A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube

Abstract

© 2014, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic. We compare the forcing-related properties of a complete Boolean algebra B with the properties of the convergences λs (the algebraic convergence) and λls on B generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that λls is a topological convergence iff forcing by B does not produce new reals and that λls is weakly topological if B satisfies condition (ħ) (implied by the t-cc). On the other hand, if λls is a weakly topological convergence, then B is a 2h-cc algebra or in some generic extension the distributivity number of the ground model is greater than or equal to the tower number of the extension. So, the statement “The convergence λls on the collapsing algebra (formula presented) is weakly topological” is independent of ZFC.