Fuzzy Neighborhood Spaces Research Articles

FTS 0: The epireflective hull of the Sierpinski object in FTS

In the category TOP of topological spaces, the epireflective hull of the standard two-point Sierpinski space is known to coincide with the subcategory of T 0 topological spaces. In this note, we determine the epireflective hulls of the recently found ‘Sierpinski objects’ in the categories FTS, FNS and ω(TOP) of, respectively, fuzzy topological spaces, fuzzy neighborhood spaces and topologically generated fuzzy topological spaces. We thereby identify the corresponding ‘T 0-objects’ in these categories and note that the categories FTS, FNS and ω(TOP) are ‘universal’.

Research on fuzzy neighbourhood spaces in Egypt

Research on fuzzy neighbourhood spaces in Egypt

Reflectors and coreflectors in the category of fuzzy topological spaces

Using a certain number of very elementary functors, explicit formulas are given for the bireflection or/and bicoreflection of TOP (topological spaces), FNS (fuzzy neighborhood spaces) and MAX (maximal, with respect to the level topologies, fuzzy topological spaces) in the category FTS of fuzzy topological spaces. A subcategory of FTS, minimal with respect to certain associated topologies and dual in a certain sense to MAX, arises in a natural way, and is proved to be nothing else than FNS.

SIERPINSKI OBJECTS IN SUBCATEGORIES OF FTS

Abstract E.G. Manes gave a characterization of the category of topological spaces as a suitable category of sets with structure having an appropriate “Sierpinski object”. By identifying an appropriate Sierpinski object, A.K. Srivastava and R. Srivastava obtained an analogous characterization of the category of fuzzy topological spaces. In the present paper, we extend these ideas further and give similar characterizations of the categories of fuzzy neighbourhood spaces and topological fuzzy topological spaces by identifying appropriate Sierpinski objects in these categories.

A short note on fuzzy neighbourhood spaces

Lowen introduced the concept of fuzzy neighbourhood spaces (fns's), and proved that the family of level topologies of a fns is an antichain. Wuyts characterized fns's as being those fully stratified fuzzy topological spaces (fts's) whose level topologies form antichains and whose fuzzy topologies are the finest possible for thier level topologies. Mashhour et al. introduced the axiom of structure A on fts's, and derived for it several characterizations and fuzzy topological consequences. We use those characterizations of structure A and Wuyts' characterization of fns's to prove that a fts is a fns iff it is of structure A and fully stratified.

Fuzzy topologies with specified level properties

Recently the author and P. Wuyts have studied the problem of defining fuzzy topologies when one is given the family of α-closure operators or the family of level topologies. This paper is concerned with the relationship between these two problems. When the family of topologies is a chain with the LT-property, there is a natural family of closure operators that generates the Wuyts maximum, i.e., the largest fuzzy topology with the given level topologies. As a corollary, it is shown that the fuzzy neighborhood spaces are precisely the α-pseudo-regular spaces. In general, the relationship between the two problems is weak. For example, given an LT-family of topologies and an L-FTP family of semi-closure operators, there is a fuzzy topology associated with both if and only if the 0-level closure generates the 0-level topology and the α-closure of a set is always closed in the α-level topology. Lastly, it is shown that, when a function's domain has a Wuyts maximum as its topology, fuzzy continuity is equivalent to level continuity.

On local and global measures of separation in fuzzy topological spaces

This paper introduces - as far as separation properties are concerned - a new way of studying a fuzzy topological space. We define separation functions which 'measure' the degree to which points in a given space can be separated in a T0-, T1- or T2-like fashion. Both local and global measures are introduced, and their interrelation, their relation with previously defined properties, their behaviour with regard to subspaces and products and their behaviour in the special classes of fuzzy neighborhood spaces and fuzzy uniform spaces are studied.

Linear fuzzy neighborhood spaces, II

Linear fuzzy neighborhood spaces, II

Linear fuzzy neighborhood spaces

Some of the properties of linear fuzzy neighborhood spaces are studied. A special class of these spaces is the family of all fuzzy linear spaces whose topology is given by probabilistic seminorms. As in the classical case, it is shown that a linear fuzzy neighborhood space is probabilistic normable iff it is Hausdorff and it has an n-bounded convex fuzzy neighborhood of zero.

On the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level-topologies

In this paper we discuss the problem of the reconstruction of a fuzzy topological space or a fuzzy neighbourhood space from an a priori given family of level-topologies. Necessary and sufficient conditions for the existence of a solution are given, and it is proved that in the particular case of fuzzy neighbourhood spaces this solution is always unique.

On separation axioms in fuzzy topological spaces, fuzzy neighborhood spaces, and fuzzy uniform spaces

A certain number of separation axioms for fuzzy topological spaces are provided, all of which are good extensions of the topological (T 0), (T 1), or (T 2). All valid implications between the different axioms are studied and counterexamples are given for the nonvalid ones.

Compactness notions in fuzzy neighborhood spaces

In the class of fuzzy neighborhood spaces we demonstrate which implications hold and give counterexamples to the implications which do not hold between the notions of compactness, α-compactness, strong compactness and ultra compactness.

Fuzzy neighborhood spaces

In this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory. The main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies.