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.

**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.)