Universal Chord Theorem and a Topological Analysis

TL;DR

This paper explores the topological properties of chords of continuous functions on [0,1], characterizing which chords can be isolated points and analyzing maximal Hopf sets.

Contribution

It introduces a topological framework for understanding the set of chords of continuous functions, including characterization and analysis of isolated chords and Hopf sets.

Findings

Characterization of which chords can be isolated points

Introduction and analysis of maximal Hopf sets

Examples illustrating the topological properties of chords

Abstract

We study the set of chords of a real-valued continuous function on [0,1] with f(0)=f(1)=0. We describe which chords may appear as isolated points and provide examples illustrating our characterization. Maximal Hopf sets are introduced and analyzed.