Tame topology in Hensel minimal structures

TL;DR

This paper explores the topology of Hensel minimal structures on valued fields, establishing key properties of definable functions and sets, including limits, closedness, inequalities, and space embeddings, extending tame geometric behavior.

Contribution

It introduces new topological results for Hensel minimal structures satisfying a definability condition, including limit existence, closedness, inequalities, and space embeddings.

Findings

Existence of limits for definable functions of one variable

A closedness theorem for definable sets

Non-Archimedean Lojasiewicz inequalities

Abstract

We are concerned with topology of Hensel minimal structures on non-trivially valued fields $K$, whose axiomatic theory was introduced in a recent paper by Cluckers-Halupczok-Rideau. We additionally require that every definable subset in the imaginary sort $RV$, binding together the residue field $Kv$ and value group $vK$, be already definable in the plain valued field language. This condition is satisfied by several classical tame structures on Henselian fields, including Henselian fields with analytic structure, V-minimal fields, and polynomially bounded o-minimal structures with a convex subring. In this article, we establish many results concerning definable functions and sets; among others, existence of the limit for definable functions of one variable, a closedness theorem, several non-Archimedean versions of the Lojasiewicz inequalities, an embedding theorem for regular definable spaces, and the definable ultranormality and ultraparacompactness of definable Hausdorff LC-spaces.