  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.
