Hutton [ 0 , 1 ] -quasi-uniformities induced by fuzzy (quasi-)metric spaces

• Published in: Fuzzy sets and systems 2006-03, Vol.157 (6), p.755-766
• Main Authors: Gutiérrez García, J., de Prada Vicente, M.A.
• Format: Article
• Language: English
• Subjects: [formula omitted]-quasi-uniformities; Algebraic topology; Exact sciences and technology; Fuzzy metric space; Fuzzy quasi-metric space; General topology; Mathematics; Probabilistic metric space; Quasi-uniformities; Residuation; Sciences and techniques of general use; t-norm; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Uniformities
• ISSN: 0165-0114; 1872-6801
• DOI: 10.1016/j.fss.2005.11.002

It is well known that given a probabilistic metric space (Menger space) with continuous t-norm T there is a Hausdorff topology associated. This association factorizes through strong uniformities (or ( ε , λ ) -uniformities). Similarly, any fuzzy metric space ( X , M , * ) can be endowed with a Hausdorff topology τ M (in the case of fuzzy quasi-metric spaces, a T 1 topology), and again this association factorizes through (quasi-)uniform spaces. In this paper we associate to each fuzzy (quasi-)metric space a Hutton [ 0 , 1 ] -quasi-uniformity U M . This allows us to give a factorization of the previous association via Hutton [ 0 , 1 ] -quasi-uniformities. It is also proved that the topology τ M is exactly the image under Lowen's functor ι of the [ 0 , 1 ] -topology induced by U M . As a consequence, we get a class of Hutton [ 0 , 1 ] -quasi-uniformities which are probabilistic metrizable.