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)