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$\mathbb {A}$. (shrink)

An important application of ultrafilters is in the ultraproduct construction in model theory. In this paper we study ultraproducts of countable structures, whose universe we assume is ω , using ultrafilters on a countable index set, which we also assume to be ω . Many of the properties of the ultraproduct are in fact inherent properties of the ultrafilter. For example, if we take a sequence of countable linear orders without maximal element, then their ultraproduct will have no maximal element, (...) and we can ask what its cofinality is. This cardinal depends only on the ultrafilter; it does not depend on what linear orders comprise the factors. (shrink)