Probabilistic Powerdomains and Quasi-Continuous Domains

The probabilistic powerdomain $\mathbf V X$ on a space $X$ is the space of all continuous valuations on $X$. We show that, for every quasi-continuous domain $X$, $\mathbf V X$ is again a quasi-continuous domain, and that the Scott and weak topologies then agree on $\mathbf V X$. This also applies to the subspaces of probability and subprobability valuations on $X$. We also show that the Scott and weak topologies on the $\mathbf V X$ may differ when $X$ is not quasi-continuous, and we give a simple, compact Hausdorff counterexample.