arXiv subjects

Jacek Hejduk

Publications and source records attributed to Jacek Hejduk.

3 recordsLinked to original sources

arXiv2026

On the family of measurable sets having the upper positive density

The essence of the density topology lies in the family of Lebesgue measurable sets where each point of a set is a density point of that set. The motivation of this work is to investigate the family of measurable sets for which, at every point within a set belonging to this family, the upper density of that set is positive. We obtain a strong generalized topology, and its essential properties are demonstrated in comparison with those of the classical density topology.

math.GM↗

arXiv2022

Baire Category Lower Density Operators with Borel Values

We prove that the lower density operator associated with the Baire category density points in the real line has Borel values of class $\pmb Π^0_3$ which is analogous to the measure case. We also introduce the notion of the Baire category density point of a subset with the Baire property of the Cantor space, and we prove that it generates a lower density operator with Borel values of class $\pmb Π^0_3$.

math.GN↗

arXiv2021

On Urysohn's Lemma for generalized topological spaces in ZF

A strong generalized topological space is an ordered pair $\mathbf{X}=\langle X, \mathcal{T}\rangle$ such that $X$ is a set and $\mathcal{T}$ is a collection of subsets of $X$ such that $\emptyset, X\in \mathcal{T}$ and $\mathcal{T}$ is stable under arbitrary unions. A necessary and sufficient condition for a strong generalized topological space $\mathbf{X}$ to satisfy Urysohn's lemma or its appropriate variant is shown in $\mathbf{ZF}$. Notions of a U-normal and an effectively normal generalized topological space are introduced. It is observed that, in $\mathbf{ZF}+\mathbf{DC}$, every U-normal generalized topological space satisfies Urysohn's lemma. It is shown that every effectively normal generalized topological space satisfies Csaszár's modification of Urysohn's Lemma. A $\mathbf{ZF}$- example of a strong generalized topological normal space which satisfies the Tietze-Urysohn Extension Theorem and fails to satisfy Urysohn's Lemma is shown.

math.GN↗