Endpoints of smooth plane dendroids

Abstract

We show that each endpoint of a smooth plane dendroid $X$ is accessible, and that the endpoint set $E(X)$ is circle-like in that every two of its points are separated by two other points. Also if $E(X)$ is totally disconnected and $1$-dimensional, then $X$ must contain an uncountable collection of pairwise-disjoint arcs. An example is constructed to show that this is false outside the plane.