Chessboard and level sets of continuous functions
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913816695013376 |
|---|---|
| author | Dybowski, Michał Górka, Przemysław |
| author_facet | Dybowski, Michał Górka, Przemysław |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13774 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chessboard and level sets of continuous functions Dybowski, Michał Górka, Przemysław General Topology Combinatorics 54D05 (Primary) 54C05, 05C15, 51M99 (Secondary) 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. |
| title | Chessboard and level sets of continuous functions |
| topic | General Topology Combinatorics 54D05 (Primary) 54C05, 05C15, 51M99 (Secondary) |
| url | https://arxiv.org/abs/2406.13774 |