Completeness, regular opens, and order continuity

A Boolean algebra $B$ is complete, or Dedekind complete, if every subset $A\subseteq B$ has a least upper bound $\bigvee A$. Complements then give greatest lower bounds by $\bigwedge A=\mathrm{\neg }\bigvee \{\mathrm{\neg }a:a\in A\}$. The empty bounds are $\bigvee \mathrm{\varnothing }=0$ and $\bigwedge \mathrm{\varnothing }=1$. The one-element Boolean algebra is allowed.

For a topological space $X$, $\mathrm{int}E$ is the union of the open subsets of $E$, and $\bar{E}$ is the intersection of the closed subsets containing $E$. An open set $U$ is regular open if $U=\mathrm{int}\bar{U}$. Write $\mathrm{RO}(X)$ for the set of regular open subsets, ordered by inclusion. Its proposed operations, justified by Regular open algebra in ZF ↗, are

$\bigvee \mathcal{U}=\mathrm{int}\overline{\bigcup \mathcal{U}},\bigwedge \mathcal{U}=\mathrm{int}\bigcap \mathcal{U},\mathrm{\neg }U=\mathrm{int}(X\setminus U)=X\setminus \bar{U}\mathrm{.}$

Here the empty union is $\mathrm{\varnothing }$ and the empty intersection is $X$; finite meets will be ordinary finite intersections, and the bounds will be $\mathrm{\varnothing },X$. No separation axiom on $X$ is assumed.

A Boolean homomorphism $h:B\to C$ is order-continuous if it preserves every supremum and infimum that exists in $B$: whenever $b={\bigvee }_{B}A$ exists, $h(b)={\bigvee }_{C}h[A]$, and likewise for infima. This includes empty families. An order-dense Boolean embedding $i:B\to C$ is an injective Boolean homomorphism such that each $0<c\in C$ has some $b\in B$ with $0<i(b)\le c$. A Boolean completion is such an embedding into a complete Boolean algebra. This definition asserts neither existence of a completion nor extension of a homomorphism.