Smooth fans that are endpoint rigid
Rodrigo Hernández-Gutiérrez, Logan C. Hoehn
TL;DR
This work investigates endpoint rigidity phenomena in smooth fans, showing that one can construct endpoint-rigid smooth fans with endpoint sets $E(X)$ homeomorphic to $ω$, the irrationals, or $C×ω$, while establishing a sharp obstruction: if $E(X)$ is homeomorphic to complete Erdős space then the fan must be the Lelek fan. Central to the approach is representing smooth fans via upper semicontinuous functions $\varphi: C\to[0,1]$ through the constructions $G_0^\varphi$ and $L_0^\varphi$, yielding $E(X)\simeq G_0^\varphi$ and enabling careful control of endpoint behavior. The paper also proves a Lelek-fan characterization: the Lelek fan is the only smooth fan whose endpoint set is complete Erdős space, giving several equivalent formulations involving density of $E(X)$, connectedness with the vertex, and the behavior of confluent and monotone images. Together, these results deepen understanding of rigidity and endpoint topology in continua and connect USC-function representations to concrete endpoint structures.
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 Erdős space, then $X$ is necessarily homeomorphic to the Lelek fan; this adds to a 1989 result by Włodzimierz Charatonik.