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It is our main purpose in this paper to approach the quasi-pseudometrization problem in (bi)topological spaces in a way which generalizes all the well-known results on the subject naturally, and which is close to a "Bing-Nagata-Smirnov style" characterization of quasi-pseudometrizability.

Different types of quasi-uniform completeness are shown to coincide in certain uniformly weakly regular spaces. This result generalizes Deák’s result in [4] which answers to the question of Fletcher and Hunsaker in [8].

The different notions of Cauchy sequence and completeness proposed in the literature for quasi-pseudometric spaces do not provide a satisfactory theory of completeness and completion for all quasi-pseudometric spaces. In this paper, we introduce a notion of completeness which is classical in the sense that it is made up of equivalence classes of Cauchy sequences and constructs a completion for any given 0 quasi-pseudometric space. This new completion theory extends the existing completion theory for metric spaces and satisfies the requirements posed by Doitchinov for a nice theory of completeness.

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