Let $X$ be a non-empty topological space. Then we have the following concepts for the topological space $X $:
1) We say $X $ has property $*$, if 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 $A \subseteq V\subseteq U$( This is the definition of zero dimensional topological space in the sense of " Zero-Dimensional Spaces - Springer, www.springer.com › content › document "
2) $X$ is called extremally disconnected if $X$ is a Hausdorff space and for every open set $U \subseteq X$ the closure $U$ is open in $X$.
Are these two concepts equivalent for a Hausdorff topological space?