Extensions of uniform space notions (Q2702349)

One of the most developed approaches to the concept of uniformity in fuzzy context is the approach based on systems of certain prefilters defined on the fuzzy powerset \([0,1]^{X\times X}\) (cf. the diagonal approach in classical theory of uniform spaces). This approach was originated by R. Lowen (and for this reason we refer to it as Lowen fuzzy uniformities) and later it was developed by Lowen and other authors, in particular by the authors of the work under review.NEWLINENEWLINENEWLINEThis paper is mainly a survey-type work. The authors expound the fundamentals of the theory of (Lowen) fuzzy uniformities with special accent to the uniform properties of fuzzy subsets of fuzzy uniform spaces as well as to the convergence structure of such spaces defined by means of prefilters. The paper contains also some new results as well as new proofs of previously known theorems.NEWLINENEWLINENEWLINEThe structure of the paper is as follows: Introduction; Section 1: Preliminaries (elements of fuzzy set theory); Section 2: \([0,1]\)-fuzzy uniform spaces (elements of the theory of prefilters, basic definitions related to fuzzy uniform spaces, some results about convergence in fuzzy uniform spaces, relations between the category of fuzzy uniform spaces and the category of ordinary uniform spaces); Section 3: Cauchy filters (extensions of the concept of a Cauchy filter to the case of fuzzy uniformities are discussed); Section 4: Precompactness (properties of compactness and precompactness of fuzzy subsets in fuzzy uniform spaces are defined and studied by means of Cauchy filters); Section 5: Boundedness (bounded fuzzy subsets of fuzzy uniform spaces are characterized by means of ultrafilters and by so-called weak Cauchy filters; properties of bounded sets are studied); Section 6: Completeness (complete fuzzy subsets of fuzzy uniform spaces are defined by means of Cauchy filters; among other results it is proved here that a fuzzy subset of a fuzzy uniform space \((X,{\mathcal U})\) is \({\mathcal U}\)-compact iff it is \({\mathcal U}\)-precompact and \({\mathcal U}\)-complete).NEWLINENEWLINEFor the entire collection see [Zbl 0942.00008].