Weak completeness of the Bourbaki quasi-uniformity

• Sánchez Granero, M.A. ^[1]
  1. [1] Universidad de Almería

     Universidad de Almería

     Almería, España

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 2, Nº. 1, 2001, págs. 101-112
• Idioma: inglés
• DOI: 10.4995/agt.2001.3018
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• Resumen

  • 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.