Let $X$ be a totally ordered set, considered as a topological space with the order topology. The regular open subsets of $X$ (i.e., the sets $U = \operatorname{int} \operatorname{cl} U$) form a complete Boolean algebra—call this $R(X)$. Let $D$ be a closed discrete subset of $X$. Claim: the set of all unions of open intervals in $X$ with endpoints in $D$ is a complete subalgebra of $R(X)$.
Is this claim true? If so, can you suggest a reference for it?
This claim is false. I'm not sure what exactly I was thinking originally, but there are obvious counterexamples. E.g., the integers are closed and discrete in the real line, and the union of intervals with integer endpoints $(0, 1) \cup (1, 2)$ is not a regular open set.