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On closed definable subsets in Hensel minimal structures

Krzysztof Jan Nowak

Abstract

This paper deals with Hensel minimal structures on non-trivially valued fields $K$. The main aim is to establish the following two properties of closed 0-definable subsets $A$ in the affine spaces $K^{n}$. Every such subset $A$ is the zero locus of a continuous 0-definable function $f:K^{n} \to K$, and there exists a 0-definable retraction $r: K^{n} \to A$. While the former property is a non-Archimedean counterpart of the one from o-minimal geometry, the former does not hold in real geometry in general. The proofs make use of a model-theoretic compactness argument and ubiquity of clopen sets in non-Archimedean geometry.

On closed definable subsets in Hensel minimal structures

Abstract

This paper deals with Hensel minimal structures on non-trivially valued fields . The main aim is to establish the following two properties of closed 0-definable subsets in the affine spaces . Every such subset is the zero locus of a continuous 0-definable function , and there exists a 0-definable retraction . While the former property is a non-Archimedean counterpart of the one from o-minimal geometry, the former does not hold in real geometry in general. The proofs make use of a model-theoretic compactness argument and ubiquity of clopen sets in non-Archimedean geometry.
Paper Structure (2 sections, 6 theorems, 55 equations)

This paper contains 2 sections, 6 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Proofs of the main results

Key Result

Theorem 1.1

Every closed 0-definable subset $A$ of $K^n$ is the zero locus $\mathcal{Z}(g) := \{ x \in K^{n}: \ g(x)=0 \}$ of a continuous 0-definable function $g$ on $K^n$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 2 more