On the structure of ultrafilters and properties related to convergence in topological spaces

We consider properties of broadly understood measurable spaces that provide the preservation of maximality when ultrafilters are restricted to filters of the corresponding subspace. We study conditions that guarantee the convergence of images of ultrafilters consisting of open sets under continuous mappings.     

Ultrafilters of measurable spaces as generalized solutions in abstract attainability problems

We consider problems of asymptotic analysis that arise, in particular, in the formalization of effects related to an approximate observation of constraints. We study nonsequential (generally speaking) variants of asymptotic behavior that can be formalized in the class of ultrafilters of an appropriate measurable space. We construct attraction sets in a topological space that are realized in the class of ultrafilters of the corresponding measurable space and specify conditions under which ultrafilters of a measurable space are sufficient for constructing the “complete” attraction set corresponding to applying ultrafilters of the family of all subsets of the space of ordinary solutions. We study a compactification of this space that is constructed in the class of Stone ultrafilters (ultrafilters of a measurable space with an algebra of sets) such that the attraction set is realized as a continuous image of the compact set of generalized solutions; we also study the structure of this compact set in terms of free ultrafilters and ordinary solutions that observe the constraints of the problem exactly. We show that, in the case when there are no exact ordinary solutions, this compact set consists of free ultrafilters only; i.e., it is contained in the remainder of the compactifier (an example is given showing that the similar property may be absent for other variants of the extension of the original problem).     

Extension of an abstract attainability problem with the use of the stone representation space

We consider an attainability problem in a complete metric space on values of an objective operator h. We assume that the latter admits a uniform approximation by mappings which are tier with respect to a given measurable space with an algebra of sets. Let asymptotic-type constraints be defined as a nonempty family of sets in this measurable space. We treat ultrafilters of the measurable space as generalized elements; we equip this space of ultrafilters with a topology of a zero-dimensional compact (the Stone representation space). On this base we construct a correct extension of the initial problem, realizing the set of attraction in the form of a continuous image of the compact of feasible generalized elements. Generalizing the objective operator, we use the limit with respect to ultrafilters of the measurable space. This provides the continuity of the generalized version of h understood as a mapping of the zero-dimensional compact into the topological space metrizable with a total metric.     

Representation of attraction elements in abstract attainability problems with asymptotic constraints

We consider an abstract attainability problem with asymptotic constraints in a topological space. We construct an extension in the class of ultrafilters of widely interpreted measurable spaces. We study an example of a static problem on the asymptotic attainability in the class of layer functions.     

On the maximal resolvability of monotonically normal spaces

We continue the work started in [6] and show that all monotonically normal (in short: MN) spaces are maximally resolvable if and only if all uniform ultrafilters are maximally decomposable. As a consequence we get that the existence of an MN space which is not maximally resolvable is equi-consistent with the existence of a measurable cardinal. We also show that it is consistent (modulo the consistency of a measurable cardinal) that there is an MN space X with |X| = Δ(X) = ? _ω which is not ω ₁-resolvable. It follows from the results of [6] that this is best possible.     

Properties of ultraproducts

A direct proof is given of the completeness of ultraproduct spaces when the factor spaces are Banach spaces. The method of proof also shows the completeness of the ultraproduct even when the factor spaces are not complete for many (possibly all) ultrafilters. It is also shown that ultraproduct spaces are separable only when they are finite dimensional.     

Compactable semilattices

We characterize those semilattices that give rise to Boolean spaces on their associated spaces of ultrafilters. The class of 0-disjunctive semilattices, important in the theory of congruence-free inverse semigroups, plays a distinguished role in this theory.     

Open Ultrafilters and Separability with the Use of the Operation of Closure

We study ultrafilters of topologies as well as sets of ultrafilters that each time dominate the open neighborhood filter of some fixed point in a topological space. The sets of ultrafilters are considered as “enlarged points” of the original space. We study conditions that provide the distinguishability of (enlarged) “points” of this type. We use nontraditional separability axioms and study their connection with the known axioms T₀, T₁, and T₂.     

Closure spaces and characterizations of filters in terms of their stone images

Fréchet, strongly Fréchet, productively Fréchet, weakly bisequential and bisequential filters (i.e., neighborhood filters in spaces of the same name) are characterized in a unified manner in terms of their images in the Stone space of ultrafilters. These characterizations involve closure structures on the set of ultrafilters. The case of productively Fréchet filters answers a question of S. Dolecki and turns out to be the only one involving a non topological closure structure.     

Homogeneous sets from several ultrafilters

We consider Ramsey-style partition theorems in which homogeneity is asserted not for subsets of a single infinite homogeneous set but for subsets whose elements are chosen, in a specified pattern, from several sets in prescribed ultrafilters. We completely characterize the sequences of ultrafilters satisfying such partition theorems. (Non-isomorphic selective ultrafilters always work, but, depending on the specified pattern, weaker hypotheses on the ultrafilters may suffice.) We also obtain similar results for analytic partitions of the infinite sets of natural numbers. Finally, we show that the two P-points obtained by applying the maximum and minimum functions to a union ultrafilter are never nearly coherent.     

