A cardinal trichotomy for topological equivalence classes of countable trees
Abstract
For a tree T, let [T] be the set of isomorphism classes of trees that are mutually topological minors of T. Bruno and Szeptycki proved that every locally finite tree satisfies |[T]|∈{1,2ℵ0} and that every tree with a ray containing infinitely many vertices of degree at least $3$ has at least 2ℵ0 topological twins. Hence equality holds in the latter case when the tree is countable. We treat the complementary countable case, without any bound on the degrees. For every such tree T, we prove that |[T]|∈{1,ℵ0,2ℵ0} and give a well-founded recursive description based on Schmidt rank. The proof uses a finite canonical subtree extracted from the subtree generated by the branching vertices and a counting theorem for countable multisets over a well-quasi-order. The same trichotomy follows for every countable tree and hence for every tree all of whose vertices have countable degree.
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BibTeX
@misc{wu2026cardinal,
title = {{A cardinal trichotomy for topological equivalence classes of countable trees}},
author = {Qi Wu and Yong Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27872},
eprint = {2609.27872},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27872},
note = {CC0},
}
RIS
TY - UNPB
TI - A cardinal trichotomy for topological equivalence classes of countable trees
AU - Qi Wu
AU - Yong Lu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27872
UR - https://arxiv.org/abs/2609.27872
AB - For a tree T, let [T] be the set of isomorphism classes of trees that are mutually topological minors of T. Bruno and Szeptycki proved that every locally finite tree satisfies |[T]|∈{1,2^ℵ_0} and that every tree with a ray containing infinitely many vertices of degree at least $3$ has at least 2^ℵ_0 topological twins. Hence equality holds in the latter case when the tree is countable. We treat the complementary countable case, without any bound on the degrees. For every such tree T, we prove that |[T]|∈{1,ℵ_0,2^ℵ_0} and give a well-founded recursive description based on Schmidt rank. The proof uses a finite canonical subtree extracted from the subtree generated by the branching vertices and a counting theorem for countable multisets over a well-quasi-order. The same trichotomy follows for every countable tree and hence for every tree all of whose vertices have countable degree.
ER -