A closedness theorem over Henselian valued fields with analytic structure

TL;DR

This paper proves a closedness theorem for Henselian valued fields with analytic structure, ensuring certain projections are definably closed maps, and applies this to results on limits and piecewise continuity.

Contribution

It establishes a closedness theorem over Henselian valued fields with analytic structure, extending previous results to a broader class of valued fields.

Findings

Projections with projective fibers are definably closed maps.

The theorem applies to classical and Tate algebra valued fields.

Results include theorems on limits and piecewise continuity.

Abstract

The main purpose of the paper is to establish a closedness theorem over Henselian valued fields $K$ of equicharacteristic zero (not necessarily algebraically closed) with separated analytic structure. It says that every projection with a projective fiber is a definably closed map. This remains valid also for valued fields with analytic structure induced by a strictly convergent Weierstrass systems, including the classical, complete rank one valued fields with the Tate algebra of strictly convergent power series. As application, we prove two theorems on existence of the limit and on piecewise continuity.