Fuzzy Convergence Research Articles

A new extension of fuzzy convergence

Fuzzy net-theory and fuzzy ideal-theory are both very important tools in fuzzy topology. In this paper, a new extension of fuzzy convergence is presented. The N-limit point and N-cluster point of an L-fuzzy net (resp. an L-fuzzy ideal), N-closed set and N-open set in an L-fuzzy topological space are defined by means of the near N-compactness, and their various properties and mutual relationships are discussed. The N-convergence theory which is a proper generalization of the Moore–Smith convergence theory in fuzzy setting is established.

Convergences in fuzzy topological spaces

In this paper we introduce the notions of fuzzy upper limit, fuzzy lower limit and the fuzzy continuous convergence on the set of fuzzy continuous functions. In examining these aforementioned notions in the present paper we find on the one hand many properties of them whilst on the other, the following applications take place: (α) the characterization of fuzzy compact spaces through the contribution of fuzzy upper limit and (β) the characterization of the fuzzy continuous convergence through the assistance of fuzzy upper limit.

Fuzzy regular convergence structures

We define a fuzzy regular modification and fuzzy regular series. And we investigate some properties of fuzzy regular modification with respect to fuzzy initial convergence structure.

Relative compact fuzzy subsets in fuzzy convergence spaces

Besides compactness the weaker notion of relative compactness plays an important role in topology. This paper generalizes a definition of relative compactness given by Chadwick [1] for fuzzy topological spaces to fuzzy convergence spaces as studied by the author ([4–6,8]). We prove that it is a “good extension”, develop the necessary theory and finally give two applications to the theory of the fuzzy Richardson compactification and to the theory of fuzzy function spaces.

On fuzzy function spaces

In [3], we started the investigation of compactness in fuzzy function spaces in FCS, the category of fuzzy convergence spaces as defined by Lowen/Lowen/Wuyts [8]. This paper goes somewhat deeper in the investigation of fuzzy function spaces using the notion of splitting and conjoining structures on fuzzy subsets. We discuss the connection to the exponential law and give several examples of such structures. As a special case, we study a notion of fuzzy compact open topology.

Fuzzy metric and convergences based on the symmetric difference

Fuzzy metric and convergences based on the symmetric difference

The Richardson compactification for fuzzy convergence spaces

In [4] we generalized the notion of fuzzy convergence space as defined by Lowen et al. [8] to fuzzy subsets in order to obtain an absolute definition of compactness for fuzzy convergence spaces. Here, as an application, a Richardson compactification is constructed which generalizes in a natural sense the classical Richardson compactification for limit spaces.

Compactness in fuzzy convergence spaces

Lowen and Lowen [Applications of category theory to fuzzy subsets (Kluwer, 1992) p. 153] and Lowen et al. [Fuzzy Sets and Systems 40 (1991) 347] recently introduced the category FCS of fuzzy convergence spaces, a topological quasitopos which is a supercategory of FTS, the category of fuzzy topological spaces. In this paper, compactness in FCS is examined. Doing so we found that to define compactness as an absolute property we had to generalize the definition of fuzzy convergence space to fuzzy subsets. All basic theorems are proved including the Tychonoff product theorem. Based on the theory developed here, in a following publication, a Richardson compactification for fuzzy convergence spaces will be given.

Fuzzy neighborhood convergence spaces

We introduce a supercategory of the category NFS of fuzzy neighborhood spaces. This supercategory is defined by means of local axioms similar to those of FNS, but the objects can also nicely be characterized internally via a notion of convergence. In future work it shall be shown that this new category allows for a good treatment of compactifications.

Neoclassical analysis: Fuzzy continuity and convergence

The neoclassical analysis is a field in which fuzzy continuous functions are investigated. For this purpose, new measures of continuity and discontinuity (or defects of continuity) are introduced and studied. Based on such measures, classes of fuzzy continuous functions are defined and their properties are obtained. The class of fuzzy continuous functions may be considered as a fuzzy set of continuous functions. Its support consists of all functions on some topological space X into a metric space Y and its membership function is the corresponding continuity measure. Such an expansion provides a possibility to complete some important classical results. Connections between boundedness and fuzzy continuity are investigated. Criteria of boundedness and local boundedness are obtained for functions on Eucleadean spaces. Besides, such new concepts as fuzzy convergence and fuzzy uniform convergence are introduced and investigated. Their properties and connections with fuzzy continuous functions are explicated. Some results, which are obtained here, are similar to the results of classical mathematical analysis, while others differ essentially from those that are proved in classical mathematics. In the first case classical results are consequences of the corresponding results of neoclassical analysis.

