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)

A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property $\unicode{x3b3} $, a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property $\unicode{x3b3} $ that is not (...) null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin–Miller and Bartoszyński–Recław, to obtain sets with analogous properties. We also consider products of Sierpiński sets in the context of combinatorial covering properties. (shrink)

A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated with arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the corresponding forcing axioms and the corresponding fragments of the strong reflection principle, are analyzed, and consequences are presented. Some of these consequences are “exact” versions of diagonal stationary reflection principles of sets of ordinals. We also separate some of these diagonal strong reflection principles from related axioms.

Let E be an analytic equivalence relation on a Polish space. We introduce a framework for studying the possible “reasonable” complete classifications and the complexity of possible classifying invariants for E, such that: (1) the standard results and intuitions regarding classifications by countable structures are preserved in this framework; (2) this framework respects Borel reducibility; and (3) this framework allows for a precise study of the possible invariants of certain equivalence relations which are not classifiable by countable structures, such as (...) E1. In this framework we show that E1 can be classified, with classifying invariants which are κ-sequences of E0-classes where κ=b, and it cannot be classified in such a manner if κshrink)

Traditionally, the role of general topology in model theory has been mainly limited to the study of compacta that arise in first-order logic. In this context, the topology tends to be so trivial that it turns into combinatorics, motivating a widespread approach that focuses on the combinatorial component while usually hiding the topological one. This popular combinatorial approach to model theory has proved to be so useful that it has become rare to see more advanced topology in model-theoretic articles. Prof. (...) Franklin D. Tall has led the re-introduction of general topology as a valuable tool to push the boundaries of model theory. Most of this thesis is directly influenced by and builds on this idea.The first part of the thesis will answer a problem of T. Gowers on the undefinability of pathological Banach spaces such as Tsirelson space. The topological content of this chapter is centred around Grothendieck spaces.In a similar spirit, the second part will show a new connection between the notion of metastability introduced by T. Tao and the topological concept of pseudocompactness. We shall make use of this connection to show a result of X. Caicedo, E. Dueñez, J. Iovino in a much simplified manner.The third part of the thesis will carry a higher set-theoretic content as we shall use forcing and descriptive set theory to show that the well-known theorem of M. Morley on the trichotomy concerning the number of models of a first-order countable theory is undecidable if one considers second-order countable theories instead.The only part that did not originate from model-theoretic questions will be the fourth one. We show that $\operatorname {ZF} + \operatorname {DC} +$ “all Turing invariant sets of reals have the perfect set property” implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations. This result provides evidence in favour of a long-standing conjecture asking whether Turing determinacy implies the axiom of determinacy.Abstract prepared by Clovis HamelE-mail: chamel@math.toronto.edu. (shrink)