Property (ħ) and cellularity of complete Boolean algebras

Abstract

A complete Boolean algebra B satisfies property (h{stroke}) iff each sequence x in B has a subsequence y such that the equality lim sup zn = lim sup yn holds for each subsequence z of y. This property, providing an explicit definition of the a posteriori convergence in complete Boolean algebras with the sequential topology and a characterization of sequential compactness of such spaces, is closely related to the cellularity of Boolean algebras. Here we determine the position of property (h{stroke}) with respect to the hierarchy of conditions of the form κ-cc. So, answering a question from Kurilić and Pavlović (Ann Pure Appl Logic 148(1-3):49-62, 2007), we show that "h-cc → (h{stroke})" is not a theorem of ZFC and that there is no cardinal, definable in ZFC, such that is a theorem of ZFC. Also, we show that the set {k:each k-cc c.B.a. (h{stroke})} has is equal to [0,h) or [0,h] and that both values are consistent, which, with the known equality {k: each c.B.a.having (h{stroke}) has the k-cc} = [s,∞) completes the picture. © Springer-Verlag 2009.