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In a comment of this question, t.b. suggests that every compact, homogeneous, totally disconnected, second countable space is either a finite discrete space or the Cantor set; equivalently, it must be a countable power of a finite discrete space. How can this be proven?

Furthermore, $2^\kappa$ is compact, homogeneous, and totally disconnected. Is every compact, homogeneous and totally disconnected space (not necessarily second countable) a power of a finite discrete space?

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  • $\begingroup$ If $X$ is compact, homogeneous, totally disconnected, second countable but not discrete, then $X$ has no isolated points, so it is a Cantor set by Brouwer's characterization theorem $\endgroup$ Apr 24, 2023 at 4:45
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    $\begingroup$ It's not clear to me that a compact, homogeneous, totally disconnected space is Hausdorff. Might be that Hausdorff must be added as a prerequisite, even in case of second countable? $\endgroup$
    – Ulli
    Apr 24, 2023 at 8:18
  • $\begingroup$ totally disconnected gets you T1, so proving regular gets you Hausdorff. $\endgroup$ Apr 24, 2023 at 11:12
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    $\begingroup$ why is it regular? $\endgroup$
    – Ulli
    Apr 24, 2023 at 11:20
  • $\begingroup$ Sorry, I should have said "would get you". I have not been successful in verifying this. $\endgroup$ Apr 24, 2023 at 12:54

1 Answer 1

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The answer to the second question is "no", even in the realm of Hausdorff spaces:

Let $X$ be the Double arrow space. It is well-known that $X$ is compact, Hausdorff, zero-dimensional, first countable, but not metrizable. By Theorem 2 in this paper of Alan Dow and Elliot Pearl, $X^\omega$ is homogeneous. Of course, $X^\omega$ is compact, Hausdorff, zero-dimensional (hence totally disconnected) and first countable. Assume $X^\omega$ is homeomorphic to $D^\kappa$ for a discrete, finite space $D$ and a cardinal $\kappa$. By first countability, $\kappa = \omega$, hence $X^\omega$ is metrizable, hence $X$ is metrizable. Contradiction.

Update
I should have thought a little bit more carefully about this before, since there is a much simpler counterexample, which also avoids this very complicated theorem of Dow and Pearl:

The endpoints of the Double arrow space are isolated, hence $Z := X \setminus \{\text{min } X, \text{max } X \}$ is also compact. It is well-known (and not too difficult to see) that $Z$ is homogeneous. Of course, it is also zero-dimensional, Hausdorff, first countable. Hence it is not homeomorphic to $D^\kappa$ for a discrete, finite space $D$.

Anyway, for reference, and since it has already been accepted, I will keep the first example.

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