A posteriori convergence in complete Boolean algebras with the sequential topology

Abstract

A sequence x = 〈 xn : n ∈ ω 〉 of elements of a complete Boolean algebra (briefly c.B.a.) B converges to b ∈ B a priori (in notation x → b) if lim inf x = lim sup x = b. The sequential topology τs on B is the maximal topology on B such that x → b implies x →τs b, where →τs denotes the convergence in the space 〈 B, τs 〉 - the a posteriori convergence. These two forms of convergence, as well as the properties of the sequential topology related to forcing, are investigated. So, the a posteriori convergence is described in terms of killing of tall ideals on ω, and it is shown that the a posteriori convergence is equivalent to the a priori convergence iff forcing by B does not produce new reals. A property (h{stroke}) of c.B.a.'s, satisfying t-cc ⇒ (h{stroke}) ⇒ s-cc and providing an explicit (algebraic) definition of the a posteriori convergence, is isolated. Finally, it is shown that, for an arbitrary c.B.a. B, the space 〈 B, τs 〉 is sequentially compact iff the algebra B has the property (h{stroke}) and does not produce independent reals by forcing, and that s = ω1 implies P (ω) is the unique sequentially compact c.B.a. in the class of Suslin forcing notions. © 2007 Elsevier B.V. All rights reserved.