Convergence

This paper is devoted to the general theory of convergence structures as well as to the special case of fuzzy convergence. Convergence structures are defined in general by means of a covariant set functor ϕ In the classical case, ϕ is the filter functor. In this paper, we are especially interested in the case of ϕ being a fuzzy filter functor. The specific types of convergence structures considered here do not depend only on ϕ For instance, they also depend on a fixed natural transformation η:id → φ from the identity set functor to ϕ as well as on a fixed preordering on each set ϕX such that for each mapping f: X → Y, ϕ f: ϕX → ϕY is isotone. Above all, there are two goals for this paper, namely to get more insight into the fuzzy convergence and to clarify problems concerning compactness, the fuzzy case included. The new notions investigated in this paper are related either to the general theory of convergence structures or to the fuzzy case.

Fuzzy convergence structures

We define a fuzzy convergence structure which is a correspondence between the prefilters on a given set X and the fuzzy subsets of X . The notion of fuzzy convergence structure is shown to be a generalization of a fuzzy topology.

Completeness of the standard fuzzy real line for FNS

Abstract We consider a special class of fuzzy metric neighbourhood spaces, where the fuzzy distance is induced by a fuzzy pseudo-norm. This class is contained in Lowen's well known category FNS of Fuzzy Neighbourhood Spaces. We define the fuzzy limit of a sequence, a fuzzy Cauchy sequence and fuzzy convergence in fuzzy metric neighbourhood spaces. We prove that the set of all nonnegative fuzzy real numbers R ∗ (I) under the N-Euclidean metric (the standard fuzzy real line for FNS) is complete, but that the smallest fuzzy neighbourhood real algebra M(I) including R ∗ (I) is not complete. We also characterize the fuzzy Cauchy sequences in M(I) in terms of their corresponding (real) sequences of α-cuts.

On N-convergence of fuzzy nets

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.

Fuzzy continuity of linear maps on vector spaces

This paper studies the fuzzy continuity of linear maps from one vector space into another, through a simple characterization of fuzzy convergence of fuzzy points. A fuzzy norm is defined on the set of all fuzzy continuous linear maps from a vector space into another. Convergence of a sequence of fuzzy continuous linear maps to a fuzzy continuous linear map is obtained. It is observed that unlike in the ordinary case, fuzzy normed topological vector spaces cannot be imbedded in their second duals.

The categorical topology approach to fuzzy topology and fuzzy convergence

Abstract It is the purpose of this paper to have a look at fuzzy topology from the point of view of categorical topology. We first look at interesting subcategories of FTS and determine whether they are bireflective and/or coreflective. We also describe the set of all subcategories of FTS which are at the same time bireflective and coreflective. Due to the fact that TOP is one of these subcategories of FTS, FTS itself can be neither Cartesian closed nor extensional. We therefore, secondly, construct a supercategory of FTS which is a topological quasitopos. The construction of this quasitopos is based on the theory of convergence in FTS.

On fuzzy convergence classes

Abstract The concept of fuzzy point and its quasi-coincidence neighborhood have been introduced in [1]. Using these notions, we give the theory of fuzzy convergence classes. Especially a characterization of fuzzy topology via the fuzzy convergence classes and relative analysis are given.

Fuzzy Filters – A Generalization of Credibility Measures

The purpose of this paper is to emphasize the importance of credibility measures for a theory of fuzzy filters, to which every concept of fuzzy convergence is closely related. Moreover we use fuzzy filters as a basis for the definition of fuzzy points and fuzzy subsets.

Representations of fuzzy topologies

We make use of the point space concept [2] to give some representations of fuzzy topologies by classical topologies. We show that fuzzy convergence and fuzzy continuity with respect given fuzzy topologies are reducible to the convergence and the continuity with respect to suitable associated topologies.