Ahmed M. Zahran

Modification of weak structures via hereditary classes

Abstract In this paper, we show that the construction arising from a generalized topology and a hereditary class introduced by A. Csaszar (2007) [9] remains valid, together with many applications, if the generalized topology is replaced by a weak structure.

The category of double fuzzy preproximity spaces

In this paper, we introduce the notions of double neighborhood systems and double fuzzy preproximity in double fuzzy topological spaces. We used double neighborhoods to study the initial structure of double fuzzy topological spaces, and the joins between them and the initial structures of double fuzzy preproximity spaces. Furthermore, we have proved that the category of double fuzzy preproximity spaces is a topological category, and hence a topological construct.

Double Fuzzy Preproximity Spaces

In this paper, we introduce the concept of double fuzzy preproximity spaces as a generalization of a fuzzy preproximity spaces and investigate some of their properties. Also we study the relationships between double fuzzy preproximity spaces, double fuzzy topological spaces and double fuzzy closure spaces. In addition to this was the introduction of the concept of double fuzzy neighborhood system and has been studying the connection with double fuzzy preproximity, which resulted in the definition of the concept double fuzzy preproximal neighborhood.

Regularly open sets and a good extension on fuzzy topological spaces

Singal and Rajvanshi (1992) have introduced the concepts of fuzzy almost separation axioms and they investigated some results on these concepts and the concepts of fuzzy almost continuity and fuzzy almost open mappings. In this paper we have succeeded in showing goodness of extension of fuzzy almost separation axioms and we show by producing counterexamples that some results which appeared in the work of Singal and Rajvanshi are incorrect. Also we discuss under which conditions some of these results are true and we give an improvement of some results in Singal and Rajvanshi (1992).

Almost continuity and d-continuity in fuzzifying topology

In this paper we introduce the concepts of δ-open sets, δ-closure, R-nbd system, almost continuity and δ-continuity in fuzzifying topology and we give some characterizations of almost continuity and δ-continuity.

Completely continuous functions and R-map in fuzzifying topological space

This paper considers fuzzifying topologies, a special case of I-fuzzy topologies introduced by Ying. The concepts of fuzzifying regular derived set, fuzzifying regular interior and fuzzifying regular convergence are studied and some results on above concepts are obtained. Also, the concepts of fuzzifying completely continuous functions and fuzzifying R-map are introduced and some important characterizations are obtained. Furthermore, some compositions of fuzzifying continuity with fuzzifying completely continuous functions and fuzzifying R-map are presented.

Corrigendum to “Almost continuity and δ-continuity in fuzzifying topology”: [Fuzzy Sets and Systems 116 (2000) 339–352]

Abstract In (2000), Zahran has introduced the concepts of δ-open sets, almost continuity and δ-continuity in fuzzifying topology. In this note we show that Lemma 2.2 and Theorem 2.4 are incorrect.

A note on the article ‘Fuzzy less strongly semiopen sets and fuzzy less strongly semicontinuity’

Abstract In 1992, the concepts of fuzzy strongly semiopen sets, fuzzy strongly semicontinuity and fuzzy strongly semiopen (semiclosed) mappings were introduced by Zhong (Fuzzy Sets and Systems 52 (1992) 345–351). In 1995, Ming (Fuzzy Sets and Systems 73 (1995) 279–290) has introduced the concepts of fuzzy less strongly semiopen sets, fuzzy less strongly semicontinuity and fuzzy less strongly semiopen (semiclosed) mappings. In this note we show that the concepts which due to both Zhong and Ming are equivalent and hence we show that some results in Ming are incorrect. Also we show that the other results in Ming coincide with the results in Zhong.

On Fuzzifying Nearly Compact Spaces

This paper considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies) introduced by Ying [16, (I)]. It investigates topological notions defined by means of regular open sets when these are planted into the framework of Yings fuzzifying topological spaces (in Łukasiewwicz fuzzy logic). The concept of fuzzifying nearly compact spaces is introduced and some of its properties are obtained. We use the finite intersection property to give a characterization of fuzzifying nearly compact spaces. Furthermore, we study the image of fuzzifying nearly compact spaces under fuzzifying completely continuous functions, fuzzifying almost continuity and fuzzifying R-map.

Some Fundamental Concepts in (2 ,L )-Fuzzy Topology Based on Complete Residuated Lattice-Valued Logic

In the present paper we introduce and study fundamental concepts in the framework of L-fuzzifying topology (so called (2, L)-fuzzy topology) as L-concepts where L is a complete residuated lattice. The concepts of (2, L)-derived, (2, L)-closure, (2, L)-interior, (2, L)-exterior and (2, L)-boundary operators are studied and some results on above concepts are obtained. Also, the concepts of an L-convergence of nets and an L-convergence of filters are introduced and some important results are obtained. Furthermore, we introduce and study bases and subbases in (2, L)-topology. As applications of our work the corresponding results (see [10?11]) are generalized and new consequences are obtained.