Related papers: Between reduced powers and ultrapowers, II
We study ultrafilters on countable sets and reaping families which are indestructible by Sacks forcing. We deal with the combinatorial characterization of such families and we prove that every reaping family of size smaller than the…
It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasov's question on…
A cohesive power of a structure is an effective analog of the classical ultrapower of a structure. We start with a computable structure, and consider its countable ultrapower over a cohesive set of natural numbers. A cohesive set is an…
In the absence of the Axiom of Choice, necessary and sufficient conditions for a locally compact Hausdorff space to have all non-empty second-countable compact Hausdorff spaces as remainders are given in $\mathbf{ZF}$. Among other…
We consider precompact sequential and Fr\'echet group topologies and show that some natural constructions of such topologies always result in metrizable groups answering a question of D.~Dikranjan et al. We show that it is consistent that…
We generalize the lexicographic product of first-order structures by presenting a framework for constructions which, in a sense, mimic iterating the lexicographic product infinitely and not necessarily countably many times. We then define…
The structures $\langle M,\subseteq^M\rangle$ arising as the inclusion relation of a countable model of sufficient set theory $\langle M,\in^M\rangle$, whether well-founded or not, are all isomorphic. These structures $\langle…
All constructive methods employed in modern mathematics produce only countable sets, even when designed to transcend countability. We show that any constructive argument for uncountability -- excluding diagonalization techniques --…
We show that every definable subset of an uncountably categorical pseudofinite structure has pseudofinite cardinality which is polynomial (over the rationals) in the size of any strongly minimal subset, with the degree of the polynomial…
Assuming the Generalized Continuum Hypothesis, this paper answers the question: when is the tensor product of two ultrafilters equal to their Cartesian product? It is necessary and sufficient that their Cartesian product is an ultrafilter;…
If $\mathcal P$ is a family of filters over some set $I$, a topological space $X$ is \emph{sequencewise $\mathcal P$-\brfrt compact} if, for every $I$-indexed sequence of elements of $X$, there is $F \in \mathcal P$ such that the sequence…
In this note we present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hern\'andez-Guti\'errez and Hru\v{s}\'ak. The method of the proof also allows us to obtain a…
We prove that no ultraproduct of Banach spaces via a countably incomplete ultrafilter can contain $c_0$ complemented. This shows that a "result" widely used in the theory of ultraproducts is wrong. We then amend a number of results whose…
The main aim of the article is to show, in the absence of the Axiom of Choice, relationships between the following, independent of $\mathbf{ZF}$, statements: "Every countable product of compact metrizable spaces is separable (respectively,…
Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martin's Axiom for countable posets, to distinguish non-principal ultrafilters on $\omega$ up to homeomorphism.…
We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum…
We prove that topological isomorphism on procountable groups is not classifiable by countable structures, in the sense of descriptive set theory. In fact, the equivalence relation $\ell_\infty$ expressing that two sequences of reals have a…
An infinite cardinal $\lambda$ is called Fr\'echet if the Fr\'echet filter on $\lambda$ extends to a countably complete ultrafilter. We investigate the relationship between Fr\'echet cardinals and strongly compact cardinals under a…
The ultraproduct construction is generalized to $p$-ultramean constructions ($1\leqslant p<\infty$) by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments $\mathscr L^p$ of…
A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…