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Somewhat Slightly Generalized Double Fuzzy Semicontinuous Functions

DOI: 10.1155/2014/756376

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Abstract:

The aim of this paper is to introduce the concepts of somewhat slightly generalized double fuzzy semicontinuous functions and somewhat slightly generalized double fuzzy semiopen functions in double fuzzy topological spaces. Some interesting properties and characterizations of these functions are introduced and discussed. Furthermore, the relationships among the new concepts are discussed with some necessary examples. 1. Introduction In 1968, Chang [1] was the first to introduce the concept of fuzzy topological spaces. These spaces and their generalization are later developed by Goguen [2], who replaced the closed interval by more general lattice . On the other hand, by the independent and parallel generalization of Kubiak and ?ostak’s [3, 4], made topology itself fuzzy besides their dependence on fuzzy set in 1985. Various generalizations of the concept of fuzzy set have been done by many authors. In [5–10], Atanassove introduced the notion of intuitionistic fuzzy sets. Later ?oker [11] defined intuitionistic fuzzy topology in Chang’s sense. Then, Mondal and Samanta [12] introduced the intuitionistic gradation of openness of fuzzy sets. Gutiérrez García and Rodabaugh [13], in 2005, replaced the term “intuitionistic” and concluded that the most appropriate work is under the name “double.” In 1980, Jain [14] introduced the notion of slightly continuous functions. On the other hand, Nour [15] defined slightly semicontinuous functions as a weak form of slight continuity and investigated their properties. In [16], Noiri introduced the concept of slightly -continuous functions. Sudha et al. [17] introduced slightly fuzzy -continuous functions. Also in 2004, Ekici and Caldas [18] introduced the notion of slight -continuity (slight -continuity). In this paper, the concepts of somewhat slightly generalized double fuzzy semicontinuous functions and somewhat slightly generalized double fuzzy semiopen functions are introduced. Several interesting properties and characterizations are introduced and discussed. Furthermore, the relationships among the concepts are obtained and established with some interesting counter examples. 2. Preliminaries Throughout this paper, let be a nonempty set, the unit interval , , and . The family of all fuzzy sets in is denoted by . is the family of all fuzzy points in . By and we denote the smallest and the greatest fuzzy sets on . For a fuzzy set , denotes its complement. Given a function , and defined the direct image and the inverse image of , defined by and for each , , and , respectively. All other notations are standard notations

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