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## Fundamenta Mathematicae

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## There are exactly $\omega _1$ topological types of locally finite trees with countably many rays

### Tom 256 / 2022

Fundamenta Mathematicae 256 (2022), 243-259 MSC: 05C05, 05C63, 06A06, 03E05, 54A35. DOI: 10.4064/fm54-4-2021 Opublikowany online: 6 October 2021

#### Streszczenie

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.

#### Autorzy

• Jorge BrunoDepartment of Digital Technologies
University of Winchester
Winchester, UK
e-mail
• Paul SzeptyckiDepartment of Mathematics and Statistics
York University