COMPUTATIONAL MODELS OF CERTAIN HYPERSPACES OF QUASI-METRIC SPACES

In this paper, for a given sequentially Yoneda-complete T-1 quasi-metric space (X, d), the domain theoretic models of the hyperspace K-0(X) of nonempty compact subsets of (X, d) are studied. To this end, the omega-Plotkin domain of the space of formal balls B X, denoted by CB X is considered. This domain is given as the chain completion of the set of all finite subsets of B X with respect to the Egli-Milner relation. Further, a map phi : K-0(X) -> CB X is established and proved that it is an embedding whenever K-0(X) is equipped with the Vietoris topology and respectively CB X with the Scott topology. Moreover, if any compact subset of (X, d) is d(-1)-precompact, phi is an embedding with respect to the topology of Hausdorff quasi-metric H-d on K-0(X). Therefore, it is concluded that (CB X, subset of, phi) is an omega-computational model for the hyperspace K-0(X) endowed with the Vietoris and respectively the Hausdorff topology. Next, an algebraic sequentially Yoneda-complete quasi-metric D on CB X is introduced in such a way that the specialization order subset of(D) is equivalent to the usual partial order of CB X and, furthermore, phi : (K-0(X), H-d) -> (CB X, D) is an isometry. This shows that (CB X, subset of, phi, D) is a quantitative omega-computational model for (K-0(X), H-d).