The Bourbaki quasi-uniformity. (English) Zbl 0876.54022

Künzi, Hans-Peter A.; Ryser, Carolina

Topol. Proc. 20, 161-183 (1995).

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.

Reviewer: P.Fletcher (Blacksburg)

MSC:

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

Keywords:

Isbell-Burdick theorem; Bourbaki quasi-uniformity; stable filter; cluster point; K-completeness