D sets and IP rich sets in countable,cancellative abelian semigroups

We give combinatorial characterizations of IP rich sets (IP sets that remain IP upon removal of any set of zero upper Banach density) and D sets (members of idempotent ultrafilters, all of whose members have positive upper Banach density) in a general countable, cancellative abelian semigroup. We then show that the family of IP rich sets strictly contains the family of D sets.     

Computability of measurable sets via effective topologies

We investigate in the frame of TTE the computability of functions of the measurable sets from an infinite computable measure space such as the measure and the four kinds of set operations. We first present a series of undecidability and incomputability results about measurable sets. Then we construct several examples of computable topological spaces from the abstract infinite computable measure space, and analyze the computability of the considered functions via respectively each of the standard representations of the computable topological spaces constructed. The authors are supported by grants of NSFC and DFG.     

Some constructions of ultrafilters over a measurable cardinal

Some non-normal κ-complete ultrafilters over a measurable κ with special properties are constructed. Questions by A. Kanamori [4] about infinite Rudin-Frolik sequences, discreteness and products are answered.     

On weak integrability and boundedness in Banach spaces

Thin and thick sets in normed spaces were defined and studied by M.I. Kadets and V.P. Fonf in 1983. In this paper, we give a new characterization of thick sets in terms of weak integrability of Banach space valued measurable functions. We also characterize thick sets in terms of boundedness of vector measures, and explain how this concept is related to the theory of barrelled spaces.     

Subdivision Schemes of Sets and the Approximation of Set-Valued Functions in the Symmetric Difference Metric

In this work we construct subdivision schemes refining general subsets of ? ⁿ and study their applications to the approximation of set-valued functions. Differently from previous works on set-valued approximation, our methods are developed and analyzed in the metric space of Lebesgue measurable sets endowed with the symmetric difference metric. The construction of the set-valued subdivision schemes is based on a new weighted average of two sets, which is defined for positive weights (corresponding to interpolation) and also when one weight is negative (corresponding to extrapolation). Using the new average with positive weights, we adapt to sets spline subdivision schemes computed by the Lane–Riesenfeld algorithm, which requires only averages of pairs of numbers. The averages of numbers are then replaced by the new averages of pairs of sets. Among other features of the resulting set-valued subdivision schemes, we prove their monotonicity preservation property. Using the new weighted average of sets with both positive and negative weights, we adapt to sets the 4-point interpolatory subdivision scheme. Finally, we discuss the extension of the results obtained in metric spaces of sets, to general metric spaces endowed with an averaging operation satisfying certain properties.     

Foreword

A hierarchy of topological Ramsey spaces \({\mathcal{R}_\alpha}\) (\({\alpha < \omega_1}\)), generalizing the Ellentuck space, were built by Dobrinen and Todorcevic in order to completely classify certain equivalent classes of ultrafilters Tukey (resp. Rudin–Keisler) below \({\mathcal{U}_\alpha}\) \({(\alpha < \omega_1)}\), where \({\mathcal{U}_\alpha}\) are ultrafilters constructed by Laflamme satisfying certain partition properties and have complete combinatorics over the Solovay model. We show that Nash–Williams, or Ramsey ultrafilters in these spaces are preserved under countable-support side-by-side Sacks forcing. This is achieved by proving a parametrized theorem for these spaces, and showing that Nash–Williams ultrafilters localizes the theorem. We also show that every Nash–Williams ultrafilter in \({\mathcal{R}_\alpha}\) is selective.     

An Algebraic Study of Affine Real Ultrafilters

Abstract

The families of affine semi-algebraic sets over a real-closed field Kand semi-linear sets over an ordered field enjoy many closure properties with algebraic and geometric significance. This paper studies the natural closure properies of Minkowski sums and scalar dilation. It gives an extension of the underlying vector space structure that enables the study of an arithmetic on the abstract points of their associated spectra. This arithmetic satisfies certain cancellation principles that motivates an investigation into an algebraic object weaker than a group and culminates with a version of the Jordan-Hölder theorem. With the subsequent definition of dimension we show that the collection of affine real ultrafilters in K ⁿ is n-dimensional over the scalar ultrafilters.     

Sets in Prikry and Magidor generic extensions

We continue [4] and study sets in generic extensions by the Magidor forcing and by the Prikry forcing with non-normal ultrafilters.     

Ultrafilters in the constructions of attraction sets: Problem of compliance to constraints of asymptotic character

We consider the properties of generalized elements in the problem of compliance to constraints of asymptotic character; these elements are identified with ultrafilters of special families of sets in the space of ordinary solutions.     

Nonstandard characterisations of tensor products and monads in the theory of ultrafilters

We use nonstandard methods, based on iterated hyperextensions, to develop applications to Ramsey theory of the theory of monads of ultrafilters. This is performed by studying in detail arbitrary tensor products of ultrafilters, as well as by characterising their combinatorial properties by means of their monads. This extends to arbitrary sets and properties methods previously used to study partition regular Diophantine equations on $\mathbb{N}$. Several applications are described by means of multiple examples.