# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1901.09922