On the Yoneda completion of a quasi-metric space

Two approaches aimed at reconciling the partial order and the metric space approaches to Domain theory were discussed. A comparison between the Yoneda completion of generalized metric spaces and the Smyth completion was presented. The largest class of quasi-metric spaces idempotent under the Yoneda completion was precisely the class of Smyth-completable spaces. The proof of the conjecture was based on a characterization of Smyth-completability of quasi-metric spaces in terms of sequences. The preservation of the properties of total boundedness, precompactness and compactness was analyzed under the two kinds of completion in the possible absence of idempotency.