Definable retractions over Henselian valued fields with analytic structure

TL;DR

This paper establishes the existence of definable retractions onto closed sets in Henselian valued fields with analytic structure, leading to non-Archimedean analogs of classical extension theorems, using resolution of singularities and model theory.

Contribution

It proves the existence of definable retractions in Henselian valued fields with analytic structure, extending previous results to a broader class of fields.

Findings

Definable retractions exist onto any closed definable subset.

Non-Archimedean versions of Tietze--Urysohn and Dugundji theorems are established.

The results apply to Henselian fields with separated and strictly convergent analytic structures.

Abstract

Let $K$ be a Henselian, non-trivially valued field with separated analytic structure. We prove the existence of definable retractions onto an arbitrary closed definable subset of $K^{n}$. Hence directly follow definable non-Archimedean versions of the extension theorems by Tietze--Urysohn and Dugundji. This generalizes our previous paper dealing with complete non-Archimedean fields with separated power series and remains true for Henselian valued fields with strictly convergent analytic structure, because every such a structure can be extended in a definitional way to a separated analytic structure. Our proof uses a variant of the one from that paper, based on canonical resolution of singularities, and a model-theoretic compactness argument.