Let $(X,\tau)$ be a topological space. $A\subseteq X$ is said to be *regular open* if $A = \text{int}(\text{cl}(A))$ and let $\text{RO}(X,\tau)$ denote the collection of regular open sets of $X$. A standard exercise exercise shows that $(\text{RO}(X,\tau),\subseteq)$ is not only a lattice, but even a [Boolean algebra](https://en.wikipedia.org/wiki/Boolean_algebra).


**Question.** Is there a topological space $(X,\tau)$ such that $$\text{RO}(X,\tau) \cong {\cal P}(\omega)/(\text{fin})?$$


(The Boolean algebra ${\cal P}(\omega)/(\text{fin})$ is defined [here](https://mathoverflow.net/questions/200769/antichain-on-mathcalp-omega-fin-of-cardinality-2-aleph-0?rq=1).)



Can the Boolean Algebra of regular open sets be isomorphic to ${\cal P}(\omega)/(\text{fin})$?
No, $\mathrm{RO}(X)$ is complete; $\mathcal{P}(\omega)/\mathit{fin}$ is not (no strictly increasing sequence has a supremum).


