A space is extremally disconnected iff is Hausdorff and if the closure of each open set is again open. In set-theory with the axiom of choice, there exist infinite extremally disconnected spaces: for example, the Stone-Cech compactification of a discrete space i.e. the space of ultrafilters on a set, equipped with the Stone-Zariski topology.
We show that in some models of ZF, every extremally disconnected compact space is finite. For example, this results holds in the model constructed by Blass, see [Bl77].
It follows that the axiom BCED (compact extremally disconnected spaces are Baire) does not imply BC (compact Hausdorff spaces are Baire).

Abstract from Zentralblatt:
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 Blass ({\it A model without Ultrafil ters}, {\bf Bull. Acad. Pol. Sci.}, Vol. XXV, No. 4 (1977)), 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) ]