A family of $\omega_1$ many topological types of locally finite trees

TL;DR

This paper constructs explicitly a vast collection of uncountably many topological types of locally finite trees, strengthening previous non-constructive results and applying to both rooted and free trees of degree two.

Contribution

It provides a constructive proof of uncountably many topological types of locally finite trees within ZFC, extending Matthiesen's non-constructive result.

Findings

Explicit construction of !1 many topological types

Strengthens Matthiesen's result to free trees of degree two

Operates solely within ZFC without additional assumptions

Abstract

Two rooted locally finite trees are considered equivalent if both can be embedded into each other as topological minors by means of tree-order preserving mappings. By exploiting Nash-William's Theorem, Matthiesen provided a non-constructive proof of the uncountability of such equivalence classes, thus answering a question of van der Holst. As an open problem, Matthiesen asks for a constructive proof of this fact. The purpose of this paper is to provide one such construction; working solely within ZFC we illustrate a collection of !1 many topological types of rooted trees. In particular, we also show that this construction strengthens that of Matthiesen in that it also applies to free (unrooted) trees of degree two.