A Hausdorff topological space $X$ is called strongly
zero-dimensional whenever for every closed subset $A$ of $X$ and
every open subset $U$ of $X$ such that $A \subseteq U$, there
exists a **clopen** subset $V$ of $X$ such that $A \subseteq V
\subseteq U$.


Now, let $X'$ be a Hausdorff topological space such that for
every open subset $O$ of $X'$ and every closed subset $C$ of $X'$
such that $O \subseteq C$, there exists a **clopen** subset $V$
of $X'$ such that $O \subseteq V \subseteq C$. In this case, we
say $X'$ is a $\*$-space.


It is clear that a $\*$-space is strongly zero-dimensional.


I am trying to answer the following questions:


1) Is it true that a strongly zero-dimensional is $\*$-space? If not, is there an additional assumption $A $ on $X'$ such that $A $ + strongly zero-dimensional implies $X'$ is $\*$-space?


2) Are $\*$-spaces famous? ( For example, are $\*$-spaces
well-known cases of topological spaces that I do not know)



Strongly zero-dimensional topological spaces and a simillar condition
(Having posted this, I saw that all of it is in the comment by Gro-Tsen)


Taking, in the definition of \*-space, $C=\text{closure of $O$}$ shows that closure of an open set must be clopen. This is clearly also sufficient. So \*-spaces are exactly extremally disconnected Hausdorff spaces - those with closures of open sets clopen. For an example of a strongly zero-dimensional space which is not extremally disconnected, take the Stone space of any incomplete Boolean algebra, e. g. one-point compactification $S\cup\{\infty\}$ of an infinite discrete space $S$. In there, take $O\subset S$ any infinite subset with infinite complement, and $C$ its closure $O\cup\{\infty\}$. Then $C$ is not clopen (the only clopens of $S\cup\{\infty\}$ are finite subsets of $S$ and their complements), so there is obviously no clopen $V$ with $O\subseteq V\subseteq C$.


