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 $\Gamma$ 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 $\Gamma_\infty$.