Probabilistic Powerdomains and Quasi-Continuous Domains

International audienceThe probabilistic powerdomain VX on a space X is the space of all continuous valuations on X. We show that, for every quasi-continuous domain X, VX is again a quasi-continuous domain, and that the Scott and weak topologies then agree on VX. This also applies to the subspaces of probability and subprobability valuations on X, in the first case under an assumption of pointedness. We also show that the Scott and weak topologies on VX may differ when X is not quasi-continuous, and we give a simple, compact Hausdorff counterexample