Topologies on spaces of valuations: a closeness criterion

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

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