Generalized Metric Spaces: Completion, Topology, and Powerdomains via the Yoneda Embedding

Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere, 1973). Combining Lawvere’s (1973) enriched-categorical and Smyth’s (1988, 1991) topological view on generalized metric spaces, it is shown how to construct (1) completion, (2) two topologies, and (3) powerdomains for generalized metric spaces. Restricted to the special cases of preorders and ordinary metric spaces, these constructions yield, respectively: (1) chain completion and Cauchy completion; (2) the Alexandroff and the Scott topology, and the s-ball topology; (3) lower, upper, and convex powerdomains, and the hyperspace of compact subsets. All constructions are formulated in terms of (a metric version of) the Yoneda (1954) embedding.