Corollary 3.1

Let $L$ be any complete Boolean algebra. Let $\mathcal Q(L)$ consist of the inclusion-maximal nonempty subsets $B\subseteq L$ excluding $\bot$ such that any two members have a common lower bound in $B$, and equip $\mathcal Q(L)$ with the topology generated by $D(a)=\{B\in\mathcal Q(L):a\in B\}$. Then the following two conditions are equivalent: for every $a\in L$ and every subset $s\subseteq L$, $a\wedge\sup s=\sup\{a\wedge b:b\in s\}$; and for every nonempty subset $s\subseteq L$ that is directed upward within itself, meaning that each $a,b\in s$ have some $c\in s$ with $a\le c$ and $b\le c$, one has $\overline{\bigcup_{a\in s}D(a)}=D(\sup s)$. The first condition quantifies over all subsets, including the empty subset, whereas the second quantifies only over nonempty directed subsets.