Let $\tau\_X$ denote the collection of open subsets of a topological space $X$ and let $\mathsf{RO}(X)$ be the subset of $\tau\_X$ made up of regular open subsets. With this terminology, the inequality $|\mathsf{RO}(X)|\leq |\tau\_X|$ is obvious. Furthermore, if $X$ is a semiregular space, it is easy to see that $|\tau\_X|\leq 2^{|\mathsf{RO}(X)|}$.


It is well-known (see for example section 10 of Hodel's chapter in the Handbook) that every perfectly normal Hausdorff space satisfies the equality $|\mathsf{RO}(X)|=|\tau\_X|$. My question goes in two directions:


* What are some examples of Tychonoff spaces $X$ such that $|\mathsf{RO}(X)|< |\tau\_X|$?
* Can the "perfect normality" condition be relaxed in such a way that the equality remains true? E.g., is it true that every perfect (closed sets are $G\_\delta$) Tychonoff space verifies the equality $|\mathsf{RO}(X)|=|\tau\_X|$?



$|\mathsf{RO}(X)|$ vs. $|\tau_X|$ for Tychonoff spaces
The compact Hausdorff space $X = \beta\mathbb{N}$ is another example. Every regular open subset is the closure of a subset of $\mathbb{N}$ and there are only $\frak{c}$ such subsets but $X$ has $2^{\frak{c}}$ points, and for each such point $p$, the set $X \setminus \{p\}$ is an open set.


For the second question, a scattered separable space of cardinality $\frak{c}$ and scattered height $2$ (such as a Cantor Tree, or a $\Psi$-space of size $\frak{c}$) gives a space in which every closed subset is a G$\_\delta$ but in which the equality fails for the same reason that it fails in the tangent-disk space.


