Topologies on spaces of valuations: a closeness criterion

TL;DR

This paper investigates how topologies on valuation spaces relate to the underlying ordered abelian group and integral domain, providing criteria for closure properties and effects on natural topologies.

Contribution

It introduces a criterion linking topologies on the value group and valuation space, enhancing understanding of their topological structure.

Findings

Provided a criterion for the valuation space to be closed in the product topology.

Analyzed the impact of this criterion on natural topologies on the value group.

Explored the relationship between topologies on the value group and the valuation space.

Abstract

This paper is part of a program to understand topologies on spaces of valuations. We fix an ordered abelian group $\Gamma$ and an integral domain $R$. We study the relation between a topology on $\Gamma_\infty$ and the induced topology on the set $\mathcal W_\Gamma$ of all valuations of $R$ taking values in $\Gamma_\infty$. For instance, we give a criterion for $ \mathcal W_\Gamma$ to be closed in $\left(\Gamma_\infty\right)^R$. We also discuss the effect of this criterion for natural topologies on $\Gamma_\infty$.