Hausdorff, first-countable, Lindelöf topological space, and let $p$ be a family of Borel subsets of $M$, ordered by inclusion, with meet given by intersection, join by union, bottom $\varnothing$, and top $M$. Then the following are equivalent: $p$ is nonempty, excludes $\varnothing$, is upward closed among Borel sets and closed under binary intersections, and for every sequence $(A_n)_{n\in\mathbb N}$ of Borel sets and every Borel least upper bound $A$ of that sequence under inclusion, $A\in p$ implies $A_n\in p$ for some $n$; and there exists an inclusion-maximal nonempty family $B$ of Borel sets excluding $\varnothing$, in which any two members contain a common member of $B$, such that $B=p$ and the singleton $\{B\}$ is open in the space of all such maximal families. The topology on that space is generated by the sets $\{C:A\in C\}$, one for each Borel set $A$. This statement also includes $M=\varnothing$, when neither side can hold for any $p$.