On Sequences of Homomorphisms Into Measure Algebras and the Efimov Problem

Journal of Symbolic Logic 88 (1):191-218 (2023)
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Abstract

For given Boolean algebras$\mathbb {A}$and$\mathbb {B}$we endow the space$\mathcal {H}(\mathbb {A},\mathbb {B})$of all Boolean homomorphisms from$\mathbb {A}$to$\mathbb {B}$with various topologies and study convergence properties of sequences in$\mathcal {H}(\mathbb {A},\mathbb {B})$. We are in particular interested in the situation when$\mathbb {B}$is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on$\mathbb {A}$in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin’s result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on$\mathcal {H}(\mathbb {A},\mathbb {B})$for a Boolean algebra$\mathbb {B}$carrying a strictly positive measure and convergence properties of sequences of measures on$\mathbb {A}$.

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Citations of this work

Convergence of measures after adding a real.Damian Sobota & Lyubomyr Zdomskyy - 2023 - Archive for Mathematical Logic 63 (1):135-162.

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References found in this work

Small cardinals and small Efimov spaces.Will Brian & Alan Dow - 2022 - Annals of Pure and Applied Logic 173 (1):103043.
Complete metric Boolean algebras.A. N. Kolmogorov - 1995 - Philosophical Studies 77 (1):57 - 66.
Convergent sequences in topological groups.Michael Hrušák & Alexander Shibakov - 2021 - Annals of Pure and Applied Logic 172 (5):102910.
A simple indeterminate infinite game.Damian Niwinski & Eryk Kopczynski - 2014 - In Damian Niwinski & Eryk Kopczynski (eds.), A simple indeterminate infinite game. pp. 205-212.

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