Firm completions in metrically generated constructs, with applications to function spaces and sobrification

For metrically generated constructs X 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 an 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-Archimedean 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 Dieudonne 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.