There are exactly $\omega _1$ topological types of locally finite trees with countably many rays
Volume 256 / 2022
Fundamenta Mathematicae 256 (2022), 243-259
MSC: 05C05, 05C63, 06A06, 03E05, 54A35.
DOI: 10.4064/fm54-4-2021
Published online: 6 October 2021
Abstract
Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a well-quasi-order. As a consequence, Matthiesen proved that the number $\lambda $ of topological types of locally finite tree must be uncountable. Since $\aleph _1 \leq \lambda \leq \mathfrak {c}$, finding the exact value of $\lambda $ becomes non-trivial in the absence of the Continuum Hypothesis. In this paper we address this task by showing that $\lambda = \aleph _1$ for locally finite trees with countably many rays. We also partially extend this result to locally finite trees with uncountably many rays.
