English

Endpoints of smooth plane dendroids

General Topology 2024-08-27 v3

Abstract

Let XX be a smooth dendroid in the plane R2\mathbb R^2. We show that each endpoint of XX is arcwise accessible from R2X\mathbb R^2\setminus X, and that the space of endpoints E(X)E(X) has the property of a circle. In the event that E(X)E(X) is connected, we call XX a *Bellamy dendroid*. We prove that if E(X)E(X) is 1-dimensional, then XX contains a Bellamy dendroid or a Cantor set of arcs. In particular, if E(X)E(X) totally disconnected and 1-dimensional, then XX is non-Suslinian. An example is constructed to show that this is false outside the plane.

Keywords

Cite

@article{arxiv.2405.01706,
  title  = {Endpoints of smooth plane dendroids},
  author = {David S. Lipham},
  journal= {arXiv preprint arXiv:2405.01706},
  year   = {2024}
}

Comments

12 pages, 3 figures

R2 v1 2026-06-28T16:14:51.368Z