Computational Models of Certain Hyperspaces of Quasi-metric Spaces

TL;DR

This paper develops domain-theoretic models for hyperspaces of nonempty compact subsets of quasi-metric spaces, establishing embeddings and isometries that serve as computational models under various topologies.

Contribution

It introduces an $ ext{omega}$-computational model for hyperspaces of quasi-metric spaces using domain theory, embedding maps, and algebraic structures, extending prior models to quasi-metric contexts.

Findings

Established an embedding of K_0(X) into CBX with Vietoris topology

Proved CBX is an $ ext{omega}$-computational model for K_0(X)

Constructed an algebraic quasi-metric D making CBX an isometric model

Abstract

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 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 $\phi:K_0(X)\rightarrow CBX$ is established and proved that it is an embedding whenever K_0(X) is equipped with the Vietoris topology and respectively CBX 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 (CBX,\sqsubseteq,\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 CBX$ is introduced in such a way that the specialization order $\sqsubseteq_D$ is equivalent to the usual partial order of CBX and, furthermore, $\phi:({\cal K}_0(X),H_d)\rightarrow({\bf C}{\bf B}X,D)$ is an isometry. This shows that (CBX,\sqsubseteq,\phi,D) is a quantitative $\omega$-computational model for (K_(X),H_d).