Lattice-theoretic properties of quasi-metric generating spaces

There is more than one way to define a fuzzy metric space and to characterize the topologies generated by them. This paper discusses two varieties of fuzzy metric spaces and shows that they are equivalent with respect to the Vietoris topologies generated by the lattices of their open sets. In addition, a more general (fuzzier) variety of fuzzy metric spaces is introduced in which the open balls are delimited by fuzzy similitude. The relationships among these various versions of a fuzzy metric spaces are discussed, and the role of this more general fuzzy metric space as a generalization of the other varieties is explained.