Lowen uniformity via fuzzifying uniformity

The original definition of a topological space given by Hausdorff used neighborhood systems. Lattice-valued maps appear in this context when you identify a topology with a monoid in the Kleisli category of the filter monad on SET. H?hle’s notion of a lattice-valued topology [2] uses the same idea and it’s inspired in the classical lattice-valued topologies. Ltopological spaces are motivated by a crisp notion of openness of a lattice-valued map, where L is a strictly two-sided commutative quantale. On his part, L-fuzzy topological spaces are structures which are capable to express the degree of openness of a lattice-valued map. These spaces constitute topological subcategories of the category of LM -topological spaces introduced by Kubiak and Šostak [3]. As a particular case we can stand out the so-called fuzzifying topologies considered by Ying (1991) whose elements are mappings T: 2 → I satisfying a multivalued version of the topological axioms (here X is a nonempty set and I is the unit closed interval). The category of fuzzifying topological spaces is isomorphic to the category of fuzzy neighborhood spaces which in turn is a topological subcategory of stratified I-topological spaces.