Are there any special conditions under which the Stone-Čech (Hausdorff) compactification and the Alexandroff one-point (Hausdorff) comactification are zero-dimensional?
I know that the Stone-Čech (Hausdorff) compactification of a space X is zero-dimensional if and only if X is strongly zero-dimensional.
I also have learned that in my another question on zero-dimensionality (here) that subspace of zero-dimensional space is zero-dimensional and that $X \times Y$ is zero-dimensional iff both $X$ and $Y$ are zero-dimensional (and non-empty.)
But are there any other criteria on Hausdorff spaces to determine zero-dimensionality? I would like to learn, what methods I can use when deciding. I know I can look at basis and whether it is clopen, but that is usually a problem for me, I find it difficult to imagine how the basis looks.
I am asking here, because I really tried Google, Math-Sci net and other sources, but couldn´t find anything. Thanks for your insights.
Thank you for advice.
Disclaimer: By a zero-dimensional space I mean a topological space having a base of sets that are at the same time open and closed in it.
A Hausdorff space $X$ is zero-dimensional iff it can be embedded as a subspace of $C(I):=\{0,1\}^I$ (the so-called Cantor cube of weight $|I|$) for some index set $I$, where $\{0,1\}$ has the discrete topology and the power has the product topology.
It's clear that the standard base of $C(I)$ is clopen and so it's zero-dimensional and hence so is any of its subspaces. The reverse follows from the Tychonoff embedding theorem: we use the indicator functions of a clopen base as the separating family.
This is classical and covered in Engelking, e.g.
BTW it's pretty easy to see that the Aleksandrov compactification of a Hausdorff (locally compact) zero-dimensional space is again Hausdorff zero-dimensional. Try the easy proof yourself. The reverse is trivial, via subpaces. You already know the answer for the Stone-Čech compactification.