Computational Models of Certain Hyperspaces of Quasi-metric Spaces

In this paper, for a given sequentially Yoneda-complete T1 quasi-metric space (X, d), the domain theoretic models of the hyperspace K0(X) of nonempty compact subsets of (X, d) are studied. To this end, the ω-Plotkin domain of the space of formal balls BX, denoted by CBX is considered. This domain is given as the chain completion of the set of all finite subsets of BX with respect to the Egli-Milner relation. Further, a map φ : K0(X) → CBX is established and proved that it is an embedding whenever K0(X) is equipped with the Vietoris topology and respectively CBX with the Scott topology. Moreover, if any compact subset of (X, d) is d-precompact, φ is an embedding with respect to the topology of Hausdorff quasi-metric Hd on K0(X). Therefore, it is concluded that (CBX,⊑, φ) is an ω-computational model for the hyperspace K0(X) endowed with the Vietoris and respectively the Hausdorff topology. Next, an algebraic sequentially Yoneda-complete quasi-metric D on CBX is introduced in such a way that the specialization order ⊑D is equivalent to the usual partial order of CBX and, furthermore, φ : (K0(X),Hd) → (CBX,D) is an isometry. This shows that (CBX,⊑, φ,D) is a quantitative ω-computational model for (K0(X),Hd).