Definable retractions and a non-Archimedean Tietze--Urysohn theorem over Henselian valued fields

TL;DR

This paper establishes the existence of definable retractions in Henselian valued fields, leading to non-Archimedean analogues of classical theorems on extending continuous functions, using tools like resolution of singularities.

Contribution

It introduces a method to construct definable retractions onto closed sets in Henselian valued fields, enabling non-Archimedean versions of the Tietze--Urysohn and Dugundji theorems.

Findings

Existence of definable retractions onto closed subsets in Henselian valued fields.

Non-Archimedean analogues of classical extension theorems established.

Utilization of resolution of singularities and closedness theorem in proofs.

Abstract

We prove the existence of definable retractions onto arbitrary closed subsets of $K^{n}$ definable over Henselian valued fields $K$. Hence directly follows non-Archimedian analogues of the Tietze--Urysohn and Dugundji theorems on extending continuous definable functions. The main ingredients of the proof are a description of definable sets due to van den Dries, resolution of singularities and our closedness theorem.