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On Some Separation Axioms in Bitopological Spaces أحمد سعيد عبدالله العربي

:Abstract

    In this note, we introduce and study some separation axioms in bitopological spaces, depending on the classes defined in (Bose, 1981 and Tadros and Abd-Allah). These axioms are different from those given in (Maheshwri and Prasad, 1975, 1978 and 1979: Mashhour el al., 1982). The relations between these new types of axioms with each other and with other existing ones are discussed. The characterizations of pairwise Hausdorff, pairwise regular and pairwise normal bitopological spaces (Kelley, 1963) in terms of relative α-open sets are given, which agree with the results obtained in the ordinary topological spaces (Tadros, 1985). The subspaces, the image and the inverse image of some of these spaces under special mappings are studied

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On Some Mappings of Topological Spaces أحمد سعيد عبدالله العربي

:Abstract

    In this note, we give the relations between α-closure (resp. s-closure,  p-closure,  ß-closure) and the ordinary operators in topological spaces. Some properties and the comparison of a series of non-continuous and non-open mappings are established. We also generalize some results concerning these types of mappings

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On Some Strong Types of Connectedness in Bitopological Spaces أحمد سعيد عبدالله العربي

:Abstract

    In this note, we introduce and study the concepts of pairwise α-(resp. p- , ß- ) connected spaces and give the implications between them and other types of pairwise connectedness introduced by Mukherjee and Pervin in [5, 12]. We prove that the notions of pairwise connectedness and pairwise α-connectedness are equivalent. This gives a new characterization of pairwise connectedness in terms of pairwise α-open sets. Finally, we generalized most of the properties given in the α-resp. p- , ß- ) connected spaces into the pairwise α-resp. p- , ß- ) connected spaces in the bitopological spaces

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On Some Special Subsets and Mappings of Bitopological Spaces أحمد سعيد عبدالله العربي

:Abstract

    In this note, we introduce and study the concepts of pairwise  α- (resp. p- , ß- ) open sets, pairwise α-(resp. p- , ß- )  continuity and pairwise α- (resp. p- , ß- )  open mappings in bitopological spaces. The pairwise semi-continuous and pairwise semi-open mappings defined by Bose in [1] are further investigated

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Extended Fuzzy Topologies, Part I, On Fuzzy Stacks أحمد سعيد عبدالله العربي

:Abstract

    The notion of fuzzy stack is introduced and investigated. It generalized both the notions of fuzzy filter and a suitable modification of the usual notion of stack. For obtaining important lattice theoretical properties the notion of fuzzy stack is established in such a way that each fuzzy stack can be associated a finer fuzzy filter. Homogenous fuzzy stacks and fuzzy stacks with the cutting property play an exceptional role. Analogously as in the fuzzy filter case, there are two different notions of fuzzy stack bases and two different notions of principal fuzzy stack respectively. Contrary to the fuzzy filter case here, a new phenomenon appears: Homogeneous fuzzy stack my not have superior fuzzy stack bases. We show that all homogeneous fuzzy stacks with the catting property have superior fuzzy stack bases, but there are also further homogenous fuzzy stacks with superior fuzzy stack bases. By means of the notion of fuzzy stack, a related partially order monad is defined, which gives the possibility to develop the theory of extended fuzzy topologies in the framework of monadic topology. The theory of extended fuzzy topologies will be developed in detail in two subsequent papers (Part II and Part III) as well as in a  separate paper ([13 ]).

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Extended Fuzzy Topologies, Part II, Characterizing Notions أحمد سعيد عبدالله العربي

:Abstract

    Basic results on extended fuzzy topologies are presented obtained in applying the theory of fuzzy stacks developed in Part I and using results given in [13]. There are mainly investigated notions defined for extended fuzzy topologies which can be used for their characterizations. They are called characterizing notions. In some cases the notions we consider are characterizing under some assumptions on the extended fuzzy topologies. Examples of characterizing notions which do not need any assumption are those of fuzzy neighborhood of a fuzzy stack, of the interior of a fuzzy stack and of the set of inner points of a fuzzy stack. The notions of open fuzzy stack and of projectively open fuzzy stack are characterizing for idempotent extended fuzzy topologies. Further characterizing notions are presented obtained by restricting to valued and superior principal fuzzy stacks. 

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Extended Fuzzy Topologies , Part III, Closedness and Copmactness أحمد سعيد عبدالله العربي

:Abstract

 Contrary to characterizing notions for extended-fuzzy topological spaces considered in Part II here  in  Part III, are investigated notions which have some invariants properties and which , as is shown ,only depend on converging ultra fuzzy filters. Examples are the notions of adherence points, of projective closedness and of closedness of a fuzzy stack.By restricting to valued and superior principle fuzzy stacks we get analogous notions for fuzzy sets. The fuzzy stack closure are defined in some sense complementarily to the fuzzy neighborhoods of fuzzy stacks. A futher notion considered here which has some invariance properties and only depends on converging ultra fuzzy filters is that of compactness. In general, compactness is defined for fuzzy stacks. A fuzzy set is called compact if the related superior principle fuzzy stack is compact. Compactness of the whole space means the compactness of the constant fuzzy set with value 1. In the classical case of a topological space this is the usual compactness. There are generalized some classical results on compactness. For instance, a generalized Tychonoff Theorem is presented. 

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Operations and its Applications On L-fuzzy Bi topological Spaces أحمد سعيد عبدالله العربي

: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

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On Extended Fuzzy Topologies أحمد سعيد عبدالله العربي

:Abstract

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

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General Notions Related to Fuzzy Filters أحمد سعيد عبدالله العربي

:Abstract

  Basic results on L-fuzzy topology are presented which are obtained in applying the theory of fuzzy filters and the operations defined on the set of all L-fuzzy subsets endowed with an L-fuzzy topology. In particular , the category CSOF of all characterized spaces and several of its bireflective and bicoreflective subcategories are studied

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