Completeness of Hutton [0,1]-quasi-uniform spaces
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). (c) 2007 Elsevier B.V. All rights reserved.
completeness completeness hutton [0,1]-quasi-uniform space fuzzy metric space cauchy l-filter probabilistic metric space uniform space t-norm completion