关于$\mathbb R$ -树的割点序的连续分次性
On the continuous gradability of the cut-point orders of $\mathbb R$-trees
摘要：
$\mathbb R$ -树是一种度量空间树，其中每个点都可以分支。Favre和Jonsson在2004年提出了以下问题：$\mathbb R$ -树的序类是否可以由每个分支都与实区间序同构的事实来表征？在第一部分，我回答这个问题的否定：有一个“分支实树顺序”，这是不是“连续分级”。在第二部分中，我证明了一个分支实树序是连续分次的当且仅当每个良好分层的子树是$\mathbb R$ -分次的。这个链接与集合论是把工作在第三部分回答细化的主要问题，产生几个独立的结果。例如，当$κ\geq \mathfrak c$，有一个分支实树顺序，它不是连续分级的，并满足相应于$κ$ -可分离性的性质。相反地，在$κ$的马丁公理下，这样的树是不存在的。
英文摘要：
An $\mathbb R$-tree is a certain kind of metric space tree in which every point can be branching. Favre and Jonsson posed the following problem in 2004: can the class of orders underlying $\mathbb R$-trees be characterised by the fact that every branch is order-isomorphic to a real interval? In the first part, I answer this question in the negative: there is a 'branchwise-real tree order' which is not 'continuously gradable'. In the second part, I show that a branchwise-real tree order is continuously gradable if and only if every well-stratified subtree is $\mathbb R$-gradable. This link with set theory is put to work in the third part answering refinements of the main question, yielding several independence results. For example, when $κ\geq \mathfrak c$, there is a branchwise-real tree order which is not continuously gradable, and which satisfies a property corresponding to $κ$-separability. Conversely, under Martin's Axiom at $κ$ such a tree does not exist.
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