Probabilistic Powerdomains and Quasi-Continuous Domains

TL;DR

This paper proves that the probabilistic powerdomain of a quasi-continuous domain retains its quasi-continuity and that Scott and weak topologies coincide, with counterexamples when the domain is not quasi-continuous.

Contribution

It establishes the preservation of quasi-continuity in probabilistic powerdomains and clarifies the topological relationship, extending previous understanding in domain theory.

Findings

Scott and weak topologies agree on $ extbf{V}X$ for quasi-continuous domains

Counterexample shows topologies may differ without quasi-continuity

Results apply to probability and subprobability valuations

Abstract

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.