arXiv subjects

Michał Dybowski

Publications and source records attributed to Michał Dybowski.

3 recordsLinked to original sources

arXiv2026

Lebesgue Covering Theorem and level sets of continuous functions

We formulate and prove a dimension-theoretic generalization of a version of the Lebesgue Covering Theorem. A generalized $n$-dimensional version of the Steinhaus Chessboard Theorem, recently proved algorithmically by Turzański and Ziajor, is a particular case of this result. Moreover, we study two types of sets associated with a continuous function $g \colon [0,1]^n \to \mathbb{R}$. Namely, the set of all points $p \in \mathbb{R}$ such that the fiber $g^{-1}[\left\{p\right\}]$ connects $i$th opposite faces of $[0,1]^n$, and the set of all points $p \in \mathbb{R}$ such that the fiber $g^{-1}[\left\{p\right\}]$ separates $i$th opposite faces of $[0, 1]^n$.

math.CA↗

arXiv2026

Chessboard and level sets of continuous functions

We provide the following result and its discrete equivalent: Let $f \colon I^n \to \mathbb{R}^{n-1}$ be a continuous function. Then, there exist a point $p \in \mathbb{R}^{n-1}$ and a compact subset $S \subset f^{-1}\left[\left\{p\right\}\right]$ which connects some opposite faces of the $n$-dimensional unit cube $I^n$. We give an example that shows it cannot be generalized to path-connected sets. Additionally, we show that a version of the Steinhaus Chessboard Theorem and the Brouwer Fixed Point Theorem are simple consequences of this result.

math.GN↗

arXiv2024

Maximal δ-separated sets in separable metric spaces and weak forms of choice

We show that the statement ``In every separable pseudometric space there is a maximal non-strictly δ-separated set.'' implies the axiom of choice for countable families of sets. This gives answers to a question of Dybowski and Górka in [M. Dybowski and P. Górka, The axiom of choice in metric measure spaces and maximal δ-separated sets, Archive for Mathematical Logic 62, 735-749, 2023.]. We also prove several related results.

math.LO↗