# 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)

• Are you the one who asked this question earlier? Because what is called "property $*$" there is what is called "strongly $0$-dimensional" here, and what is called a $*$-space here implies what is called "extremally disconnected" there, which is insanely confusing. Where did you get this terminology? At any rate, you should check the examples from Steen & Seebach which I mentioned in the comments there, maybe one of them will help in some way, or at least, clarify the question. – Gro-Tsen Mar 26 '17 at 17:27

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 $O\subseteq V\subseteq C$.
• It's not entirely in my comment, because I hadn't noticed that "extremally disconnected" was also sufficient for what OP calls being a $*$-space. – Gro-Tsen Mar 26 '17 at 17:40
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.
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• I don't understand the containment $Z \subseteq D^c \subseteq Z$ … do you mean $Z = D^c$, or is one of those $Z$'s supposed to be something else? Reading the introduction to your paper suggests that maybe you just meant $C \subseteq Z \subseteq D^c$. – LSpice Apr 12 '18 at 19:14