Computational Models of Certain Hyperspaces of Quasi-metric Spaces

In this paper, for a given sequentially Yoneda-complete T_1 quasi-metricspace (X,d), the domain theoretic models of the hyperspace K_0(X) of nonemptycompact subsets of (X,d) are studied. To this end, the $\omega$-Plotkin domainof the space of formal balls BX, denoted by CBX is considered. This domain isgiven as the chain completion of the set of all finite subsets of BX withrespect to the Egli-Milner relation. Further, a map $\phi:K_0(X)\rightarrowCBX$ is established and proved that it is an embedding whenever K_0(X) isequipped with the Vietoris topology and respectively CBX with the Scotttopology. Moreover, if any compact subset of (X,d) is d^{-1}-precompact, \phiis an embedding with respect to the topology of Hausdorff quasi-metric H_d onK_0(X). Therefore, it is concluded that (CBX,\sqsubseteq,\phi) is an$\omega$-computational model for the hyperspace K_0(X) endowed with theVietoris and respectively the Hausdorff topology. Next, an algebraicsequentially Yoneda-complete quasi-metric D on CBX$ is introduced in such a waythat the specialization order $\sqsubseteq_D$ is equivalent to the usualpartial order of CBX and, furthermore, $\phi:({\calK}_0(X),H_d)\rightarrow({\bf C}{\bf B}X,D)$ is an isometry. This shows that(CBX,\sqsubseteq,\phi,D) is a quantitative $\omega$-computational model for(K_(X),H_d).Comment: 25 page