Fuzzy Neighborhood Spaces Research Articles

Fuzzy Convergence, Fuzzy Neighborhood Convergence and I-Tolerance Structures for Groups

We introduce a category of fuzzy convergence groups, FCONVGRP a subcategory of the category of fuzzy convergence spaces, FCONV. Viewing [Formula: see text] as a complete Heyting algebra, we prove that the category of [Formula: see text]-tolerance groups, [Formula: see text]-TOLGRP is isomorphic to a subcategory of FCONVGRP. Since FCONV is a topological universe, and thereby possesses function space structure, upon invoking this, we are able, among others, to show that FCONVGRP is topological, and more importantly, it enables us to obtain a compatible fuzzy convergence function space structure on group of homeomorphisms. It is noticeable, however, that the category of fuzzy neighborhood convergence groups, FNCONVGRP — a supercategory of the well-known category FNS, of fuzzy neighborhood spaces, as well as the category of fuzzy neighborhood groups, FNGRP — a subcategory of FNCONVGRP exhibit nice relationships with FCONVGRP. It is important to note that the objects of FCONVGRP are homogeneous, this paves the way to present two pertinent characterization theorems on fuzzy convergence groups. Finally, introducing a category PSTOPGRP, of pseudotopological groups, we reveal the embeddings of FTOPGRP and PSTOPGRP into FCONVGRP.

Categories isomorphic to (L,M)-DFTOP

Abstract The aim of this paper is to introduce the notion of ( L,M )-double fuzzy quasi-coincident neighborhood system, and investigate the relationships between ( L,M )-double fuzzy quasi-coincident neighborhood spaces and ( L,M )-double fuzzy topological spaces in the category aspect. Also, we give a characterization of ( L,M )-DFTOP, which is called compatible antichain L -double topologies and consider the categorical connections between them. Finally, to better understand ( L,M )- DFTOP , several categories were introduced.

Generalized Sequential Convergence in Fuzzy Neighborhood Spaces

The present work deals with introducing and studying a generalized sequential convergence, namely statistical convergence in fuzzy neighborhood spaces. Some properties of statistical convergence on fuzzy neighborhood spaces are discussed.

On concretely both reflective and coreflective subconstructs of Chang fuzzy topological spaces

In this paper, the concretely both reflective and coreflective subconstructs of the construct CFTS of Chang fuzzy topological spaces are characterized and their relationship to such subconstructs of stratified fuzzy topological spaces is investigated. Properties of a particular such subconstruct of CFTS, the construct of weak fuzzy neighborhood spaces, are discussed in detail. And it is proved that there exists exactly c such subconstructs between the subconstruct of weakly induced spaces and CFTS.

On the Relationship Between Several Basic Categories in Fuzzy Topology

In this paper, a process, which associates with every fibre-small topo- logical category A and every completely distributive lattice L another topological category called the co-tower extension of A, is introduced to investigate the relationship between several basic categories in fuzzy topology. It is proved that the category of stratified Šostak L-fuzzy topological spaces is the co-tower extension of the category of stratified Chang-Goguen spaces; and the category of L-fuzzifying topological spaces (which is isomorphic the category of fuzzy neighborhood spaces when L = [0, 1]) is the co-tower extension of the category of topological spaces. Thus, in some sense, we can say that properties of the category of stratified Šostak L-fuzzy topological spaces and L-fuzzifying topological spaces are determined by the categories of stratified Chang-Goguen spaces, topological spaces and the lattice L. Therefore it can be said that in fuzzy topology the category of (stratified) Chang-Goguen topological spaces is more basic and more important than the other categories.

Even continuity and equicontinuity in fuzzy topology

We study the notion of even continuity in the category FCS of fuzzy convergence spaces and in FNS, the category of fuzzy neighborhood spaces. We further compare this notion to equicontinuity in FUS, the category of fuzzy uniform spaces.

Pointwise convergence, continuous convergence and even continuity in FNS

We discuss the notions of pointwise convergence, continuous convergence and even continuity, which were previously defined for fuzzy convergence spaces using prefilters, in fuzzy neighborhood spaces and show that these concepts can there be described entirely using ordinary filters.

Lowen Spaces

An L-fuzzy topological space (X,Δ) is called a Lowen space if Δ has a basis consisting of leveled characteristic functions, that is to say, the elements in Δ of the form aU,a∈L, U⊆X are a basis for Δ. In the case L=[0,1], (X,Δ) is a Lowen space if and only if (X,Δ) is a fuzzy neighborhood space in the sense of Lowen. It is proved in this paper that the category of Lowen spaces is a simultaneously bireflective and bicoreflective subcategory of the category of L-fuzzy topological spaces, and it is isomorphic to the category of L-fuzzifying topological spaces. Several characterizations of these spaces are given.

A short note on fuzzy neighborhood convergence spaces

We show that every topological fuzzy neighborhood convergence space is a fuzzy neighborhood space. Consequently, the meet of the categories FNCS and FTS is the category FNS.

