Kantorovich-Rubinstein quasi-metrics. III: Spaces of sublinear and superlinear previsions (Q2087781)

This paper cogently demonstrates that the Kantorovich-Rubinstein quasi-metrics \(d_{KR}\) and \(d^a_{KR}\) extend naturally to various spaces of previsions. A \textit{quasi-metric} on a nonvoid set \(X\) is a map \(d:X\times X\to \bar{\mathbb{R}}_+\) satisfying \begin{gather*} x,y,z \in X.\\ d(x,x) =0.\\ d(x,y) = d(y,x) = 0\ \text{implies}\ x = y.\\ d(x,z) \leq d(x,y) + d(y,z). \end{gather*} Let \(\mathcal{L}X\) be the space of all lower continuous maps from \(X\) to \(\bar{\mathbb{R}}_+\) with its Scott topology. A \textit{Scott topology} on a poset \(X\), with ordering \(\leq\), has as opens the upward-closed subsets \(U\) so that for every directed family \((x_i)_{i\in I}\) having a least upper bound in inside \(U\) for some \(x_i\), is already in \(U\). A \textit{prevision} on a topological space \(x\) is a Scott-continuous map between posets that is monotonic and preserves existing suprema [\textit{J. Goubault-Larrecq}, Lect. Notes Comput. Sci. 4646, 542--557 (2007; Zbl 1179.68074)]. Among the many results in this paper, the following main result is given. Threorem 5.30. (\(d_{\mathcal H}\) metricizes the lower Vietoris topology). Let \(X,d\) be a continuous complete quasi-metric space. The \(d_{\mathcal H}\)-Scott topology, the \(d^a_{\mathcal H}\)-Scott topology, for every \(a\in \mathbb{R}, a>0\) and the lower Vietoris topology all coincide on \(\mathcal{H}_0 X\), resp. \(\mathcal{H} X\). This paper has 17 references that include [\textit{J. Goubault-Larrecq}, Non-Hausdorff topology and domain theory. Selected topics in point-set topology. Cambridge: Cambridge University Press (2013; Zbl 1280.54002)], which would be of interest for researchers in point-set topology. For Parts I and II see [\textit{J. Goubault-Larrecq}, Topology Appl. 295, Article ID 107673, 37 p. (2021; Zbl 1473.46036) and ibid. 305, Article ID 107885, 38 p. (2022; Zbl 1478.54020)]