A posteriori convergence in complete Boolean algebras with the sequential topology

Abstract

A sequence x = 〈 x n : 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 τ s on B is the maximal topology on B such that x → b implies x → τ s b , where → τ s denotes the convergence in the space 〈 B , τ s 〉 — 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 ( ħ ) of c.B.a.’s, satisfying t -cc ⇒ ( ħ ) ⇒ 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 , τ s 〉 is sequentially compact iff the algebra B has the property ( ħ ) and does not produce independent reals by forcing, and that s = ω 1 implies P ( ω ) is the unique sequentially compact c.B.a. in the class of Suslin forcing notions.