Ali Kandil

On fuzzy bitopological spaces

Abstract In this paper, we have used the fuzzy topologies τ1 and τ2 to generate a family τs which is a supra fuzzy topology on X. Using the family τs, we introduce and study the concepts of separation axioms, continuity (resp. openness, closedness) of a mapping and compactness for a fuzzy bitopological space (X, τ1, τ2). Our definitions preserve much of the correspondence between concepts of fuzzy bitopological spaces and the associated fuzzy topological spaces. We then investigate the relationship between these concepts and their correspondence with the fuzzy bitopological spaces (Kandil and El-Shafee, 1991).

On extended fuzzy topologies

Basic results on extended fuzzy topologies are presented which are obtained in applying the theory of fuzzy stacks. In particular, the category EFTOP of extended-fuzzy topological spaces and several of its bireflective and bicoreflective subcategories are studied.

The theory of global fuzzy neighborhood structures part I—the general case

Abstract This paper is devoted to global fuzzy neighborhood structures. We introduce three types of these structures, defined by means of fuzzy filters. In some sense, the first type is more general than the second one, and the second type is more general than the third one. In the second case, only homogeneous fuzzy filters are used. In the third case, homogeneous fuzzy filters are used which are representable by a prefilter, or equivalently, prefilters are used which represent special homogeneous fuzzy filters. All fuzzy topologies and stratified fuzzy topologies are global fuzzy neighborhood structures of the first and second type, respectively. They appear in a canonical way as interior operators. Fuzzy neighborhood structures introduced by Lowen [Fuzzy Sets and Systems 7 (1982) 165] are defined by means of prefilters. The definition of these structures is in some sense similar to a characterization of those global fuzzy neighborhood structures of the third type which can be identified with fuzzy topologies. However, the related fuzzy topological approach differs. Fuzzy neighborhood structures in sense of Lowen are characterized canonically as fuzzy closure operators. In this paper the relations between the three types of global fuzzy neighborhood structures and their associated fuzzy topologies and also some relations to the fuzzy neighborhood structures in the sense of Lowen are investigated. Moreover, this paper deals with initial and final structures of global fuzzy neighborhood structures. In two subsequent papers (Part II and Part III), fuzzy topogenous orders and fuzzy uniform structures will be investigated, respectively. All regular fuzzy topogenous orders, that is, all fuzzy topogenous structures, and in particular, all fuzzy proximities, are global fuzzy neighborhood structures. As is shown by examples there exist global fuzzy neighborhood structures which are not fuzzy topogenous structures. Hence, the notion of global fuzzy neighborhood structure is more rich. Fuzzy uniform structures, defined analogously to A. Weils definition of a uniform structure as fuzzy filters, generate in a canonical way global fuzzy neighborhood structures. Some results on global fuzzy neighborhood structures will also be published in Gahler et al. (submitted), in particular those related to stratifications.

The theory of global fuzzy neighborhood structures. II: fuzzy topogenous orders

Abstract This paper deals with fuzzy topogenous orders, in particular, with fuzzy topogenous structures and with the more special fuzzy proximities. These structures have been investigated ay Katsaras and Petalas (1983, 1984). The notion of fuzzy proximity was introduced by Katsaeas (1980). Fuzzy topogenous structures and fuzzy proximities are represented in this paper as global fuzzy neighborhood structures. Fuzzy topogenous orders, in general, are characterized by the notion of global fuzzy neighborhood prestructure, which is a weakening of that one of global fuzzy neighborhood structure. In this paper, moreover, a modification of the notion of fuzzy proximity is considered, called fuzzy proximity of the internal type. Whereas the notion of fuzzy proximity proposed by Katsaras depends on a fixed order-reversing involution of the related lattice L , the notion of fuzzy proximity of the internal type is independent on such an involution. The investigations intthis paper demonstrate that there are important global fuzzy neighborhood structures and prestructures different from fuzzy topologies and fuzzy pretopologies, respectively.

Strong and ultra separation axioms of fuzzy bitopological spaces

Given a fuzzy bitopological space (X, τ1, τ2), we introduce a new notion of fuzzy pairwise separation axioms by using the family of its level bitopologies ια(τ1), ια(τ2), α ϵ[0,1). We prove that these concepts are good extension and we compare them with its corresponding FPT; (Kandil and EI-Shafee, 1991) and FPT1∗ (Abu Safiya et al., 1994) (i = 0, 1, 2, 3, 4), respectively. We show that these notions are not equivalent and we give a number of examples which illustrate this fact.

Operations and its applications on L -fuzzy bitopological spaces: part I

Abstract In this and a subsequent paper, a new operation on the class of all L-fuzzy subsets of a universe endowed with a fuzzy topological space is introduced to generalize several characterization and properties of some fuzzy bitopological concepts such as a fuzzy interior operators and some new classes of fuzzy separation axioms. Also, the relation between these classes and the weaker and stronger forms in fuzzy bitopological spaces are investigated. Finally, we showed that these axioms are good extension in the sense of Lowen (1978) due to Lowen, in Part II this theory is applied on the mappings defined on L-fuzzy bitopological spaces.