One of the standard definitions of a compact topological space $\langle X,\mathscr{O}\rangle$ says that $X$ is compact iff every open cover of $X$ has a finite subcover. I would like to ask you if any class of spaces has been distinguished with respect to the following (or equivalent, similar) condition ($\mathrm{Cl}$ is the standard topological closure operator):
For any $A\subseteq X$ and $\mathscr{S}\subseteq\mathscr{O}$, if $A\subseteq\mathrm{Cl\,}\bigcup\mathscr{S}$, then there are $S_1,\ldots,S_n\in\mathscr{S}$ such that $A\subseteq\mathrm{Cl\,}S_1\cup\ldots\cup\mathrm{Cl\,}S_n$.
The motivation stems from studying regular open and closed sets (i.e. those that are equal to interior of their closures and the closures of interiors, respectively). In this setting the following condition is interetsing for me (let $\mathrm{r}\mathscr{O}$ and $\mathrm{r}\mathscr{C}$ be families of regular open and regular closed subsets of $X$):
For any $A\in\mathrm{r}\mathscr{O}$ and $\mathscr{S}\subseteq\mathrm{r}\mathscr{O}$, if $A\subseteq\mathrm{Cl\,}\bigcup\mathscr{S}$, then there are $S_1,\ldots,S_n\in\mathscr{S}$ such that $A\subseteq\mathrm{Cl\,}S_1\cup\ldots\cup\mathrm{Cl\,}S_n$.
In particular, w.r.t. the above, I would like to ask if there is a notion of $H$-closed set restricted to the class of regular open sets:
For any $A\in\mathrm{r}\mathscr{O}$ and $\mathscr{S}\subseteq\mathrm{r}\mathscr{O}$, if $A\subseteq\mathrm{Int\,Cl\,}\bigcup\mathscr{S}$, then there are $S_1,\ldots,S_n\in\mathscr{S}$ such that $A\subseteq\mathrm{Cl\,}S_1\cup\ldots\cup\mathrm{Cl\,}S_n$.