Completeness of Hutton [0,1]-quasi-uniform spaces

  • Authors:

  • J. Gutiérrez García;M. A. de Prada Vicente;S. Romaguera

  • Affiliations:

  • Departamento de Matemáticas, Universidad del País Vasco-Euskal Herriko Unibertsitatea, Apdo. 644, 48080 Bilbao, Spain;Departamento de Matemáticas, Universidad del País Vasco-Euskal Herriko Unibertsitatea, Apdo. 644, 48080 Bilbao, Spain;Departamento de Matemática Aplicada, IMPA-UPV, Universidad Politécnica de Valencia, 46071 Valencia, Spain

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2007

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Abstract

This paper deals with completeness of Hutton [0,1]-quasi-uniform spaces. Recently, the first two authors, [J. Gutierrez Garcia, M.A. de Prada Vicente, Hutton [0,1]-quasi-uniformities induced by fuzzy (quasi-)metric spaces, Fuzzy Sets and Systems 157 (2006), 755-766], have constructed a Hutton [0,1]-quasi-uniformity induced by a fuzzy metric space (in the sense of George and Veeramani). In this paper, we define completeness of Hutton [0,1]-quasi-uniform spaces as convergence of any stratified tight Cauchy [0,1]-filter. Our main result states the equivalence between completeness of any fuzzy metric space (X,M,*) and completeness of the induced Hutton [0,1]-quasi-uniformity U"M. Also it is proved that the Hutton [0,1]-quasi-uniform space (X,U"M) has, in this context, a kind of completion that is unique up to uniform isomorphism. The obtained results come from an appropriate definition of Cauchy L-filter (where L stands for a complete lattice, with additional properties).