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) en riched-categorical and Smyth's (1988, 1991) topological view on genera lized metric spaces, it is shown how to construct (1) completion, (2) two topologies, and (3) powerdomains for generalized metric spaces. Re stricted to the special cases of preorders and ordinary metric spaces, these constructions yield, respectively: (1) chain completion and Cau chy completion; (2) the Alexandroff and the Scott topology, and the E- ball topology; (3) lower, upper, and convex powerdomains, and the hype rspace of compact subsets. All constructions are formulated in terms o f (a metric version of) the Yoneda (1954) embedding.