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?

totally disconnected) ... $\endgroup$ – David Handelman Mar 21 '17 at 13:26Counterexamples in Topology, check the definitions at the beginning (because they might subtly differ from those found in other textbooks), look up in the tables at the end, and see if any spaces answer the question. Here we find counterexamples to both implications: S&S's nº113 ("strong ultrafilter topology") is Hausdorff extremally disconnected but not $0$-dimensional, and nº139 ("post-office metric") is Hausdorff, $0$-dimensional but not extremally disconnected. $\endgroup$ – Gro-Tsen Mar 21 '17 at 13:36