Probabilistic Powerdomains and Quasi-Continuous Domains - Goubault-Larrecq, Jean (Topology Proceedings, 2021)
The probabilistic powerdomain $V X$ on a space $X$ is the space of all continuous valuations on $X$. We show that, for every quasi-continuous domain $X, V X$ is again a quasi-continuous domain, and that the Scott and weak topologies then agree on $V X$. This also applies to the subspaces of probability and subprobability valuations on $X_1$ in the first case under an assumption of pointedness. We also show that the Scott and weak topologies on $V X$ may differ when $X$ is not quasi-continuous, and we give a simple, compact Hausdorff counterexample.