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Rendiconti del Circolo Matematico di Palermo

, Volume 50, Issue 1, pp 186–198 | Cite as

Bispaces admitting only bicomplete or only totally bounded quasi-metrics

  • H. P. A. Künzi
  • S. Romaguera
  • S. Salbany
Article

Abstract

We characterize quasi-metrizable bispaces that admit only bicomplete quasimetrics by means of doubly primitive sequences, and deduce that if (X, S, T) is a quasi-metrizable bispace admitting only bicomplete quasi-metrics and either (X, S) or (X, T) is hereditarily Lindelöf, then (X, S ∨ T) is compact. We also give an example which shows that hereditary Lindelöfness cannot be omitted in the above result. Finally, we show that a quasi-pseudometrizable bispace (X, S, T) admits only totally bounded quasi-pseudometrics if and only if (X, S ∨ T) is compact, and deduce that a quasi-pseudometrizable topological space admits only totally bounded quasi-pseudometrics if and only if it is hereditarily compact and quasi-sober (equivalently, if and only if it admits a unique quasi-uniformity).

AMS (1991) Subject classification

54E35 54E55 54D30 54E15 

Key words and phrases

quasi-pseudometrizable bispace quasi-uniformity bicomplete primitive sequence hereditarily Lindelöf totally bounded quasi-sober 

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Copyright information

© Springer 2001

Authors and Affiliations

  • H. P. A. Künzi
    • 1
  • S. Romaguera
    • 2
  • S. Salbany
    • 3
  1. 1.Dept. Math. Appl.University of Cape TownRondeboschSouth Africa
  2. 2.Escuela de Caminos Departmento de Matemática AplicadaUniversidad Politécnica de ValenciaValenciaSpain
  3. 3.Deparment of MathematicsUNISAPretoriaRepublic of South Africa

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