Chessboard and level sets of continuous functions
Abstract
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 the $n$-dimensional Steinhaus Chessboard Theorem and the Brouwer Fixed Point Theorem are simple consequences of this result.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.13774
- arXiv:
- arXiv:2406.13774
- Bibcode:
- 2024arXiv240613774D
- Keywords:
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- Mathematics - General Topology;
- Mathematics - Combinatorics;
- 54D05 (Primary) 54C05;
- 05C15;
- 51M99 (Secondary)
- E-Print:
- 20 pages