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Zbl 0997.54055
Morillon, Marianne
Extremally discontinuous compact sets are finite! (Les compacts extrêmement discontinus sont finis!) (French)
[J] Sémin. Anal., Univ. Blaise Pascal, Clermont II 9, Année 1993-1994, Exp. No.11, 13 p. (1994). ISSN 0985-0732

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 {\it 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.
[A.Dow (North York)]

MSC 2000:: *54G05 Extremally disconnected spaces, etc.
03E35 Consistency and independence results (set theory)

Keywords: extremally disconnected; definable choice

Citations: Zbl 0365.02054

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