There are exactly ω1 topological types of locally finite trees with countably many rays

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 λ of topological types of locally finite tree must be uncountable. Since ℵ1 ≤ λ ≤ c, finding the exact value of λ becomes non-trivial in the absence of the Continuum Hypothesis. In this paper we address this task by showing that λ = ℵ1 for locally finite trees with countably many rays. We also partially extend this result to locally finite trees with uncountably many rays.