Extremally disconnected spaces and the axiom of choice

axiom-of-choicegeneral-topologylogicset-theory

Is ZF consistent with "any compact Hausdorff extremally disconnected topological space is finite?"

(Motivation: it is a theorem that a compact Hausdorff extremally disconnected space is a retract of a Stone-Cech compactification of a discrete set. Now Stone-Cech compactifications depend on choice, and I believe (but am not sure) it's consistent with ZF that there exists a discrete space without a Stone-Cech compactification.)

Best Answer

Yes, it is consistent with ZF that all compact Hausdorff extremally disconnected spaces are finite. In Paul Howard and Jean E. Rubin's Consequences of the Axiom of Choice this is given as Form 371

FORM 371. There is an infinite, compact, Hausdorff, extremally disconnected topological space.

The reference they give for this is

  • Marianne Morillon; Les compacts extrêmement discontinus sont finis!; Séminaire d’Analyse; Université Blaise Pascal, Clermont-Ferrand II; 9, Année 1993-1994, Exp. No. 11, 13 p. (1994).

I have been unable to find this paper, but the zbMath review of the paper by Alan Dow gives an overview of its contents:

A space is extremally disconnected if the closure of each open set is again open. Each compact extremally disconnected space is ‘equal’ to the Stone space of some complete Boolean algebra (and conversely). We must be careful that this is a consequence of just ZF (without the Axiom of Choice) since the topic of this paper is to show that ZF does not imply that there is an infinite compact extremally disconnected space. The approach is clever: work in a model of Definable Choice (DC) plus there is no ultrafilter on the integers. It is easy to see that DC implies that each infinite complete Boolean algebra contains the power set of the integers as a subalgebra. Therefore, from DC, if any infinite extremally disconnected space is compact, there are many ultrafilters on the integers. Finally the author shows that in a model constructed by A. Blass [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 25, No. 4, 329-331 (1977; Zbl 0365.02054)], in which DC, and even BC (compact spaces are Baire) fail, every compact extremally disconnected space is again finite. Of course this establishes that BCED (compact extremally disconnected spaces are Baire) does not imply BC.

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