2
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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) ... $\endgroup$ – David Handelman Mar 21 '17 at 13:26
  • 4
    $\begingroup$ 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. $\endgroup$ – Gro-Tsen Mar 21 '17 at 13:36
  • 1
    $\begingroup$ Dear Gro-Tsen: In the 'Steen & Seebach's Counterexamples in Topology' the definition of zero dimensional space is different from mentioned $*$ property $\endgroup$ – Alesix Mar 21 '17 at 17:23
  • $\begingroup$ 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. $\endgroup$ – Gro-Tsen Mar 21 '17 at 18:15
  • $\begingroup$ Dear Gro-Tsen: Thank you very much for your help. $\endgroup$ – Alesix Mar 21 '17 at 19:11
2
$\begingroup$

These properties are not equivalent. First of all, property * is commonly referred to as ultranormality. Every ultranormal space is normal, but not every zero-dimensional space is normal. Every compact zero-dimensional space is ultranormal. However, the only compact zero-dimensional spaces which are extremally disconnected are the ones with a complete Boolean algebra of clopen sets. See my expository paper for more details on these various notions.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.