Endpoints of smooth plane dendroids

TL;DR

This paper investigates the topological properties of endpoints in smooth dendroids in the plane, showing their accessibility, characterizing the endpoint space, and exploring conditions under which the dendroid contains specific substructures.

Contribution

It establishes new properties of endpoint spaces in smooth planar dendroids, introduces Bellamy dendroids, and provides examples illustrating the limits of these properties.

Findings

Endpoints are arcwise accessible from outside the dendroid.

The endpoint space has the topological structure of a circle.

If the endpoint space is 1-dimensional and connected, the dendroid contains a Bellamy dendroid or a Cantor set of arcs.

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.