可数树的拓扑等价类的基数三分法

中文摘要 AI 辅助

对于一棵树$T$，设$[T]$为与$T$互为拓扑子式的树的同构类之集。Bruno和Szeptycki证明了每棵局部有限树满足$|[T]|\in\{1,2^{\aleph_0}\}$，并且每棵含有一条包含无穷多个度数至少为$3$的顶点的射线的树，至少有$2^{\aleph_0}$个拓扑孪生树。因此，当树可数时，后一种情形下等式成立。我们处理互补的可数情形，不对度数作任何限制。对于每棵这样的树$T$，我们证明$|[T]|\in\{1,\aleph_0,2^{\aleph_0}\}$，并给出基于Schmidt秩的良基递归描述。证明使用了从分支顶点生成的子树中提取的有限典范子树，以及关于良拟序上可数多重集的计数定理。同样的三分法适用于每棵可数树，从而适用于所有顶点度数均可数的树。

英文摘要

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 $|[T]|\in\{1,2^{\aleph_0}\}$ and that every tree with a ray containing infinitely many vertices of degree at least $3$ has at least $2^{\aleph_0}$ 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 $|[T]|\in\{1,\aleph_0,2^{\aleph_0}\}$ 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.

发表机构

• Jiangsu Normal University（江苏师范大学）

机构由 AI 辅助整理，请以论文原文为准。