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Kuratowski’s measure of non-compactness revisited. (English) Zbl 0672.54025

The author earlier introduced the category of approach spaces with contractions, which contains the topological spaces with continuous maps as a bireflective and bicoreflective subcategory and the extended pseudo- quasi-metric spaces with non-expansive maps as a bicoreflective subcategory. A measure of non-compactness is defined for approach spaces and is shown to be effectively equivalent to the Kuratowski measure of non-compactness for extended pseudo-metric spaces. For X topological, \(m(X)=0\) if and only if X is compact. For X extended pseudo-quasi-metric, \(m(X)=0\) if and only if X is totally bounded. An arbitrary product (in the category of approach spaces) of totally bounded pseudo-quasi-metric spaces is totally bounded.
Reviewer: A.J.Klein

MSC:

54E50 Complete metric spaces
54B30 Categorical methods in general topology
54D30 Compactness
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