Chessboard and level sets of continuous functions

TL;DR

This paper proves a topological result about continuous functions on cubes, showing the existence of certain connected sets, and demonstrates how classical theorems like Brouwer's follow from it.

Contribution

It introduces a new topological theorem linking level sets of continuous functions to connected subsets connecting opposite faces of a cube, with implications for classical fixed point theorems.

Findings

Existence of a connected set in the preimage of a point connecting opposite faces

Discrete analogue of the continuous result

Derivation of Steinhaus Chessboard and Brouwer Fixed Point Theorems from the main result

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.