Question

What is an example of a 0-dimensional locally compact Hausdorff space X for which the Cech-Stone compactification beta(X) is NOT 0-dimensional?

It is known that if X is a 0-dimensional locally compact Hausdorff space which is also paracompact, then beta(X) is 0-dimensional. (Engelking 1989, Th. 6.2.9). I would expect a counterexample in the non-paracompact case.

Another way of asking the question is to look at a Boolean ring L (without assuming a unit). If X is the Stone space of L then X is a 0-dimensional locally compact Hausdorff space and L is isomorphic with the ring of compact open sets in X. The Boolean algebra Cl(X) of closed-open sets is clearly a Boolean algebra extension of L, and the Stone space of Cl(X) is a certain compactification of X. I believe it is easy to see that this compactification of X is just beta(X) if and only if beta(X) is 0-dimensional.

Thus the question can be expressed in the algebraic side of the duality. Find conditions on the Boolean ring L so that beta(X) is 0-dimensional, where X is the Stone space of L.

I think this may be true if, for example, L is a Boolean sigma-ring. My original question asks for an example showing this is not true without some additional conditions on L.

Answer 1

In this paper (I hope the PDF is freely available) Jun Terasawa constructs maximal almost disjoint families on $\mathbb{N}$ whose associated spaces can have any dimension you want. Given an almost disjoint family $\mathcal{A}$ on $\mathbb{N}$ one defines a topology on the union $\mathbb{N}\cup\mathcal{A}$ by declaring each natural number to be isolated and giving each $A\in\mathcal{A}$ a countable local base by putting $$U(A,n)=\lbrace A\rbrace \cup \lbrace i\in A:i\ge n\rbrace$$ for each $n$. This space is locally compact and zero-dimensional but Terasawa could arrange it so that its Čech-Stone Compactification would contain the Hilbert cube, for example.

Dowker's example does have a locally compact version: Let us retain the notation of Dowker's paper and use $T$ to denote the set of countable ordinals and $Q_\alpha$ to denote the $\alpha$th congruence class as chosen by Dowker. In addition let $\mathbb{A}$ denote Alexandroff's double arrow space. This is the product $[0,1]\times\lbrace0,1\rbrace$, ordered lexicographically and endowed with its order topology and with the two isolated points $\langle 0,0\rangle$ and $\langle 1,1\rangle$ deleted.

Now consider the product $T\times \mathbb{A}$ and define a quotient space $X$ by identifying the points $\langle \alpha,x,0\rangle$ and $\langle \alpha,x,1\rangle$ whenever $x\notin\bigcup_{\beta\ge\alpha}Q_\beta$.

It is elementary to verify that this is an upper semicontinuous decomposition and that the resulting space is locally compact and zero-dimensional. The key observation is that for every $\alpha$ the product $T_\alpha\times\mathbb{A}$ is compact and open in the domain and its image is compact and open in $X$.

Furthermore, arguments similar to those given by Dowker will show that each finite open cover of $T\times\mathbb{A}$ has a refinement of the form $\mathcal{U}\cup\mathcal{V}$, where $\mathcal{U}$ is a disjoint open cover of $T_\alpha\times\mathbb{A}$ for some $\alpha$ and $\mathcal{V}$ consists of finitely many sets of the form $T\times C$, where $C$ is a clopen interval in $\mathbb{A}$.

The latter can then be used, just as for Dowker's $M$, to show that the bottom and top lines in $X$ cannot be separated by clopen sets and hence that $\beta X$ is not zero-dimensional.

Answer 2

Thanks to KP. This is the class $\Psi$ of Gillman-Jerison 5I, but apparently Terasawa explained more some 16 years later. There is a little history here. As pointed out by Figa-Talamanca and Franklin, "Multipliers of Distributive Lattices", Indian J. Math. 12 (1970) p. 159, John Kelley asserted in his topology book (p. 169) that the Cech-Stone compactification is always zero-dimensional. They attempt to correct this error by citing an unspecified example in the 1955 paper of Dowker, "Local Dimension of Normal Spaces", Quart. J. Math. 2 (1955) 101-120, and they say this example was pointed out to them by J. Isbell and P. Dwinger. However, I have not yet been able to verify that any of the non-strongly zero-dimensional examples in Dowker's paper are locally compact. I wonder which one they had in mind? If they were right, this would be a much earlier example than Terasawa's $\Psi$. Could all four of them been mis-reading Dowker?

Here is a link to my copy of the Dowker paper. Does it contain a counterexample?