J. Gutiérrez García

A Unified Approach To The Concept Of Fuzzy L-Uniform Space

The theory of uniform structures is an important area of topology which in a certain sense can be viewed as a bridge linking metrics as well as topological groups with general topological structures. In particular, uniformities form, the widest natural context where such concepts as uniform continuity of functions, completeness and precompactness can be extended from the metric case. Therefore, it is not surprising that the attention of mathematicians interested in fuzzy topology constantly addressed the problem to give an appropriate definition of a uniformity in fuzzy context and to develop the corresponding theory. Already by the late 1970’s and early 1080’s, this problem was studied (independently at the first stage) by three authors: B. Hutton [21], U. Hohle [11, 12], and R. Lowen [30]. Each of these authors used in the fuzzy context a different aspect of the filter theory of traditional uniformities as a starting point, related in part to the different approaches to traditional unformities as seen in [37, 2] vis-a-vis [36, 22]; and consequently, the applied techniques and the obtained results of these authors are essentially different. Therefore it seems natural and urgent to find a common context as broad as necessary for these theories and to develop a general approach containing the previously obtained results as special cases—it was probably S. E. Rodabaugh [31] who first stated this problem explicitly.

Generalised filters 1

We continue the study, started in [8], of generalised filters. Prime prefilters have played a central role in the theory of (Lowen) fuzzy uniform spaces and Lowen discovered a characterisation of the set of all minimal prime prefilters finer than a given prefilter in terms of ultrafilters. We define the notion of a prime generalised filter and describe the set of all minimal prime g-filters finer than a given g-filter in terms of ultrafilters. The relationship between prime prefilters and prime g-filters is revealed. The behaviour of the images and preimages of g-filters are investigated.

Examples of non-strong fuzzy metrics

Answering a recent question posed by Gregori et al. [On a class of completable fuzzy metric spaces, Fuzzy Sets and Systems 161 (2010), 2193-2205] we present two examples of non-strong fuzzy metrics (in the sense of George and Veeramani).

SUPER UNIFORM SPACES

Abstract A new, generalized form, of uniformity, the so called super uniformity is defined and studied. It is based on the concept of fuzzy filter, as introduced by Eklund and Gaaler [EG]. To each super uniformity, a fuzzy α-uniformities system can be associated. They will be called α-levels. These α-levels are fuzzy uniformities in the sense of Lowen, for α=1, and α-modifications with pleasant properties, for α≠1. The *-version of super uniformities is related, at level 1, with T-uniformities, as defined by Hohle [Ho]. A criterion for a given family of prefilters {F α}αeI0 on a set X to generate a fuzzy filter 𝔉 on X with {F α)αe I0 as its family of α-level prefilters, that is 𝔉α = F α is found, and extended to super uniformities. Finally, super uniformities are related with fuzzy topologies in the sense of Sostak.

Fuzzy uniform structures and continuous t-norms

We introduce and discuss a notion of fuzzy uniform structure that provides a direct link with the classical theory of uniform spaces. More exactly, for each continuous t-norm we prove that the category of all fuzzy uniform spaces in our sense (and fuzzy uniformly continuous mappings) is isomorphic to the category of uniform spaces (and uniformly continuous mappings) by means of a covariant functor. Moreover, we also describe the inverse functor and, then, we discuss completeness and completion of these fuzzy uniform structures. It follows from our results that each fuzzy uniform structure in our sense induces a Hutton [0, 1]-uniformity and, conversely, each Hutton [0, 1]-uniformity induces a fuzzy uniform structure in our sense.

EVEN FUZZY TOPOLOGIES AND RELATED STRUCTURES

Two special kinds of fuzzy topologies in the sense of the third author: the so called even and supereven fuzzy topologies are introduced. Some properties of even fuzzy topologies are established; their role in the (general) theory of fuzzy topologies is discussed. Besides, proximal and uniform counterparts of (super) even fuzzy topologies are considered.

Standard fuzzy uniform structures based on continuous t-norms

This paper deals with fuzzy uniform structures previously introduced by the authors [Fuzzy uniform structures and continuous t-norms, Fuzzy Sets Syst. 161 (2009) 1011-1021]. Our approach involves a covariant functor @J from the category of fuzzy uniform spaces and fuzzy uniformly continuous mappings (in our sense) to the category of uniform spaces and uniformly continuous mappings. We show that @J is well-behaved with respect to some significant fuzzy uniform concepts, and its behavior provides a method to introduce notions of fine fuzzy uniform structure and Stone-Cech fuzzy compactification in this context. Our method also applies to obtain fuzzy versions of some classical results on topological algebra and hyperspaces. The case of quasi-uniform structures is also analyzed.

EMBEDDINGS INTO THE CATEGORY OF SUPER UNIFORM SPACES

Abstract The category US of uniform spaces has been generalised in various ways. The category FUS, of fuzzy uniform spaces and the category GUS, of generalised uniform spaces have both been shown to be good extensions in the sense that US can be embedded into them. We show here that the category SUS, of super uniform spaces also enjoys this property and furthermore, the categories FUS and GUS can be embedded into SUS.

An identification theorem for the completion of the Hausdorff fuzzy metric

Abstract We prove that given a fuzzy metric space (in the sense of Kramosil and Michalek), the completion of its Hausdorff fuzzy metric space is isometric to the Hausdorff fuzzy metric space of its completion, when the Hausdorff fuzzy metrics are defined on the respective collections of non-empty closed subsets. As a consequence, we deduce the corresponding result for metric spaces by using the standard fuzzy metric. An application to the extension of multivalued mappings and some illustrative examples are also given.

Characteristic values of T-filters

In this paper we extend the notions of characteristic set and characteristic value of a prefilter (cf. [Lowen, Convergence in fuzzy topological spaces, Gen. Topol. Appl. 10 (1979) 147-160]) to the setting of @?-filters and @?-filter bases (cf. [Hohle, Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Math. 38 (1982) 289-323]) in the lattice valued context of complete MV-algebras. This notion allows us to reinterpret the @k-condition and consequently to give an alternative but equivalent definition of @?-filters and @?-filter bases. Some important properties of the characteristic value are also studied, which allows us to establish a relation between @?-filters and both L-filters and L-filters of ordinary subsets.