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Wieslaw Kubiś

Publications and source records attributed to Wieslaw Kubiś.

14 recordsLinked to original sources
arXiv2023

Universal monoid actions: the power of freedom

Motivated by a recent work of Balcerzak and Kania [Proc. Amer. Math. Soc. 151 (2023) 3737--3742], we show that every countable monoid has a universal action on the free object over a countable infinite set. This is a general result concerning concrete categories with a left adjoint (free) functor. On the way, we introduce an abstract concept of ``being generated by a set". At the same time we obtain a simpler proof of the result of Darji and Matheron concerning a surjectively universal operator on the classical Banach space $\ell_1$ of summable sequences.

math.CT↗
arXiv2022

Ultrafilter selection and Corson compacta

We study the question which Boolean algebras have the property that for every generating set there is an ultrafilter selecting maximal number of its elements. We call it the ultrafilter selection property. For cardinality aleph-one the property is equivalent to the fact that the space of ultrafilters is not Corson compact. We also consider the pointwise topology on a Boolean algebra, proving a result on the Lindelöf number in the context of the ultrafilter selection property. Finally, we discuss poset Boolean algebras, interval algebras, and semilattices in the context of ultrafilter selection properties.

math.LO↗
arXiv2021

Weak Fraisse categories

We develop the theory of weak Fraisse categories, where the crucial concept is the weak amalgamation property, discovered relatively recently in model theory. We show that, in a suitable framework, every weak Fraisse category has its unique limit, a special object in a bigger category, characterized by certain variant of injectivity. This significantly extends the present theory of Fraisse limits.

math.CT↗
arXiv2019

A note on universal operators between separable Banach spaces

We compare two types of universal operators constructed relatively recently by Cabello Sánchez, and the authors. The first operator $Ω$ acts on the Gurarii space, while the second one $P_S$ has values in a fixed separable Banach space $S$. We show that if $S$ is the Gurarii space, then both operators are isometric. We also prove that, for a fixed space $S$, the operator $P_S$ is isometrically unique. Finally, we show that $Ω$ is generic in the sense of a natural infinite game.

math.FA↗
arXiv2018

Chain intersection closures

We study spherical completeness of ball spaces and its stability under expansions. We introduce the notion of an ultra-diameter, mimicking diameters in ultrametric spaces. We prove some positive results on preservation of spherical completeness involving ultra-diameters with values in narrow partially ordered sets. Finally, we show that in general, chain intersection closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness.

math.LO↗
arXiv2015

Banach-Mazur game played in partially ordered sets

We present a version of the Banach-Mazur game, where open sets are replaced by elements of a fixed partially ordered set. We show how to apply it in the theory of Fraisse limits and beyond, obtaining simple proofs of universality of certain objects and classes.

math.LO↗
arXiv2012

Injective objects and retracts of Fraïssé limits

We present a purely category-theoretic characterization of retracts of Fraïssé limits. For this aim, we consider a natural version of injectivity with respect to a pair of categories (a category and its subcategory). It turns out that retracts of Fraïssé limits are precisely the objects that are injective relatively to such a pair. One of the applications is a characterization of non-expansive retracts of Urysohn's universal metric space.

math.CT↗
arXiv2011

Complementation in Spaces of Continuous Functions on Compact Lines

We characterize order preserving continuous surjections between compact linearly ordered spaces which admit an averaging operator, together with estimates of the norm of such an operator. This result is used to the study of strengthenings of the separable complementation property in spaces of continuous functions on compact lines. These properties include in particular continuous separable complementation property and existence of a projectional skeleton.

math.FA↗
arXiv2007

Valdivia compact Abelian groups

Let R denote the smallest class of compact spaces containing all metric compacta and closed under limits of continuous inverse sequences of retractions. Class R is striclty larger than the class of Valdivia compact spaces. We show that every compact connected Abelian group which is a topological retract of a space from class R is necessarily isomorphic to a product of metric groups. This completes the result of V. Uspenskij and the author, where a compact connected Abelian group outside class R has been described.

math.GN↗
arXiv2005

Small Valdivia compact spaces

We prove a preservation theorem for the class of Valdivia compact spaces, which involves inverse sequences of ``simple'' retractions. Consequently, a compact space of weight $\loe\aleph_1$ is Valdivia compact iff it is the limit of an inverse sequence of metric compacta whose bonding maps are retractions. As a corollary, we show that the class of Valdivia compacta of weight at most $\aleph_1$ is preserved both under retractions and under open 0-dimensional images. Finally, we characterize the class of all Valdivia compacta in the language of category theory, which implies that this class is preserved under all continuous weight preserving functors.

math.GN↗
arXiv2003

A compact group which is not Valdivia compact

A compact space $K$ is {\em Valdivia compact} if it can be embedded in a Tikhonov cube $I^A$ in such a way that the intersection $K\capΣ$ is dense in $K$, where $Σ$ is the sigma-product (= the set of points with countably many non-zero coordinates). We show that there exists a compact connected Abelian group of weight $ø_1$ which is not Valdivia compact, and deduce that Valdivia compact spaces are not preserved by open maps.

math.GN↗