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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 Cauchydoi:10.1016/s0304-3975(97)00042-x fatcat:z3xewd6tzzf63nmkgklayqu63u