# Extremally disconnectedness and 0-dimensional space

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?

• When $X$ is compact (and Hausdorff), $C(X)$ is an AW*-algebra iff $X$ is extremally disconnected (see any book on W* or AW* algebras). In particular, a separable extremally disconnected compact space is finite. On the other hand, Cantor sets and finite-point compactifications of infinite discrete spaces are zero-dimensional (totally disconnected) ... Mar 21 '17 at 13:26
• For this sort of questions, the meta-answer is to open Steen & Seebach's Counterexamples 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. Mar 21 '17 at 13:36
• Dear Gro-Tsen: In the 'Steen & Seebach's Counterexamples in Topology' the definition of zero dimensional space is different from mentioned $*$ property Mar 21 '17 at 17:23
• All right, your property $*$ seems stronger than $0$-dimensional, but since I gave counterexamples in both directions, one of them should survive; and you should probably check the other to see if, perchance, it might have the property $*$ anyway. Mar 21 '17 at 18:15
• Dear Gro-Tsen: Thank you very much for your help. Mar 21 '17 at 19:11