Smooth fans that are endpoint rigid

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 $h:X\to X$, 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 Erdős space, then $X$ is necessarily homeomorphic to the Lelek fan; this adds to a 1989 result by Włodzimierz Charatonik.

15 pages, 4 figures