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.