On closed definable subsets in Hensel minimal structures

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 Kⁿ. Every such subset A is the zero locus of a continuous 0-definable function f:Kⁿ → K, and there exists a 0-definable retraction r: Kⁿ → 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.