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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.

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2 Answers 2

up vote 10 down vote accepted

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.

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OK, the PDF is behind the paywall, mail me if you can't get it. –  KP Hart Apr 11 '12 at 18:00
Got it. This example corrects an old error in Kelley's book General Topology, which apparently went uncorrected for over 22 years, but Terasawa was evidently unaware of the error. See my comment below. –  Fred Dashiell Apr 12 '12 at 13:35
Very nice. We now have at least 2 substantially different examples. –  Fred Dashiell Apr 15 '12 at 2:25
The Dowker example is the space called $\Delta_1$, described in the text of Gillman-Jerison, 16M. I think KP's modification is nontrivial, and the example should count as "new". –  Fred Dashiell Apr 15 '12 at 19:53
Another comment: I think KP's example is "constructive": no choice principle is required. (Is this true?) The spaces $\Psi$ used by Terasawa are described using a maximal principle. –  Fred Dashiell Apr 15 '12 at 20:14
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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?

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The space $M$ is definitely not locally compact, hence neither are the others as these are made by adding one point to $M$ or to a sum of copies of $M$. –  KP Hart Apr 13 '12 at 20:00
I added a locally compact version of Dowker's space above. –  KP Hart Apr 14 '12 at 20:19
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