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 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).

💡 Research Summary

The paper investigates domain‑theoretic computational models for the hyperspace K₀(X) consisting of all non‑empty compact subsets of a given sequentially Yoneda‑complete T₁ quasi‑metric space (X,d). The authors introduce the ω‑Plotkin domain CBX, defined as the chain completion of the set of all finite subsets of the formal ball space BX with respect to the Egli‑Milner relation. BX itself consists of pairs (x,r) with x∈X and r≥0, ordered by (x,r)⊑(y,s) iff d(x,y)≤r−s; this order makes BX a directed complete poset precisely when (X,d) satisfies the Yoneda completeness condition.

The construction proceeds by taking the abstract basis (P_fin BX,≺_EM) where ≺_EM is the Egli‑Milner relation on finite subsets, and then forming its chain completion, yielding a continuous ω‑dcpo CBX in which every ω‑chain has a least upper bound. Elements of CBX can be viewed as round ideals generated by increasing chains of finite subsets of formal balls.

A central contribution is the definition of a map φ:K₀(X)→CBX given by φ(K)=⋂_{(x,r)∈K}↑↑(x,r), where ↑↑ denotes the upward closure under ≺_EM. The map is shown to be injective, and when K₀(X) carries the Vietoris topology τ_V while CBX carries the Scott topology σ, φ becomes a topological embedding. Moreover, φ preserves maximal elements, satisfying the maximal‑point condition required for a computational model.

If every compact subset of X is d⁻¹‑precompact, the Hausdorff quasi‑metric H_d on K₀(X) is a T₁ quasi‑metric, and φ is also an embedding with respect to the topology induced by H_d. Consequently (CBX,⊑,φ) serves as an ω‑computational model for K₀(X) equipped either with the Vietoris topology or with the Hausdorff topology.

To obtain a quantitative model, the authors adopt the quasi‑metric q on BX introduced by Romaguera and Valero: q((x,r),(y,s))=max{d(x,y), r−s}. This metric makes (BX,q) a quantitative computational model for (X,d). They extend q to a Hausdorff quasi‑metric H_q on finite subsets of BX, which induces the Egli‑Milner order on P_fin BX. Lifting H_q to CBX yields a quasi‑metric D on CBX such that the specialization order ⊑_D coincides with the original order ⊑. D is shown to be Yoneda‑complete and, in fact, the Yoneda completion of (P_fin BX, H_q). With this construction, φ becomes an isometry from (K₀(X),H_d) into (CBX,D). Hence (CBX,⊑,φ,D) constitutes a quantitative ω‑computational model for the hyperspace equipped with the Hausdorff quasi‑metric.

In the final section the paper compares the classical Plotkin powerdomain PBX with the newly introduced ω‑Plotkin domain CBX. It is proved that if (X,d) is Smyth‑complete and every compact subset is d⁻¹‑precompact, or if (X,d) is ω‑algebraic Yoneda‑complete, then PBX and CBX are order‑isomorphic. This demonstrates that the ω‑Plotkin construction generalizes the traditional powerdomain approach and works in settings where symmetry (required in metric spaces) is absent.

Overall, the work extends domain‑theoretic techniques to non‑symmetric distance spaces, providing both topological and quantitative computational models for hyperspaces of compact sets. It bridges the gap between quasi‑metric topology and domain theory, showing that even without metric symmetry one can obtain robust representations of hyperspaces suitable for applications in denotational semantics, complexity analysis, and other areas where quasi‑metrics naturally arise.