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

TL;DR

This paper explores how certain convergence properties of Boolean algebras relate to forcing and the addition of new reals, generalizing known convergence on the Cantor and Aleksandrov cubes.

Contribution

It establishes conditions under which algebraic convergences are topological or weakly topological, linking these to forcing properties and cardinal invariants.

Findings

$oldsymbol{ ext{$oldsymbol{ ext{ls}}}$}$ convergence is topological iff no new reals are added by forcing.

$oldsymbol{ ext{$oldsymbol{ ext{ls}}}$}$ is weakly topological if the algebra satisfies condition $( ext{ extbar})$.

The weak topological nature of $oldsymbol{ ext{$oldsymbol{ ext{ls}}}$}$ on the collapsing algebra is independent of ZFC.

Abstract

We compare the forcing related properties of a complete Boolean algebra B with the properties of the convergences $\lambda_s$ (the algebraic convergence) and $\lambda_{ls}$ on B generalizing the convergence on the Cantor and Aleksandrov cube respectively. In particular we show that $\lambda_{ls}$ is a topological convergence iff forcing by B does not produce new reals and that $\lambda_{ls}$ is weakly topological if B satisfies condition $(\hbar)$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $\lambda_{ls}$ is a weakly topological convergence, then B is a $2^{\mathfrak h}$-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 $\lambda_{ls}$ on the collapsing algebra $B=\ro ((\omega_2)^{<\omega})$ is weakly topological" is independent of ZFC.