I know that when $X$ is a normal and totally disconnected space, the Stone-Cech compactification $\beta(X)$ is totally disconnected. But I can't find a counterexample when considering $X$ totally disconnected only.
2026-03-26 14:42:57.1774536177
If $X$ is a totally disconnected space, then is $\beta(X)$ totally disconnected?
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No. Any totally disconnected compact Hausdorff space is zero-dimensional (clopen sets generate the topology), and any subspace of a zero-dimensional space is zero-dimensional. So, if $X$ is completely regular and $\beta(X)$ is totally disconnected, then $X$ must be zero-dimensional. This means any completely regular space which is totally disconnected but not zero-dimensional is a counterexample. Such a space is complicated to construct but one well-known example is Cantor's teepee (also known as the deleted Knaster-Kuratowski fan). Note that this example is also normal (it is a subspace of $\mathbb{R}^2$), so your statement about the case that $X$ is normal is incorrect.