Fibers of continuous functions that connect or separate some opposite faces of the unit cube, and a generalization of the Lebesgue Covering Theorem

TL;DR

This paper generalizes the Lebesgue Covering Theorem using dimension theory, explores fibers of continuous functions on the unit cube, and characterizes sets where fibers connect or separate opposite faces.

Contribution

It introduces a dimension-theoretic generalization of the Lebesgue Covering Theorem and characterizes fibers connecting or separating faces of the cube.

Findings

A dimension-theoretic generalization of the Lebesgue Covering Theorem.

Necessary and sufficient conditions for fibers connecting or separating faces.

A generalized version of the Steinhaus Chessboard Theorem as a consequence.

Abstract

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 Turza\'nski 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.