atomic-open-quasipoints

Let $M$ be a Hausdorff topological space, and let $\mathcal O(M)$ be its lattice of open subsets, ordered by inclusion. A quasipoint is an inclusion-maximal nonempty family of open sets that excludes $\varnothing$ and, for any two of its members, contains an open set included in both. Let $\mathcal Q(\mathcal O(M))$ be the set of these quasipoints, with the topology generated by $D(U)=\{C:U\in C\}$ for $U\in\mathcal O(M)$, and fix $B\in\mathcal Q(\mathcal O(M))$. Then both of the following equivalences hold: $\{B\}$ is open if and only if there exists exactly one open set $U$ such that $U$ is a nonempty atom of $\mathcal O(M)$—every open subset of $U$ is either $\varnothing$ or $U$—and $B=\{V\in\mathcal O(M):U\subseteq V\}$; and $\{B\}$ is open if and only if there exists exactly one point $x\in M$ such that $\{x\}$ is open in $M$ and $B=\{U\in\mathcal O(M):x\in U\}$. Each uniqueness assertion concerns the object satisfying its entire stated conjunction.