Endpoints of smooth plane dendroids

Abstract

Let $X$ be a smooth dendroid in the plane $\mathbb R^2$. We show that each endpoint of $X$ is arcwise accessible from $\mathbb R^2\setminus X$, and that the space of endpoints $E(X)$ has the property of a circle. In the event that $E(X)$ is connected, we call $X$ a *Bellamy dendroid*. We prove that if $E(X)$ is 1-dimensional, then $X$ contains a Bellamy dendroid or a Cantor set of arcs. In particular, if $E(X)$ totally disconnected and 1-dimensional, then $X$ is non-Suslinian. An example is constructed to show that this is false outside the plane.