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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 𝒰 on a set X and the Bourbaki quasi-uniformity 𝒰 * on the collection 𝒫 0 (X) of all nonempty subsets of X. The authors prove that (𝒫 0 (X),𝒰 * ) is precompact (totally bounded) if, and only if, (X,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 𝒰 * is right K-complete if, and only if, each stable filter on (X,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.

MSC:
54E15Uniform structures and generalizations
54B20Hyperspaces (general topology)