arXiv subjects

Piotr Szewczak

Publications and source records attributed to Piotr Szewczak.

At least 19 recordsLinked to original sources

arXiv2026

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. Sets of this kind have been extensively investigated in products of spaces with combinatorial covering properties such as the Hurewicz, Scheepers, Menger, and Rothberger properties. We will complete this picture by focusing on properties from the second row of the Scheepers diagram. We solve in the affirmative a problem due to Bartoszyński-Tsaban, whether all finite powers of a $\mathfrak{b}$-scale set satisfy the strong covering property $\mathsf{S}_1(Γ,Ω)$. We also show that, in the Miller model, the product of finitely many $\mathfrak{d}$-concentrated sets satisfies $\mathsf{S}_1(Γ,Ω)$. We further provide product counterexamples demonstrating limitations of the methods used.

math.CO↗

arXiv2026

Productively Scheepers spaces and their relatives

We prove that assuming $\mathfrak{b}=\mathfrak{d}$, in the class of hereditarily Lindelöf spaces, each productively Scheepers space is productively Hurewicz. The above statement remains true in the class of all general topological spaces assuming that $\mathfrak{d}=\aleph_1$. To this end we use combinatorial methods and the Menger covering property parametrized by ultrafilters. We also show that if near coherence of filters holds, then the Scheepers property is equivalent to a Menger property parametrized by any ultrafilter.

math.GN↗

arXiv2026

On disjoint tilings of $\ell_1(κ)$ by star-shaped bodies

We employ the set-theoretic methods yielding the consistency of the existence of nontrivial pairwise disjoint covers of the unit interval (or equivalently $\mathbb{R}^n$ for $n\in\mathbb{N}\setminus\{0\}$) by less than $2^ω$ closed sets in the context of covers of infinite dimensional Banach spaces $\ell_1(κ)$ for infinite $κ$ under geometric conditions arising in tiling theory and in approximation theory. Specifically, for $κ=ω_1, ω_2$, using side-by-side Sacks forcing we prove the consistency of an arbitrarily large continuum above $κ$ with the existence of a highly disconnected proximinal set in $\ell_1(κ)$, built from norm-compact pieces separated by a common positive distance, and the existence of a normal disjoint tiling of $\ell_1(κ)$ by star-shaped bodies of the form $X+B$, where $X$ is norm compact and $B$ is the unit ball. We also prove that if $κ$ is an uncountable cardinal of countable cofinality, then $\ell_1(κ)$ does not admit any normal disjoint tiling by bodies of the form $X+B$, where $X$ is closed and norm-separable and $B$ is the unit ball. These results complement a result of Klee of 1981 obtained for $κ$ satisfying $κ^ω=κ$, recent results of De Bernardi, Russo, Sezgek and Somaglia, and a $\mathsf{ZFC}$ a machine-discovered result (included in the appendix) that there are no nontrivial discrete Chebyshev sets in $\ell_1(κ)$ when $κ<2^ω$.

math.GN↗

arXiv2026

Concentrated sets and the Hurewicz property

A set of reals $X$ is $\mathfrak{b}$-concentrated if it has cardinality at least $\mathfrak{b}$ and it contains a countable set $D\subseteq X$ such that each closed subset of $X$ disjoint with $D$ has size smaller than $\mathfrak{b}$. We present ZFC results about structures of $\mathfrak{b}$-concentrated sets with the Hurewicz covering property using semifilters. Then we show that assuming that the semifilter trichotomy holds, then each $\mathfrak{b}$-concentrated set is Hurewicz and even productively Hurewicz. We also show that the appearance of Hurewicz $\mathfrak{b}$-concentrated sets under the semifilter trichotomy is somewhat specific and the situation in the Laver model for the consitency of the Borel Conjecture is different.

math.GN↗

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↗

arXiv2026

Small Hurewicz and Menger sets which have large continuous images

We provide new techniques to construct sets of reals without perfect subsets and with the Hurewicz or Menger covering properties. In particular, we show that if the Continuum Hypothesis holds, then there are such sets which can be mapped continuously onto the Cantor space. These results allow to separate the properties of Menger and $\mathsf{S}_1(Γ,\mathrm{O})$ in the realm of sets of reals without perfect subsets and solve a problem of Nowik and Tsaban concerning perfectly meager subsets in the transitive sense. We present also some other applications of the mentioned above methods.

math.GN↗

arXiv2025

Universally meager sets in the Miller model and similar ones

We work in the realm of sets of reals. We prove that in the Miller model and in a model constructed by Goldstern-Judah-Shelah all universally meager sets have size at most $ω_1$. Some relations between combinatorial covering properties in these models allow to obtain the same limitations for sizes of Rothberger spaces and Hurewicz spaces with no homeomorphic copy of the Cantor set inside. It follows from our results that the existence of a strong measure zero set of size $ω_2$ does not imply the existence of a Rothberger space of size $ω_2$. We also prove that in the Miller model all strong measure zero sets have size at most $ω_1$.

