# Article

 Title: A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube (English) Author: Kurilić, Miloš S. Author: Pavlović, Aleksandar Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 64 Issue: 2 Year: 2014 Pages: 519-537 Summary lang: English . Category: math . Summary: We compare the forcing-related properties of a complete Boolean algebra ${\mathbb B}$ with the properties of the convergences $\lambda _{\mathrm s}$ (the algebraic convergence) and $\lambda _{\mathrm {ls}}$ on ${\mathbb B}$ generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that $\lambda _{\mathrm {ls}}$ is a topological convergence iff forcing by ${\mathbb B}$ does not produce new reals and that $\lambda _{\mathrm {ls}}$ is weakly topological if ${\mathbb B}$ satisfies condition $(\hbar )$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $\lambda _{\mathrm {ls}}$ is a weakly topological convergence, then ${\mathbb B}$ 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 $\lambda _{\mathrm {ls}}$ on the collapsing algebra ${\mathbb B}=\mathop {\mathrm {ro}} (^{<\omega }\omega _2)$ is weakly topological“ is independent of ZFC. (English) Keyword: complete Boolean algebra Keyword: convergence structure Keyword: algebraic convergence Keyword: forcing Keyword: Cantor cube Keyword: Aleksandrov cube Keyword: small cardinal MSC: 03E17 MSC: 03E40 MSC: 06E10 MSC: 54A20 MSC: 54D55 idZBL: Zbl 06391510 idMR: MR3277752 DOI: 10.1007/s10587-014-0117-6 . Date available: 2014-11-10T09:57:35Z Last updated: 2016-07-01 Stable URL: http://hdl.handle.net/10338.dmlcz/144014 . 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