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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 ${\cal U}$ on a set $X$ and the Bourbaki quasi-uniformity ${\cal U}_*$ on the collection ${\cal P}_0 (X)$ of all nonempty subsets of $X$. The authors prove that $({\cal P}_0 (X), {\cal U}_*)$ is precompact (totally bounded) if, and only if, $(X, {\cal U})$ 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 ${\cal U}_*$ is right K-complete if, and only if, each stable filter on $(X, {\cal U})$ 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.

54E15Uniform structures and generalizations
54B20Hyperspaces (general topology)
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