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Def: A Tychonoff space $X$ is said to be strongly zero-dimensional if its Stone-Čech compactification $\beta X$ is totally disconnected (that is if the only connected subspaces of $\beta X$ are singletons).

I have heard that strong zero-dimensionality of $X$ provides that disjoint zero-sets are separated by clopen sets. My question is: why is this equivalent? Could you provide a source for proof or anything?

Thank you.

(By $\beta X$, we mean the Stone-Čech compactification of $X$).

Tereza Tizkova
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    Engelking, General topology 6.2.4, 6.2.12 and 6.2.10. – Ulli Sep 03 '22 at 21:26
  • @Ulli Wouldn´t the theorem 6.2.4 be enough? Or maybe I just dont see how these three theorems put together to obtain the result in my question. – Tereza Tizkova Sep 04 '22 at 08:01
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    No, you need the others as well: 6.2.12 and 6.2.10 to show that your definition of strong zero-dimensionality is the same as Engelking's. – Ulli Sep 04 '22 at 20:26
  • @Ulli I am sorry, I still cant see it. Engelking defines strong zero-dimensionality via open refinements, which is something different than the two equivalent definitions I have written. Hence, I dont understand, how 6.2.10 and 6.2.12 imply that my two definitions are equivalent? I dont even see how it leads to Theorem 6.2.4. – Tereza Tizkova Sep 05 '22 at 14:24
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    Ok, step by step: All spaces are Tychonoff. Let's say $X$ is szd, if $\beta X$ is totally disconnected, i.e. exactly your definition. $X$ is called E-szd, if it is strongly zero-dimensional as defined in Engelking's book. Then by 6.2.10: $X$ szd $\Leftrightarrow$ $\beta X$ totally disonnected $\Leftrightarrow$ $\beta X$ E-szd. By 6.2.12: $\beta X$ E-szd $\Leftrightarrow$ $X$ E-szd. Hence $X$ szd $\Leftrightarrow$ $X$ E-szd. – Ulli Sep 05 '22 at 19:41
  • 6.2.4: $X$ E-szd $\Leftrightarrow$ every $A, B$ completely separated subsets of $X$ can be separated by clopen subsets, which is obviously equivalent to "disjoint zero-sets are separated by clopen sets", i.e. exactly the condition you are asking for. – Ulli Sep 05 '22 at 19:56

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