A posteriori convergence in complete Boolean algebras with the sequential topology

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Abstract

A sequence x=xn:nω of elements of a complete Boolean algebra (briefly c.B.a.) B converges to bB a priori (in notation xb) if lim infx=lim supx=b. The sequential topology τs on B is the maximal topology on B such that xb implies xτsb, 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 (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,τs is sequentially compact iff the algebra B has the property (ħ) 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.

Keywords

Boolean algebra
Sequential topology
Sequential compactness
Chain conditions
Small cardinals
Forcing
Tall ideal
Independent real

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