Weak completeness of the Bourbaki quasi-uniformity

M.A. Sánchez Granero


The concept of semicompleteness (weaker than half-completeness) is defined for the Bourbaki quasi-uniformity of the hyperspace of a quasi-uniform space. It is proved that the Bourbaki quasi-uniformity is semicomplete in the space of nonempty sets of a quasi-uniform space (X,U) if and only if each stable filter on (X,U*) has a cluster point in (X,U). As a consequence the space of nonempty sets of a quasi-pseudometric space is semicomplete if and only if the space itself is half-complete. It is also given a characterization of semicompleteness of the space of nonempty U*-compact sets of a quasi-uniform space (X,U) which extends the well known Zenor-Morita theorem.


Bourbaki quasi-uniformity; Hausdorff quasi-uniformity; Half-completeness

Subject classification

54E15; 54E35; 54B20

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1. The Hausdorff fuzzy quasi-metric
J. Rodríguez-López, S. Romaguera, J.M. Sánchez-Álvarez
Fuzzy Sets and Systems  vol: 161  issue: 8  first page: 1078  year: 2010  
doi: 10.1016/j.fss.2009.09.019

Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt