On Sequences of Homomorphisms Into Measure Algebras and the Efimov Problem

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 $\mathbb {A}$ .

On measures induced by forcing names for ultrafilters

“…The following simple fact implies in particular that the Stone topology is a pointwise convergence topology. To some extend this proposition can be adapted to a more general situation of sequence of homomorphisms to measure algebras and convergence in the random models (see [BNS21]).…”

“…the measure algebra of type κ (see Section 1 for the precise definitions). In the article [BNS21] due to the first author and Damian Sobota it was used to study sequences of ultrafilters in the forcing extensions by measure algebras.…”