M.A. de Prada Vicente

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.

t -prefilter theory

Abstract In the first part of this paper, the notion of t -prefilter is introduced and studied, as are the applications λ t and ω t , their properties and relations between them. We characterize the minimal prime t -prefilters finer than a given t -prefilter, by using ultrafilters on X : and we give equivalents conditions of the t -maximality. In the second part, we define t -convergence, and we obtain a characterization of t -compactness (Gantner-Steinlage) by means of the maximal t -prefilters. Finally, some applications of these concepts, related to the continuity, separation axioms and fuzzy compactness are studied.

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.

Strong separation and strong countability in fuzzy topological spaces

Given a fuzzy topological space (X, δ), we introduce new notions of fuzzy separation and fuzzy countability, using the family of its level-topologies: {ιt(σ): t ϵ [0, 1)}. We check that these are well-defined fuzzy topological concepts and we compare them with the analogous fuzzy ones introduced in the literature. We verify that these notions are not equivalent, and we give a large number of examples which illustrate this fact.

A representation theorem for fuzzy pseudometrics

In this paper, we show that there exists a one to one correspondence between a certain class of fuzzy pseudometrics (in the sense of Kramosil and Michalek) and [0,1)-indexed families of ordinary pseudometrics satisfying a property of lower semicontinuity. The aforementioned bijection is proved to be independent of the t-norm and it provides a representation theorem for a large class of fuzzy pseudometric spaces. Further, the relations between the uniformities and topologies both generated by the fuzzy pseudometric and by the corresponding family of ordinary pseudometrics are also investigated.

Some questions in fuzzy topology

The purpose of this paper is to discuss some basic questions related (mainly) to Huttons L-fuzzy unit interval. Some of those questions are stated here for the first time, while some of them are long-standing. Related results are surveyed in some cases.

Fuzzy pseudometric spaces vs fuzzifying structures

In this paper, we apply the representation theorem established in Mardones-Perez and de Prada Vicente (2012) 12 to define and study the degree in which some topological-type properties of fuzzy pseudometric spaces are fulfilled. Fuzzifying structures which appear naturally are also investigated, and the relation between these structures and fuzzy pseudometric spaces is explored.

On fuzzy subspaces

Abstract Throughout the literature, if (X, δ) is an f.t.s., and Y ⊂ X, the induced fuzzy topological subspace (Y, δY) is defined so that δY = {ν ∧ XY: ν ∈ δ}. Since a fuzzy set isan element of Ix, it seems more reasonable to think about fuzzy topological subspaces as structures induced over each μ ∈ Ix from the fuzzy topology δ. In this sense, we introduce a new concept of fuzzy topological subspace, which coincides with the usual definition in the case that μ = χY, for Y ⊂ X. We characterize some fuzzy topological concepts, which generalize standard situations, using a certain type of prefilters.

On N-convergence of fuzzy nets

Abstract In this paper, we define the N-convergence of fuzzy nets in a fuzzy topological space and we use it to give a characterization of some fuzzy topological concepts. We give necessary conditions for the N-convergence of fuzzy nets in a fuzzy topological space and we characterize this N-convergence in a topologically generated fuzzy topological space. We also study some characterizations of fuzzy ultranets and we introduce the concept of subordination between fuzzy nets. We associate to each fuzzy net in X a prefilter on X and conversely, and we give an exhaustive description of the connections between these two theories. We give a theory of fuzzy convergence classes and we characterize fuzzy topologies via some special types of fuzzy convergence classes.

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.