Smooth fans that are endpoint rigid

TL;DR

This paper constructs smooth fans with prescribed endpoint sets where all homeomorphisms fix the endpoints, and characterizes fans with Erdős space endpoints as the Lelek fan, extending previous results.

Contribution

It introduces new examples of smooth fans with endpoint rigidity and characterizes fans with Erdős space endpoints as the Lelek fan.

Findings

Constructed smooth fans with endpoint sets homeomorphic to natural, irrational, or Cantor product spaces with endpoint rigidity.

Proved that fans with Erdős space endpoints are necessarily Lelek fans.

Extended Charatonik's 1989 result on the uniqueness of the Lelek fan.

Abstract

Let $X$ be a smooth fan and denote its set of endpoints by $E(X)$. Let $E$ be one of the following spaces: the natural numbers, the irrational numbers, or the product of the Cantor set with the natural numbers. We prove that there is a smooth fan $X$ such that $E(X)$ is homeomorphic to $E$ and for every homeomorphism $h \colon X \to X$, the restriction of $h$ to $E(X)$ is the identity. On the other hand, we also prove that if $X$ is any smooth fan such that $E(X)$ is homeomorphic to complete Erd\H{o}s space, then $X$ is necessarily homeomorphic to the Lelek fan; this adds to a 1989 result by W{\l}odzimierz Charatonik.