Related papers: Ultrafilter selection and Corson compacta
Two of the pillars of combinatorics are the notion of choosing an arbitrary subset of a set with $n$ elements (which can be done in $2^n$ ways), and the notion of choosing a $k$-element subset of a set with $n$ elements (which can be done…
We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property…
We introduce a general notion of covering property, of which many classical definitions are particular instances. Notions of closure under various sorts of convergence, or, more generally, under taking kinds of accumulation points, are…
The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free…
We initiate the study of Cartan subalgebras in C*-algebras, with a particular focus on existence and uniqueness questions. For homogeneous C*-algebras, these questions can be analysed systematically using the theory of fibre bundles. For…
It is shown that there exists a complete, atomless, sigma-centered Boolean algebra, which does not contain any regular countable subalgebra if and only if there exist a nowhere dense ultrafilter. Therefore the existence of such algebras is…
We consider, for infinite cardinals kappa and alpha <= kappa^+, the group Pi(kappa,< alpha) of sequences of integers, of length kappa, with non-zero entries in fewer than alpha positions. Our main result tells when Pi(kappa,< alpha) can be…
We determine multiplication and convolution topological algebras for classes of $\omega$-ultradifferentiable functions of Beurling type. Hypocontinuity and discontinuity of the multiplication and convolution mappings are also investigated.
Selective ultrafilters are characterized by many equivalent properties, in particular the Ramsey property that every finite colouring of unordered pairs of integers has a homogeneous set in U, and the equivalent property that every function…
We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that irr(B_0 times B_1)= max(irr(B_0),irr(B_1)). We prove…
Right and left thick, syndetic, piecewise syndetic, and fat sets in groups are studied. The main concern is the interplay between such sets in Boolean groups. Natural topologies closely related to fat sets are also considered, which leads…
I isolate a combinatorial property of a poset $\mathbb{P}$ that I call the strong Prikry property, which implies the existence of an ultrafilter on the complete Boolean algebra $\mathbb{B}$ of $\mathbb{P}$ such that one inclusion of the…
We study a commutant-closed collection of von Neumann algebras acting on a common Hilbert space indexed by a poset with an order-reversing involution. We give simple geometric axioms for the poset which allow us to construct a braided…
If $A$ is an infinite Boolean algebra the cardinal invariant $\mathfrak{a}(A)$ is defined as the smallest size of an infinite partition of $A$. The cardinal $\mathfrak{a}(A\oplus B)$, where $A\oplus B$ is the free product of the Boolean…
Let $X$ be an unbounded metric space, $B(x,r) = \{y\in X: d(x,y) \leqslant r\}$ for all $x\in X$ and $r\geqslant 0$. We endow $X$ with the discrete topology and identify the Stone-\v{C}ech compactification $\beta X$ of $X$ with the set of…
A set of natural numbers will be called \emph{harmonic} if the reciprocals of its elements form a divergent series. An ultrafilter of the natural numbers will he called \emph{harmonic} if all each members are harmonic sets. The harmonic…
This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity…
We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with…
We survey some connections between topological dynamics, semigroups of ultrafilters, and combinatorics. As an application, we give a proof, based on ideas of Bergelson and Hindman, of the Hales-Jewett partition theorem.
We prove a strong dichotomy for the number of ultrapowers of a given countable model associated with nonprincipal ultrafilters on N. They are either all isomorphic, or else there are $2^{2^{\aleph_0}}$ many nonisomorphic ultrapowers. We…