arXiv subjects

Piotr Zakrzewski

Publications and source records attributed to Piotr Zakrzewski.

8 recordsLinked to original sources

arXiv2025

On Sierpiński sets, Hurewicz spaces and Hilgers functions

The Hurewicz property is a classical generalization of $σ$-compactness and Sierpiński sets (whose existence follows from CH) are standard examples of non-$σ$-compact Hurewicz spaces. We show, solving a problem stated by Szewczak and Tsaban, that for each Sierpiński set S of cardinality at least $\mathfrak b$ there is a Hurewicz space H with $S\times H$ not Hurewicz. Some other questions in the literature concerning this topic are also answered.

math.GN↗

arXiv2025

A note on uniform continuity of monotone functions

We prove that it is consistent with ZFC that for every non-decreasing function $f:[0,1]\to [0,1]$, each subset of $[0,1]$ of cardinality $\mathfrak c$ contains a set of cardinality $\mathfrak c$ on which $f$ is uniformly continuous. We show that this statement follows from the assumptions that $\mathfrak d^* < \mathfrak c$ and $\mathfrak c$ is regular, where $\mathfrak d^*\leq \mathfrak d$ is the smallest cardinality $κ$ such that any two disjoint countable dense sets in the Cantor set can be separated by sets each of which is an intersection of at most $κ$-many open sets in the Cantor set. We establish also that $\mathfrak d^*=\min\{\mathfrak u, \mathfrak d\}=\min\{\mathfrak r, \mathfrak d\}$, thus giving an alternative proof of the latter equality established by J. Aubrey in 2004.

math.LO↗

arXiv2024

On two consequences of CH established by Sierpinski. II

We continue a study of the relations between two consequences of the Continuum Hypothesis discovered by Waclaw Sierpinski, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum.

math.LO↗

arXiv2023

On two consequences of CH established by Sierpiński

We study the relations between two consequences of the Continuum Hypothesis discovered by Wacław Sierpiński, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum.

math.LO↗

arXiv2023

Countably perfectly meager and countably perfectly null sets

We study a strengthening of the notion of a universally meager set and its dual counterpart that strengthens the notion of a universally null set. We say that a subset $A$ of a perfect Polish space $X$ is countably perfectly meager (respectively, countably perfectly null) in $X$, if for every perfect Polish topology $τ$ on $X$, giving the original Borel structure of $X$, $A$ is covered by an $F_σ$-set $F$ in $X$ with the original Polish topology such that $F$ is meager with respect to $τ$ (respectively, for every finite, non-atomic, Borel measure $μ$ on $X$, $A$ is covered by an $F_σ$-set $F$ in $X$ with $μ(F)=0$). We prove that if $2^{\aleph_0}\leq \aleph_2$, then there exists a universally meager set in $2^{\mathbb N}$ which is not countably perfectly meager in $2^{\mathbb N}$ (respectively, a universally null set in $2^{\mathbb N}$ which is not countably perfectly null in $2^{\mathbb N}$).

math.LO↗

arXiv2021

Countably perfectly meager sets

We study a strengthening of the notion of a perfectly meager set. We say that that a subset $A$ of a perfect Polish space $X$ is countably perfectly meager in $X$, if for every sequence of perfect subsets $\{P_n: n \in {\mathbb N}\}$ of $X$, there exists an $F_σ$-set $F$ in $X$ such that $A \subseteq F$ and $F\cap P_n$ is meager in $P_n$ for each $n$. We give various characterizations and examples of countably perfectly meager sets. We prove that not every universally meager set is countably perfectly meager correcting an earlier result of Bartoszyński.

math.LO↗

arXiv2019

On Mazurkiewicz's sets, thin σ-ideals of compact sets and the space of probability measures on the rationals

We shall establish some properties of thin $σ$-ideals of compact sets in compact metric spaces (in particular, the $σ$-ideals of compact null-sets for thin subadditive capacities), and we shall refine the celebrated theorem of David Preiss that there exist compact non-uniformly tight sets of probability measures on the rationals. Both topics will be based on a construction of Stefan Mazurkiewicz from his 1927 paper containing a solution of a Urysohn's problem in dimension theory.

math.GN↗

arXiv2017

On Borel maps, calibrated $σ$-ideals and homogeneity

Let $μ$ be a Borel measure on a compactum $X$. The main objects in this paper are $σ$-ideals $I(dim)$, $J_0(μ)$, $J_f(μ)$ of Borel sets in $X$ that can be covered by countably many compacta which are finite-dimensional, or of $μ$-measure null, or of finite $μ$-measure, respectively. Answering a question of J. Zapletal, we shall show that for the Hilbert cube, the $σ$-ideal $I(dim)$ is not homogeneous in a strong way. We shall also show that in some natural instances of measures $μ$ with non-homogeneous $σ$-ideals $J_0(μ)$ or $J_f(μ)$, the completions of the quotient Boolean algebras $Borel(X)/J_0(μ)$ or $Borel(X)/J_f(μ)$ may be homogeneous. We discuss the topic in a more general setting, involving calibrated $σ$-ideals.

math.LO↗