Quasi-valuations -- topology and the weak approximation theorem

Abstract

Suppose $F$ is a field with a nontrivial valuation $v$ and valuation ring $O_v$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study the topology induced by $w$. We prove that the quasi-valuation ring determines the topology, independent of the choice of its quasi-valuation. Moreover, we prove the weak approximation theorem for quasi-valuations.