A short note on fuzzy neighbourhood convergence spaces

We show that every fuzzy topological neighbourhood convergence space is a fuzzy neighbourhood space. Consequently, the meet of the two categories FNCS and FTS is the category FNS.

Categories isomorphic to FNS

The main result in this paper is that the category of fuzzy neighborhood spaces is isomorphic to that of fuzzifying topological spaces.

Net-convergence in fuzzy neighborhood spaces

Our aim is to introduce a concept of net convergence in the well-behaved subcategory of Lowen-fuzzy neighborhood spaces of the category of fuzzy topological spaces. In doing so, we notice that our present notion of net convergence, in addition to Katsaras and Papageorgeou net convergence, is generalized sequential convergence. We also apply this notion of net convergence in fuzzy neighborhood groups and fuzzy neighborhood linear spaces.

Strictly minimal fuzzy neighborhood division rings

The aim of this article is, on the one hand, to determine suitable classes of fuzzy neighborhood division rings that can be used as scalar domains for linear fuzzy neighborhood spaces, so as to accommodate: 1. (a) a nonvanishing linear transformation T is continuous on the space if and only if the kernel of T is N-hyperclosed; 2. (b) a nonvanishing linear transformation T is continuous on the space if and only if it has N-hyperclosed graph; 3. (c) every finite-dimensional linear space admits exactly one admissible fuzzy neighborhood system; and, on the other hand, to examine several elegant features of those classes of fuzzy neighborhood division rings.

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.

On co-optimal lifts in the category of fuzzy neighbourhood spaces

The category FNS of fuzzy neighbourhood spaces and their continuous functions is closed under the formation of optimal and co-optimal lifts in the larger topological category FTS of fuzzy topological spaces. But, while the fuzzy neighbourhood system and the level topologies of an optimal lift of a source in FNS have simple descriptions in terms of the fuzzy neighbourhood systems and the level topologies, respectively, of the source spaces, analogous descriptions are yet to be given for both constructs for a co-optimal lift of a sink in FNS. We here establish such descriptions, with the help of a functor, introduced earlier by one of us, with codomain FNS and domain another full subcategory of FTS, the category FNBS of fuzzy neighbourhood-base spaces.

A fuzzy R1 separation axiom

We introduce a new R 1 separation axiom for fuzzy topological spaces. We identify, from the literature, which lower fuzzy separation axioms are compatible with our R 1, axiom. We provide alternative descriptions for this R 1 axiom in the special setting of fuzzy neighborhood spaces. This special case has standard higher fuzzy separation axioms. In it, we establish aspects of compatibility of our R 1 axiom with those higher separation axioms, as well as with fuzzy compactness.

On sequentially closed graph mappings in fuzzy neighbourhood spaces

Two notions of closed graph and strongly closed graph in fuzzy neighbourhood spaces have been introduced; one in [ Fuzzy Sets and System 48 (1992) 377–382], and the other by the author. We prove that those notions are equivalent. We introduce the notion of sequential ⊖-convergence in fuzzy neighbourhood space.

Fuzzy proximities induced by functional fuzzy separation

Mashhour and Morsi introduced an axiom of N-complete regularity to characterize the Lowen fuzzy uniformizable fuzzy neighbourhood spaces (fns's). Guided by that axiom we supply a Sierpinski object for the Artico-Moresco category FP of fuzzy proximity spaces, which Katsaras has shown to be a topological category. We define the degree of functional fuzzy separatedness of a pair of subsets of a fns. We use that notion to define the Čech fuzzy proximity of an N-completely regular fns ( X, ϱ). We establish that it is the finest fuzzy proximity that induces the fns ( X, ϱ). We study in some detail the Čech fuzzy proximities obtained from N-normal fns's.

A level-topologies criterion for Lowen fuzzy uniformizability

We study some relationships between a fuzzy uniformity and its α-level uniformaties. We prove that a fuzzy neighbourhood space is fuzzy uniformizable iff its α-level spaces, for all 0< α<1, are uniformizable. We supply an explicit formula for a fuzzy uniformity inducing such a space. Similar criteria are established for the axioms of N-normality NT 2, NT 1, WNT 1, NT 0, WNT 0 and NR 0. We show that the two notions of the α-level uniformity of a fuzzy uniformity induced by a fuzzy metric, and the uniformity generated by the α-level metric of that fuzzy metric, are compatible. We also supply an explicit formula for a fuzzy pseudo-metric which induces a fuzzy neigbourhood space whose α-level topologies are pseudo-metrizable for all 0< α<1.

On the Metrizability of Fuzzy Neighborhood Spaces

A fuzzy neighborhood space is said to be pseudometrizable if, and only if, the associated fuzzy topology is induced by a probabilistic pseudometric under. By the aid of the concept of quasi-inverse functions it will be shown that a fuzzy neighborhood space is pseudometrizable if, and only if, every associated level-topology is pseudometrizable and a family of generating pseudometrics is monotone decreasing.