# Locally finite trees and the topological minor relation

## TL;DR

This paper investigates the diversity of topological types of locally finite trees and the sizes of their equivalence classes under the topological minor relation, providing results within ZFC that clarify these set-theoretic questions.

## Contribution

It proves that the number of topological types of locally finite trees is always  within ZFC and characterizes the possible sizes of equivalence classes.

## Key findings

- Number of topological types of locally finite trees is 
- Equivalence class sizes are either 1 or 
- Results hold within ZFC, independent of the Continuum Hypothesis

## Abstract

A well-known theorem of Nash-Williams shows that the collection of locally finite trees under the topological minor relation results in a BQO. Set theoretically, two very natural questions arise: (1) What is the number $\lambda$ of topological types of locally finite trees? (2) What are the possible sizes of an equivalence class of locally finite trees? For (1), clearly, $\omega \leq \lambda \leq \mathfrak{c}$ and Matthiesen refined it to $\omega_1 \leq \lambda \leq \mathfrak{c}$. Thus, this question becomes non-trivial when the Continuum Hypothesis is not assumed. In this paper we address both questions by showing that - entirely within ZFC - for a large collection of locally finite trees that includes those with countably many rays: the answer for (1) is $\lambda = \omega_1$, and that for (2) the size of an equivalence class can only be either $1$ or $\mathfrak{c}$.