Fuzzy uniform structures and continuous t-norms

Standard fuzzy uniform structures based on continuous t-norms

“…The following concepts and results on fuzzy uniform structures and fuzzy uniform spaces may be found in [13]. DEFINITION 2.3.…”

“…In this paper we deal with a notion of fuzzy uniform space introduced by the authors in [13]. They proved in this previous paper that, for each continuous tnorm, the category of all fuzzy uniform spaces in authors' sense (and fuzzy uniformly continuous mappings) is isomorphic to the category of uniform spaces (and uniformly continuous mappings) by means of a covariant functor Ψ which leaves mappings unchanged.…”

“…In this context, a question deserving further exploration is whether the functor Ψ carries natural fuzzy uniform concepts to natural uniform concepts. A first step in this direction is presented in [13,Section 4] where it is shown that the image by Ψ of a complete fuzzy uniform space is a complete uniform space. This fact permits us to show that a Hausdorff fuzzy uniform space has a unique Hausdorff fuzzy completion up to a fuzzy uniform isomorphism (see [13,Theorem 4.3 and Remark 4.4]).…”

“…Extensions of our approach to other settings are left to the interested reader. For a contextualization of our approach in the realm of theories of fuzzified uniformities, see Section 1 and Remark 4.5 in [13].…”

“…The paper is organized as follows: in Section 2 we borrow from [13] the basic facts on fuzzy uniform structures that are useful in the sequel. Section 3 is devoted to discuss the fine fuzzy uniformity and the Stone-Čech fuzzy compactification (for This research is supported by the Plan Nacional I+D+i, under grants MTM2009-12872-C02-01 and MTM2009-12872-C02-02.…”