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The Bourbaki quasi-uniformity. (English) Zbl 0876.54022
This paper initiates the systematic study of the preservation of quasi-uniform properties between a quasi-uniformity \({\mathcal U}\) on a set \(X\) and the Bourbaki quasi-uniformity \({\mathcal U}_*\) on the collection \({\mathcal P}_0 (X)\) of all nonempty subsets of \(X\). The authors prove that \(({\mathcal P}_0 (X), {\mathcal U}_*)\) is precompact (totally bounded) if, and only if, \((X, {\mathcal U})\) is precompact (totally bounded), and they give examples to show that the corresponding results hold neither for compactness nor hereditary precompactness. The principal result is an extension of the Isbell-Burdick Theorem: The Bourbaki quasi-uniformity \({\mathcal U}_*\) is right K-complete if, and only if, each stable filter on \((X, {\mathcal U})\) has a cluster point. As might be expected, along the way the authors provide a good many interesting results and examples concerning both right K-completeness and the related property that each stable filter has a cluster point.

54E15 Uniform structures and generalizations
54B20 Hyperspaces in general topology
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