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)

thereis what is called "strongly $0$-dimensional" here, and what is called a $*$-spacehereimplies what is called "extremally disconnected" there, which is insanely confusing. Where did you get this terminology? At any rate, you should check the examples from Steen & Seebach which I mentioned in the comments there, maybe one of them will help in some way, or at least, clarify the question. $\endgroup$ – Gro-Tsen Mar 26 '17 at 17:27