The Bourbaki Quasi-Uniformity
by
Hans-Peter Künzi
University of Berne
Coauthors: Caroline Ryser

We study properties of the Bourbaki quasi-uniformity U^* defined on the collection P₀(X) of nonempty subsets of a given quasi-uniform space (X, U).

We note that U^* is precompact (totally bounded, respectively) if and only if U is precompact (totally bounded, respectively). Examples are given to show that for the properties of compactness and hereditary precompactness the corresponding statement does not hold.

Furthermore we establish that the Bourbaki quasi-uniformity U^* on P₀(X) of a quasi-uniform space (X, U) is right K-complete if and only if each stable filter on (X, U) has a cluster point. This theorem generalizes the well-known Isbell-Burdick theorem for uniform spaces to the quasi-uniform setting. The paper ends with a related theorem characterizing bicompleteness of (P₀(X), U^*) in terms of a property of (X, U).

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-60.