Strongly zero-dimensional topological spaces and a simillar condition 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)
 A: What you call a strongly zero-dimensional space is commonly known as an ultranormal space. An ultranormal space is a Hausdorff space where for every pair of disjoint closed sets $C,D$ there is a clopen set $Z$ with $C⊆Z⊆D^{c}⊆Z⊆D^c$. A strongly zero-dimensional space is a space whose Stone-Cech compactification is zero-dimensional. The ultranormal spaces are precisely the strongly zero-dimensional normal spaces. See my expository for more details on these discussions. So it is true that the $*$-spaces are called extremally disconnected spaces. Finally, there are extremally disconnected spaces which are Hausdorff but not normal (nor even regular) such as this one. In fact, the absolute of a normal space is always completely regular and extremally disconnected but not necessarily normal.
A: (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 $O\subseteq V\subseteq C$.
