The converse to Borsuk's result on fans fails

TL;DR

This paper demonstrates that the converse of Borsuk's theorem on fans does not hold by presenting a more general result, expanding understanding of continuum structures.

Contribution

It provides a counterexample to the converse of Borsuk's result, showing that not all 1-dimensional continua formed as unions of arcs are fans.

Findings

Counterexample to the converse of Borsuk's theorem

Shows existence of continua not fitting the fan structure

Expands the classification of 1-dimensional continua

Abstract

A fan is an arc-wise connected hereditarily unicoherent continuum with exactly one branching point. By a result of Borsuk, every fan is a 1-dimensional continuum that can be expressed as the union of a family of arcs, each pair of which intersects in the branching point. In this paper, we prove that the converse does not hold by providing a more general result.