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Approach spaces - a common supercategory of TOP and MET. (English) Zbl 0676.54012
Axiomatizing the concept of “distance between points and sets” in a rather natural way the author creates the concept of approach spaces. He demonstrates that the arising category AP of approach spaces and contracting maps is a topological construct, which contains Top as a simultaneously bireflective and bicoreflective subconstruct, and the construct of pseudo-quasi-metric spaces and non-expansive maps as a bicoreflective subconstruct. Thus, in particular, topological spaces and metric spaces can be regarded as objects of the same type and continuous resp. non-expansive maps as morphisms of the same type.
The author provides several completely different but equivalent descriptions of AP. Moreover he demonstrates convincingly that certain probabilistic metric spaces, spaces of measures and spaces of random variables can be viewed most naturally as approach spaces.
Reviewer: H.Herrlich

MSC:
54B30 Categorical methods in general topology
18B30 Categories of topological spaces and continuous mappings (MSC2010)
54A05 Topological spaces and generalizations (closure spaces, etc.)
54E35 Metric spaces, metrizability
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