math.LO↗

arXiv2025

Totally paracompact spaces and the Menger covering property

A topological space is totally paracompact if any base of this space contains a locally finite subcover. We focus on a problem of Curtis whether in the class of regular Lindelöf spaces total paracompactness is equivalent to the Menger covering property. To this end we consider topological spaces with certain dense subsets. It follows from our results that the above equivalence holds in the class of Lindelöf GO-spaces defined on subsets of reals. We also provide a game-theoretical proof that any regular Menger space is totally paracompact and show that in the class of first-countable spaces the Menger game and a partial open neighborhood assignment game of Aurichi are equivalent. We also show that if $\mathfrak{b}=ω_1$, then there is an uncountable subspace of the Sorgenfrey line whose all finite powers are Lindelöf, which is a strengthening of a famous result due to Michael.

math.GN↗

arXiv2025

Menger and consonant sets in the Sacks model

Using iterated Sacks forcing and topological games, we prove that the existence of a totally imperfect Menger set in the Cantor cube with cardinality continuum is independent from ZFC. We also analyze the structure of Hurewicz and consonant subsets of the Cantor cube in the Sacks model.

math.LO↗

arXiv2024

Concentrated sets and $γ$-sets in the Miller model

Using combinatorial covering properties, we show that there is no concentrated set of reals of size $ω_2$ in the Miller model. The main result refutes a conjecture of Bartoszyński and Halbeisen. We also prove that there are no $γ$-set of reals of size $ω_2$ in the Miller model.

math.GN↗

arXiv2021

Abstract colorings, games and ultrafilters

The main result provide a common generalization for Ramsey-type theorems concerning finite colorings of edge sets of complete graphs with vertices in infinite semigroups. We capture the essence of theorems proved in different fields: for natural numbers due to Milliken--Tylor, Deuber--Hindman, Bergelson--Hindman, for combinatorial covering properties due to Scheepers and Tsaban, and local properties in function spaces due to Scheepers. To this end, we use idempotent ultrafilters in the Čech--Stone compactifications of discrete infinite semigroups and topological games. The research is motivated by the recent breakthrough work of Tsaban about colorings and the Menger covering property.

math.CO↗

arXiv2021

Null sets and combinatorial covering properties

A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property $γ$, a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property $γ$ that is not null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin--Miller and Bartoszyński--Recław, to obtain sets with analogous properties. We also consider products of Sierpiński sets in the context of combinatorial covering properties.

math.GN↗

arXiv2020

Finite powers and products of Menger sets

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 the Blass--Shelah model for arbitrary values of the ultrafilter and dominating number.

math.GN↗

arXiv2019

Unbounded towers and products

We investigate products of sets of reals with combinatorial covering properties. A topological space satisfies $\mathsf{S}_1(Γ,Γ)$ if for each sequence of point-cofinite open covers of the space, one can pick one element from each cover and obtain a point-cofinite cover of the space. We prove that, if there is an unbounded tower, then there is a nontrivial set of reals satisfying $\mathsf{S}_1(Γ,Γ)$ in all finite powers. In contrast to earlier results, our proof does not require any additional set-theoretic assumptions. A topological space satisfies $Ω\chooseΓ$ (also known as Gerlits--Nagy's property $γ$) if every open cover of the space such that each finite subset of the space is contained in a member of the cover, contains a point-cofinite cover of the space. We investigate products of sets satisfying $Ω\chooseΓ$ and their relations to other classic combinatorial covering properties. We show that finite products of sets with a certain combinatorial structure satisfy $Ω\chooseΓ$ and give necessary and sufficient conditions when these sets are productively $Ω\chooseΓ$.

math.GN↗

arXiv2019

Strongly sequentially separable function spaces, via selection principles

A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continous and Borel real-valued functions on Tychonoff spaces, with the topology of pointwise convergence. Our results solve a problem stated by Gartside, Lo, and Marsh.

math.GN↗

arXiv2019

Conceptual proofs of the Menger and Rothberger games

We provide conceptual proofs of the two most fundamental theorems concerning topological games and open covers: Hurewicz's Theorem concerning the Menger game, and Pawlikowski's Theorem concerning the Rothberger game.

math.GN↗

arXiv2019

Products of Luzin-type sets with combinatorial properties

We construct Luzin-type subsets of the real line in all finite powers Rothberger, with a non-Menger product. To this end, we use a purely combinatorial approach which allows to weaken assumptions used earlier to construct sets with analogous properties. Our assumptions hold, e.g., in the Random model, where already known category theoretic methods fail.

math.GN↗

arXiv2018

The Haar Measure Problem

An old problem asks whether every compact group has a Haar-nonmeasurable subgroup. A series of earlier results reduce the problem to infinite metrizable profinite groups. We provide a positive answer, assuming a weak, potentially provable, consequence of the Continuum Hypothesis. We also establish the dual, Baire category analogue of this result.

math.GN↗