Related papers: Between reduced powers and ultrapowers, II

A finite relational structure A is called compact if for any infinite relational structure B of the same type, the existence of a homomorphism from B to A is equivalent to the existence of homomorphisms from all finite substructures of B to…

In two papers we noted that in common practice many algebraic constructions are defined only `up to isomorphism' rather than explicitly. We mentioned some questions raised by this fact, and we gave some partial answers. The present paper…

We show that any II$_1$ factor that has the same 4-quantifier theory as the hyperfinite II$_1$ factor $\mathcal{R}$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results…

This article concerns the Herrlich-Chew theorem stating that a Hausdorff zero-dimensional space is $\mathbb{N}$-compact if and only if every clopen ultrafilter with the countable intersection property in this space is fixed. It also…

We isolate a new class of ultrafilters on N, called "quasi-selective" because they are intermediate between selective ultrafilters and P-points. (Under the Continuum Hypothesis these three classes are distinct.) The existence of…

We answer a question of Moore by building a forcing extension satisfying measuring together with CH. The construction works over any model of ZFC and can be described as a forcing iteration with countable structures as side conditions and…

We consider a filtration on the cohomology of the structure sheaf indexed by (not necessarily reduced) divisors ``at infinity''. We show that the filtered pieces have transfers morphisms, fpqc descent, and are so called cube invariant. In…

A topological monoid is isomorphic to an endomorphism monoid of a countable structure if and only if it is separable and has a compatible complete ultrametric such that composition from the left is non-expansive. We also give a topological…

Several important conjectures in Fractal Geometry can be summarised as follows: If the dimension of a self-similar measure in $\mathbb{R}$ does not equal its expected value, then the underlying iterated function system contains an exact…

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…

We study ultrafilters from the perspective of the algebra in the \v{C}ech-Stone compactification of the natural numbers, and idempotent elements therein. The first two results that we prove establish that, if $p$ is a Q-point (resp. a…

A combinatorial characterization of measurable filters on a countable set is found. We apply it to the problem of measurability of the intersection of nonmeasurable filters.

We construct, using mild combinatorial hypotheses, a real Menger set that is not Scheepers, and two real sets that are Menger in all finite powers, with a non-Menger product. By a forcing-theoretic argument, we show that the same holds in…

We develop the theory of cofinal types of ultrafilters over measurable cardinals and establish its connections to Galvin's property. We generalize fundamental results from the countable to the uncountable, but often in surprisingly…

Let C be a separable unital C*-algebra, not isomorphic to the complex numbers, equipped with a faithful tracial state. Let A be a unital direct limit of one dimensional NCCW complexes, also equipped with a faithful tracial state. Suppose…

Answering a question raised by V. V. Tkachuk, we present several examples of $\sigma$-compact spaces, some only consistent and some in ZFC, that are not countably tight but in which the closure of any discrete subset is countably tight. In…

Let f be a continuous map of a complete separable metric space E onto the irrationals. We show that if a complete separable metric space M contains isometric copies of every closed relatively discrete set in E, then M contains also an…

We study various combinatorial properties, and the implications between them, for filters generated by infinite-dimensional subspaces of a countable vector space. These properties are analogous to selectivity for ultrafilters on the natural…

Indecomposable continua with one composant are $\textit{large}$ in the sense of being non-metrisable. We adapt the method of Smith $[18]$ to construct an example which is $\textit{small}$ in the sense of being separable.

We consider quasiconformal deformations of $\mathbb{C}\setminus\mathbb{Z}$. We give some criteria for infinitely often punctured planes to be quasiconformally equivalent to $\mathbb{C}\setminus\mathbb{Z}$. In particular, we characterize the…