Spaces of valuations as quasimetric domains

Abstract

We define a natural quasimetric on the set of continuous valuations of a topological space and investigate it in the spirit of quasimetric domain theory. It turns out that the space of valuations of an (ordinary) algebraic domain D is an algebraic quasimetric domain. Moreover, it is precisely the lower powerdomain of D, where D is regarded as a quasimetric domain. The essential tool for proving these results is a generalization of the Splitting Lemma which characterizes the quasimetric for simple valuations and holds for valuations on arbitrary topological spaces.