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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.

Dedicated to Professor S. Naimpally on the occasion of his 70 th birthday.Abstract. It is our aim in this note to take a closer look at some separation axioms in the construct AP of approach spaces and contractions. Whereas lower separation axioms seem to be qualitative, the higher ones seem to have a quantitative nature. Also some characterizations for the corresponding epireflectors will be given.2000 AMS Classification: 54A05, 54D10, 54D15, 54B30.

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<p>We continue the investigation of suitable structures for quantified functional analysis, by looking at the notion of local convexity in the setting of approach vector spaces as introduced in [6]. We prove that the locally convex objects are exactly the ones generated (in the usual approach sense) by collections of seminorms. Furthermore, we construct a quantified version of the projective tensor product and show that the locally convex objects admitting a decent exponential law with respect to it are precisely the seminormed spaces.</p>

The aim of this paper is twofold: first we investigate the role of proximity notions in the framework of approach theory [R. Lowen, Approach Spaces. The Missing Link in the Topology-UniformityMetric Triad, Clarendon Press, Oxford, 1997] and show how quantified proximity structures can be used to obtain a canonical intrinsic version of the Smirnov compactification in this setting. Second we introduce a hyperspace structure which commutes with this type of Smirnov compactification, solving a 'dual' problem to the one treated concerning approach completion in [R. Lowen, M. Sioen,