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

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.

References in corpus (4)

• Some results of algebraic geometry over Henselian rank one valued fields
• Some results of geometry over Henselian fields with analytic structure
• Definable transformation to normal crossings over Henselian fields with separated analytic structure
• Definable retractions over complete fields with separated power series

Cited by in corpus (3)

• Definable transformation to normal crossings over Henselian fields with separated analytic structure
• Some results of geometry over Henselian fields with analytic structure
• Definable retractions over complete fields with separated power series