Weak completeness of the Bourbaki quasi-uniformity

Abstract

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.