The converse to Borsuk's result on fans fails

Published: April 14, 2026PDF arXiv

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.

The converse to Borsuk's result on fans fails

Benjamin Vejnar

math.GN

Published: April 14, 2026PDF arXiv

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.