Searcharxiv⌕ Search

arXiv subjects

Joanna Jureczko

Publications and source records attributed to Joanna Jureczko.

At least 19 recordsLinked to original sources
arXiv2026

On Banach and Kuratowski Theorem and generalized strong sequences

In 1929, Banach and Kuratowski proved under CH a combinatorial theorem, which implies that there is not a non-vanishing $σ$-additive finite measure on $\mathbb{R}$ which is defined for every set of reals. In 2003 Bartoszyński and Halbeisen proved that Banach and Kuratowski theorem is equivalent to the existence of a K-Lusin set of the cardinality continuum an the existence of such sets is independent of $ZFC + \neg CH$. On the other hand in 1965 Efimov introduced the strong sequences method which used to prove some well-known theorems in dyadic spaces. The aim of this paper is to show that all this notions can be generalized, i.e. considered in the generalized Baire space and to show some equivalences among them. Moreover, some applications in the direction of calibres, boundedness and partitions relations are also shown.

math.LO↗
arXiv2023

A note on Kuratowski partitions in metric spaces

We prove that the existence of a complete metric space of cardinality at most $2^κ$ admitting Kuratowski partition is a consequence of $κ$ being the smallest real-valued measurable cardinal not greater than $ 2^{\aleph_0}$.

math.LO↗
arXiv2023

On equivalences of polarized partition relations

The paper deals with two notions: polarized partition relations and product of generalized strong sequences. Strong sequences were introduced by Efimov in 1965 as a usefull tool for proving famous theorems in dyadic spaces, i.e. continuous images of Cantor cube. In this paper we introduce the notion of product of generalized strong sequences and give pure combinatorial proof that existence of product of generalized strong sequences is equivalent to polarized partition relations.

math.LO↗
arXiv2023

Weak $P$-points for regulars

The main result of this paper is to show that for any compact space $X$ and any family $\{f_α\colon α< 2^κ, f_α\colon X \stackrel{onto}{\longrightarrow} βκ\}$ of open mappings there exists a point $x \in X$ such that for each $α< 2^κ$ either $f_α(x)$ belongs to $κ$ or $f_α(x)$ is a weak $P_κ$-point in $κ^*$, whenever $κ$ is regular.

math.GN↗
arXiv2023

Nonmeasurable sets in tree structures and Ellentuck topology

The aim of this paper is to provide the results that answer the Kuratowski problem posed in 1935 concerning the existence of nonmeasurable sets. The Kuratowski problem was considered for partitions, here we provide a generalization to point-finite families for tree structures and structures in Ellentuck space. The main result of this paper generalizes previous results in the topic of nonmeasurable sets.

math.LO↗
arXiv2023

Some remarks on polarized partition relations

This paper deals with two notions: a polarized partition relations $\left( \begin{array}{c} α β \end{array} \right) \to \left( \begin{array}{cc} γ& η δ& λ \end{array} \right)$ and product of generalized strong sequences. Strong sequences were introduced by Efimov in 1965 as a usefull tool for proving famous theorems in dyadic spaces, i.e. continuous images of Cantor cube. In this paper we introduce the notion of product of generalized strong sequences and give the pure combinatorial proof that $\left( \begin{array}{c} α β \end{array} \right) \to \left( \begin{array}{cc} γ& η δ& λ \end{array} \right)$ is a consequence of the existence of product of generalized strong sequences.

math.LO↗
arXiv2021

Remarks on the existence of measurable selectors

A classical theorem from measure theory that gives a sufficient condition for a multifunction to have a measurable selection is Kuratowski and Ryll-Nardzewski Selection Theorem. The aim of this paper is to show some generalizations of this result.

math.LO↗
arXiv2021

A note on asymptotic density

It is proved that $P(ω)/\triangle_d$, where $\triangle_d$ is the ideal of sets of asymptotic density zero, is universal in the sense of embeddings.

math.LO↗
arXiv2020

On special partitions of metric spaces

The main result of this paper is to show that, if $κ$ is the smallest real-valued measurable cardinal not greater than $ 2^{\aleph_0}$, then there exists a complete metric space of cardinality not greater than $ 2^κ$ admitting a Kuratowski partition.

math.LO↗