Suppose that the family $\mathrm{RO}(X)$ of regular open subsets of $(X,\mathscr{O})$ is a basis of $X$. Let the density of $\mathrm{RO}(X)$ (considered as boolean algebra) be $\aleph_0$.

Does $X$ have to be second-countable? If not, what if we add regularity of $X$ (both $T_1$ and $T_3$ separation axioms)? If answers to both questions are negative, what is the maximal cardinality (relative to $|X|$) of the set of points not covered by dense countable $D\subseteq\mathrm{RO}(X)$?

Suppose that the family $\mathrm{RO}(X)$ of regular open subsets of $(X,\mathscr{O})$ is a basis of $X$. Let the density of $\mathrm{RO}(X)$ (considered as Boolean algebra) be $\aleph_0$.

Does $X$ have to be second-countable? If not, what if we add regularity of $X$ (both $T_1$ and $T_3$ separation axioms)? If answers to both questions are negative, what is the maximal cardinality (relative to $|X|$) of the set of points not covered by dense countable $D\subseteq\mathrm{RO}(X)$?

Suppose that the family $\mathrm{RO}(X)$ of regular open subsets of $(X,\mathscr{O})$ is a basis of $X$. Let the density of $\mathrm{RO}(X)$ (considered as boolean algebra) be $\aleph_0$.

Does $X$ have to be second-countable? If not, what if we add regularity of $X$? If answers to both questions are negative, what is the maximal cardinality (relative to $|X|$) of the set of points not covered by dense countable $D\subseteq\mathrm{RO}(X)$?

Suppose that the family $\mathrm{RO}(X)$ of regular open subsets of $(X,\mathscr{O})$ is a basis of $X$. Let the density of $\mathrm{RO}(X)$ (considered as boolean algebra) be $\aleph_0$.

Does $X$ have to be second-countable? If not, what if we add regularity of $X$ (both $T_1$ and $T_3$ separation axioms)? If answers to both questions are negative, what is the maximal cardinality (relative to $|X|$) of the set of points not covered by dense countable $D\subseteq\mathrm{RO}(X)$?