Hutton [0,1]-quasi-uniformities induced by fuzzy (quasi-)metric spaces
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 (epsilon, lambda)-uniformities). Similarly, any fuzzy metric space (X, M,*) can be endowed with a Hausdorff topology tau(M) (in the case of fuzzy quasi-metric spaces, a T(1) topology), and again this association factorized 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 tau(M) is exactly the image under Lowen's functor iota 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. (C) 2005 Elsevier B.V. All rights reserved.
[0,1]-quasi-uniformities uniformities probabilistic metric space t-norm fuzzy metric space fuzzy quasi-metric space quasi-uniformities residuation