Chessboard and level sets of continuous functions

2024/06/19 by Dybowski, Michał, Górka, Przemysław
#05C15 #51M99 (Secondary) #54D05 (Primary) 54C05 #Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2406.13774

Abstract

We provide the following result and its discrete equivalent: Let f \colon Iⁿ → ℝ^n-1 be a continuous function. Then, there exist a point p ∈ ℝ^n-1 and a compact subset S ⊂ f^-1[\p\] which connects some opposite faces of the n-dimensional unit cube Iⁿ. 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.

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