arXiv subjects

Michał Pawlikowski

Publications and source records attributed to Michał Pawlikowski.

5 recordsLinked to original sources

arXiv2026

Combinatorial covering properties in an uncountable setting: canonical examples

We provide examples of spaces satisfying generalized combinatorial covering properties such as the Hurewicz, Menger, and $γ$-properties in an uncountable setting. Our approach is motivated by canonical constructions from the classical countable case, including the examples of Bartoszyński and Shelah separating the Hurewicz property from $σ$-compactness, the examples of Tsaban and Zdomskyy separating the Hurewicz and Menger properties, and Tsaban's construction of a nontrivial set of reals with the $γ$-property. We focus on the genuinely nontrivial aspects of these higher-cardinal generalizations, uncovering several open problems whose nature appears substantially different from that of their countable counterparts.

math.GN↗

arXiv2025

Scales, products and the second row of the Scheepers diagram

We consider products of sets of reals with a combinatorial structure based on scales parameterized by filters. This kind of sets were intensively investigated in products of spaces with combinatorial covering properties as Hurewicz, Scheepers, Menger and Rothberger. We will complete this picture with focusing on properties from the second row of the Scheepers diagram. In particular we show that in the Miller model a product space of two $\mathfrak{d}$-concentrated sets has a strong covering property $\mathsf{S}_1(Γ,Ω)$. We provide also counterexamples in products to demonstrate limitations of used methods.

math.CO↗

arXiv2023

On strong algebrability of families of non-measurable functions of two variables

Recently Tomasz Natkaniec in [On lineability of families of non-measurable functions of two variable. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 115(1):Paper No. 33, 10, 2021] studied the lineability problem for several classes of non-measurable functions in two variables. In this note we improve his results in the direction of algebrability. In particular, we show that most of the classes considered by Natkaniec contain free algebras with $2^{\mathfrak{c}}$ many generators.

math.FA↗

arXiv2023

Free group of Hamel functions

We construct a free group of continuum many generators among those autobijections of $\mathbb{R}$ which are also Hamel bases of $\mathbb{R}^2$, with identity function included. We also observe two new cases when a real function is a composition of two real functions which are Hamel bases of $\mathbb{R}^2$.

math.GR↗

arXiv2022

An inverse Fraïssé limit for finite posets and duality for posets and lattices

We consider a category of all finite partial orderings with quotient maps as arrows and construct a Fraïssé sequence in this category. Then we use commonly known relations between partial orders and lattices to construct a sequence of lattices associated with it. Each of these two sequences has a limit object -- an inverse limit, which is an object of our interest as well. In the first chapter there are some preliminaries considering partial orders, lattices, topology, inverse limits, category theory and Fraïssé theory, which are used later. In the second chapter there are our results considering a Fraïssé sequence in category of finite posets with quotient maps and properties of inverse limit of this sequence. In the third chapter we investigate connections between posets and order ideals corresponding to them, getting an inductive sequence made of these ideals; then we study properties of the inverse limit of this sequence.

math.CO↗