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In this paper, for metrically generated constructs X in the sense of [E. Colebunders, R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (2005) 1547-1556] we study completion as a U -reflector R on the subconstruct X 0 of all T 0 -objects, for U some class of embeddings. Roughly speaking we deal with constructs X that are generated by the subclass of their metrizable objects and for various types of completion functors R available in that context, we obtain internal descriptions of the largest class U for which completion is unique. We apply our results to some well known situations. Completion of uniform spaces, of proximity spaces or of non-Archimedian uniform spaces is unique with respect to the class of all epimorphic embeddings, and this class is the largest one. However the largest class of morphisms for which Dieudonné completion of completely regular spaces or of zero dimensional spaces is unique, is strictly smaller than the class of all epimorphic embeddings. The same is true for completion in quantitative theories like uniform approach spaces for which the largest U coincides with the class of all embeddings that are dense with respect to the metric coreflection. Our results on completion for metrically generated constructs explain these differences.

In [2] the subconstruct Sob of sober approach spaces was introduced and it was shown to be a reflective subconstruct of the category Ap 0 of T 0 approach spaces. The main result of this paper states that moreover Sob is firmly U-reflective in Ap 0 for the class U of epimorphic embeddings. 'Firm U-reflective' is a notion introduced in [3] by G.C.L. Brümmer and E. Giuli and is inspired by the exemplary behaviour of the usual completion in the category Unif 0 of Hausdorff uniform spaces with uniformly continuous maps. It means that Sob is U-reflective in Ap 0 and that the reflector is such that f : X → Y belongs to U if and only if ( f ) is an isomorphism. Firm U-reflectiveness implies uniqueness of completion in the sense that whenever f : X → Y is a map with f ∈ U and Y sober, the associated f * : (X ) → Y is an isomorphism. Our result generalizes the fact that in the category Top 0 the subconstruct of sober topological spaces is firmly reflective for the class U b of b-dense embeddings in Top 0 . Also firmness in some other subconstructs of Ap 0 will be easily obtained.

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In this paper we will study completeness in symmetric metrically generated constructs via nearness spaces. Our approach consists in associating an appropriate regular nearness space with a given symmetric metered space. The completion theory known for regular nearness space has some convenient properties on which our completion of symmetric metered spaces will be based. This technique appears to be suitable for most symmetric metrically generated constructs and leads to a firm completion theory.