A.A. Allam

New approach for basic rough set concepts

The standard rough set theory has been introduced in 1982 [5]. In this paper we use a topological concepts to investigate a new definitions for the lower and upper approximation operators. This approach is a generalization for Pawlak approach and the generalizations in [2, 7, 10, 12, 13, 14, 15, 16]. Properties of the suggested concepts are obtained. Also comparison between our approach and previous approaches are given. In this case, we show that the generalized approximation space is a topological space for any reflexive relation.

On fuzzy d-continuity and a-near compactness in fuzzy topological spaces

Abstract The concept of fuzzy δ-continuity was introduced by Saha [18]. Mashhour et al. have introduced the concept of α-near compactness in fuzzy setting. In this paper we give a characterization of fuzzy δ-continuity and fuzzy α-nearly compactness. Also we investigate the image of fuzzy α-nearly compact spaces under some types of functions. Finally a refinement is made to some results in [15, 17].

On fuzzy neighbourhood spaces

Abstract We study the behaviour of the fuzzy closure operator of a fuzzy neighbourhood space (fns, for short) with regard to the two operations of sum ‘+’ and conjunction ‘&’ on fuzzy subsets. We supply sufficient conditions on a Cech fuzzy closure operator to be the fuzzy closure operator of a fns. Finally, we study quotient fuzzy neighbourhood systems, with the help of the notion of dual fuzzy neighbourhood spaces of [11].

Fuzzy syntopogenous structures and order

The study of the relationship between fuzzy topological and order structures was initiated by Katsaras in [J. Math. Anal. Appl. 84 (1981) 44–58]. In [M.Y. Bakier et al. Proc. of Assiut First Intern. Conf., Part VIII (1990) 41–61] some properties of these spaces are studied. In [M.Y. Bakier and K. El-Saady, Fuzzy Sets and Systems 54 (1993) 213–220] is, by similar manner introduced the idea of order in fuzzy topological vector spaces. In another work of A.S. Mashhour et al. the study of the relationship between order and Lowen-fuzzy uniformities is introduced and many results are studied. In this paper, by similar manner, we will introduce the idea of order in both the fuzzy proximity space and the fuzzy syntopogenous space and we study some properties of these spaces.

Functions and aS-closedness in fuzzy topological spaces

Abstract In this paper, we discuss S-closedness in fuzzy topological spaces, and we give some characterizations of αS-closedness. Also, we investigate the image and the inverse image of αS-closed spaces under some types of functions and we improve and generalize some results on αS-closed spaces in fuzzy setting.

Functions with closed graphs and strongly closed graphs in fuzzy neighbourhood spaces

Abstract Functions with closed graphs, strongly closed graphs, weakly closed graphs and weakly strongly closed graphs are defined and studied. Many properties of these concepts are investigated.

Functions and aI-compactness on fuzzy topological spaces

Abstract The concept of αI-compactness is introduced and studied in the light of the concept of α-shading in a fuzzy setting. We give some characterization of αI-compactness in terms of fuzzy regular semiopen, fuzzy semiopen and fuzzy β-open sets. We investigated the image and the inverse image of αI-compact spaces under some types of functions and we show that in a regular extremely disconnected fuzzy space, the concepts α-compactness, α-almost compactness, α-near compactness, αS-closedness and αI-compactness are equivalent. Also we improve and generalize in fuzzy setting some results on αI-compact spaces.

On pairwise a S -closed spaces

Abstract In this piece of work, the notion of pairwise αS-closed space is introduced in a sense based on other definitions of pairwise fuzzy semiopenness and pairwise fuzzy regular closedness different from that given by M.E. Abd El-monsef and A.E. Ramdan [Bull. Fac. Sci. Alex. 26A (1) (1986) 18–27]. Also the image and the inverse image of pairwise αS-closed spaces under some types of functions are investigated. For some special cases a good agreement is achieved between the present investigation and that obtained in 1986 by Abd El-Monsef and Ramdan.