We construct a nontrivial definable type V field topology on any dp-minimal field [Formula: see text] that is not strongly minimal, and prove that definable subsets of [Formula: see text] have small boundary. Using this topology and its properties, we show that in any dp-minimal field [Formula: see text], dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and [Formula: see text] is finite for all [Formula: see text]. Additionally, by combining the existence of the topology with results of Jahnke, Simon and Walsberg [Dp-minimal valued fields, J. Symbolic Logic 82(1) (2017) 151–165], it follows that dp-minimal fields that are neither algebraically closed nor real closed admit nontrivial definable Henselian valuations. These results are a key stepping stone toward the classification of dp-minimal fields in [Fun with fields, Ph.D. thesis, University of California, Berkeley (2016)].

中文翻译：

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dp-minimal 域上的规范拓扑

我们在任何非强最小的 dp-minimal 域 [Formula: see text] 上构造一个非平凡的可定义类型 V 域拓扑，并证明 [Formula: see text] 的可定义子集具有小边界。使用这种拓扑及其性质，我们表明在任何 dp-minimal 域 [公式：参见文本]，可定义集合的 dp-rank 在族中可定义地变化，完整类型的 dp-rank 以代数闭包为特征，并且 [公式：见正文]对所有[公式：见正文]都是有限的。此外，通过将拓扑的存在与 Jahnke、Simon 和 Walsberg [Dp-minimal valued fields, J. Symbolic Logic 82(1) (2017) 151-165] 的结果相结合，可以得出 dp-minimal fields 既不是代数封闭或实封闭承认非平凡的可定义亨斯估值。这些结果是 [Fun with fields, Ph.D. 中 dp 最小字段分类的关键垫脚石。论文，加州大学伯克利分校 (2016)]。