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

Michał Dybowski

TL;DR

This paper develops a dimension-theoretic generalization of the Lebesgue Covering Theorem, connecting dimension, connectivity, and separation properties for families of subsets of the unit cube. It introduces and analyzes the fiber-connection sets $ extsf{Conn}_i(g)$ and $ extsf{Sep}_i(g)$ for continuous maps $g:I^n\to\mathbb{R}$, providing necessary and sufficient conditions for realizing prescribed data $igl(A_i,B_iigr)$ as these fiber-connection/separation sets. A parametric extension of the Poincaré–Miranda theorem is proven, establishing the existence of connected zero-sets with boundary behavior, and the paper proves that at least one fiber must connect or separate opposite faces. The results generalize the Steinhaus Chessboard Theorem to higher dimension and offer a complete characterization (for $n\ge2$) of when a continuous function can be chosen to realize given Conn and Sep data, with a remark on higher-dimensional targets indicating the potential for arbitrary compact sets to arise as Conn or Sep. These findings deepen the understanding of how dimension, connectivity, and separation interact in fiber structures over the unit cube, with implications for dimension theory and related combinatorial-topological results.

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 Turzański 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.

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 develops a dimension-theoretic generalization of the Lebesgue Covering Theorem, connecting dimension, connectivity, and separation properties for families of subsets of the unit cube. It introduces and analyzes the fiber-connection sets and for continuous maps , providing necessary and sufficient conditions for realizing prescribed data as these fiber-connection/separation sets. A parametric extension of the Poincaré–Miranda theorem is proven, establishing the existence of connected zero-sets with boundary behavior, and the paper proves that at least one fiber must connect or separate opposite faces. The results generalize the Steinhaus Chessboard Theorem to higher dimension and offer a complete characterization (for ) of when a continuous function can be chosen to realize given Conn and Sep data, with a remark on higher-dimensional targets indicating the potential for arbitrary compact sets to arise as Conn or Sep. These findings deepen the understanding of how dimension, connectivity, and separation interact in fiber structures over the unit cube, with implications for dimension theory and related combinatorial-topological results.

Abstract

We formulate and prove a dimension-theoretic generalization of the Lebesgue Covering Theorem. A generalized -dimensional version of the Steinhaus Chessboard Theorem, recently proved by Turzański and Ziajor, is a simple consequence of this result. Moreover, we study two types of sets associated with a continuous function . Namely, the set of all points such that the fiber connects th opposite faces of , and the set of all points such that the fiber separates th opposite faces of . We provide necessary and sufficient conditions for the existence of a continuous function in terms of these sets.
Paper Structure (11 sections, 65 theorems, 14 equations)

This paper contains 11 sections, 65 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subsets that connect or separate some opposite faces of the unit cube
  4. The Lebesgue covering theorem and generalized Steinhaus chessboard theorem
  5. Fibers of continuous functions that connect or separate some opposite faces of the unit cube
  6. Basic definitions and observations
  7. Parametric extension of the Poincaré-Miranda Theorem
  8. There exists some fiber that connects or separates $i$th opposite faces
  9. The sets $\textsf{Conn}_{i}(g)$ and $\textsf{Sep}_{i}(g)$ are compact and connected
  10. Characterization of the existence of a continuous function $g$ with given, for all $i$, sets $\textsf{Conn}_i(g)$ and $\textsf{Sep}_i(g)$
  11. A remark on the $m$-dimensional case

Key Result

Theorem 1.1

Let$I^n_{i ,-} = \left\{z \in I^n \colon\, z_i = 0\right\}$, $I^n_{i ,+} = \left\{z \in I^n \colon\, z_i = 1\right\}$.$n \in \mathbb{N}$ and $\left\{F_i\right\}_{i=1}^n$ be a family of closed sets that covers $I^n$. Then, for some $i \in [n]$, there exists a connected component of $F_i$ that interse

Theorems & Definitions (124)

  • Theorem 1.1: Lebesgue Covering Theorem, karasev
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: engelkingdimension
  • Lemma 2.2: engelkingdimension
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 114 more