O-极小结构中Lipschitz胞腔的Hausdorff极限的可定义性
Definability of Hausdorff Limits for Lipschitz Cells in O-minimal Structures
浏览论文内容
中文总结 AI 辅助
本文研究实闭域o-极小扩张模型上可定义族的Hausdorff极限可定义性，针对固定表示和一致Lipschitz界的Lipschitz胞腔族，证明其Hausdorff极限在稠密完备化中可定义且构成可定义族，给出Marker--Steinhorn定理的非阿基米德类比。
中文摘要 AI 辅助
我们研究实闭域的o-极小扩张的任意模型上可定义族的Hausdorff极限。在实域上，van den Dries证明了可定义族的Hausdorff极限是可定义的，这给出了Marker--Steinhorn定理的几何解释。我们针对具有固定胞腔表示和一致Lipschitz界的Lipschitz胞腔构成的可定义族，证明了一个非阿基米德类比结果。该证明将$\boldsymbol{\text{R}}^n$中闭有界子集的紧性替换为稠密完备化和长柯西序列，并将Hausdorff距离视为取值于有序完备化的度量。我们证明这类族的Hausdorff极限是驯服扩张上外部纤维的标准部分，且利用驯服对的稳定嵌入性证明每个此类极限在基模型的稠密完备化中是可定义的。我们还证明了一个一致版本：这类Hausdorff极限的集合在稠密完备化中构成一个可定义族。
英文摘要
We study Hausdorff limits of definable families over arbitrary models of o-minimal expansions of real closed fields. Over the real field, van den Dries proved that Hausdorff limits of definable families are definable, giving a geometric interpretation of the Marker--Steinhorn theorem. We prove a non-Archimedean analogue for definable families which are Lipschitz cells with a fixed cell presentation and a uniform Lipschitz bound. The proof replaces compactness of closed and bounded subsets of $\mathbb R^n$ by dense completions and long Cauchy sequences, and treats the Hausdorff distance as a metric valued in an ordered completion. We show that Hausdorff limits of such families are standard parts of external fibers over tame extensions, and use stable embeddedness of tame pairs to prove that every such limit is definable in the dense completion of the base model. We also prove a uniform version: the collection of these Hausdorff limits forms a definable family in the dense completion.
