Abstract. Many examples are known of natural functors K describing the transition from categories C of generalized metric spaces to the "metrizable" objects in some given topological construct X . If K preserves initial morphisms and if K(C) is initially dense in X , then we say that X is C-metrically generated. Our main theorem proves that X is C-metrically generated if and only if X can be isomorphically described as a concretely coreflective subconstruct of a model category with objects sets structured by collections of generalized metrics in C and natural morphisms. This theorem allows for a unifying treatment of many well-known and varied theories. Moreover, via suitable comparison functors, the various relationships between these theories are studied.

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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 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.

The main purpose of this paper is to explore normality in terms of distances between points and sets. We prove some important consequences on realvalued contractions, i.e. functions not enlarging the distance, showing that as in the classical context of closures and continuous maps, normality in terms of distances based on an appropriate numerical notion of γ-separation of sets, has far reaching consequences on real valued contractive maps, where the real line is endowed with the Euclidean metric. We show that normality is equivalent to (1) separation of γ-separated sets by some Urysohn contractive map, (2) to Katětov-Tong's interpolation, stating that for bounded positive realvalued functions, between an upper and a larger lower regular function, there exists a contractive interpolating map and (3) to Tietze's extension theorem stating that certain contractions defined on a subspace can be contractively extended to the whole space.The appropriate setting for these investigations is the category of approach spaces, but the results have (quasi)-metric counterparts in terms of non-expansive maps. Moreover when restricted to topological spaces, classical normality and its equivalence to separation by a Urysohn continuous map, to Katětov-Tong's interpolation for semicontinuous maps and to Tietze's extension theorem for continuous maps are recovered.

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In this paper we study approach structures on dcpo's. A dcpo (X, 6) will be endowed with several other structures: the Scott topology; an approach structure generated by a collection of weightable quasi metrics on X; and a collection W of weights corresponding to the quasi metrics. Understanding the interaction between these structures on X will eventually lead to some fixed-point theorems for the morphisms in the category of approach spaces, which are called contractions. Existing fixed-point theorems on both monotone and non-monotone maps are obtained as special cases.

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