For metrically generated constructs X we give an internal characterization of the regular closure operator on X, determined by the subconstruct X 0 , consisting of its T 0 objects. This allows us to describe the epimorphisms in X 0 and to show that all the constructs of that type are cowellpowered. We capture many known results but our method also gives solutions in cases where the epimorphism problem was still open.

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.

In a topological construct X endowed with a proper (E, M)-factorization system and a concrete functor : X → Prap, we study F -compactness and FHausdorff separation, where F is a class of "closed morphisms" in the sense of Clementino et al. (A functional approach to general topology. In: Categorical by . In particular, we point out under which conditions on , the notion of F -compactness of an object X of X coincides with 0-compactness of the image (X) in Prap. Our results will be illustrated by some examples: except for some well-known ones, like b -compactness of a topological space, we also capture some compactness notions that were not considered before in the literature. In particular, we obtain a generalization of bcompactness to the setting of approach spaces. This notion is shown to play an important role in the study of uniformizability.

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