Fibers of continuous functions that connect or separate some opposite faces of the unit cube, and a generalization of the Lebesgue Covering Theorem
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911449553567744 |
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| author | Dybowski, Michał |
| author_facet | Dybowski, Michał |
| contents | We formulate and prove a dimension-theoretic generalization of the Lebesgue Covering Theorem. A generalized $n$-dimensional version of the Steinhaus Chessboard Theorem, recently proved by Turzański and Ziajor, is a simple consequence of this result. Moreover, we study two types of sets associated with a continuous function $g \colon I^n \to \mathbb{R}$. 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 $I^n$, 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 $I^n$. We provide necessary and sufficient conditions for the existence of a continuous function $g \colon I^n \to \mathbb{R}$ in terms of these sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13883 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fibers of continuous functions that connect or separate some opposite faces of the unit cube, and a generalization of the Lebesgue Covering Theorem Dybowski, Michał Classical Analysis and ODEs General Topology 54F15 (Primary), 54C05, 54F45, 51M99 (Secondary) We formulate and prove a dimension-theoretic generalization of the Lebesgue Covering Theorem. A generalized $n$-dimensional version of the Steinhaus Chessboard Theorem, recently proved by Turzański and Ziajor, is a simple consequence of this result. Moreover, we study two types of sets associated with a continuous function $g \colon I^n \to \mathbb{R}$. 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 $I^n$, 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 $I^n$. We provide necessary and sufficient conditions for the existence of a continuous function $g \colon I^n \to \mathbb{R}$ in terms of these sets. |
| title | Fibers of continuous functions that connect or separate some opposite faces of the unit cube, and a generalization of the Lebesgue Covering Theorem |
| topic | Classical Analysis and ODEs General Topology 54F15 (Primary), 54C05, 54F45, 51M99 (Secondary) |
| url | https://arxiv.org/abs/2602.13883 |