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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
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Category: math
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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
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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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