Metrizability of spaces of valuation domains associated to pseudo-convergent sequences

TL;DR

This paper investigates the topological properties, specifically metrizability, of spaces of valuation domains associated with pseudo-convergent sequences over a valuation domain of rank one, using Zariski and constructible topologies.

Contribution

It provides necessary conditions for the Zariski space to be metrizable, linking topological properties with algebraic invariants like the value group and residue field.

Findings

Necessary conditions for metrizability under the constructible topology.

Analysis of subspaces with prescribed pseudo-limits or breadth.

Topological characterization of valuation domain spaces.

Abstract

Let $V$ be a valuation domain of rank one with quotient field $K$. We study the set of extensions of $V$ to the field of rational functions $K(X)$ induced by pseudo-convergent sequences of $K$ from a topological point of view, endowing this set either with the Zariski or with the constructible topology. In particular, we consider the two subspaces induced by sequences with a prescribed breadth or with a prescribed pseudo-limit. We give some necessary conditions for the Zariski space to be metrizable (under the constructible topology) in terms of the value group and the residue field of $V$.