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 (ε,λ)-uniformities). Similarly, any fuzzy metric space (X,M,*) can be endowed with a Hausdorff topology τM (in the case of fuzzy quasi-metric spaces, a T1 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 UM. This allows us to give a factorization of the previous association via Hutton [0,1]-quasi-uniformities. It is also proved that the topology τM is exactly the image under Lowen's functor ι of the [0,1]-topology induced by UM. As a consequence, we get a class of Hutton [0,1]-quasi-uniformities which are probabilistic metrizable.