
Several Types of Totally Continuous Functions in Double Fuzzy Topological SpacesDOI: 10.1155/2014/361398 Abstract: We introduce the notions of totally continuous functions, totally semicontinuous functions, and semitotally continuous functions in double fuzzy topological spaces. Their characterizations and the relationship with other already known kinds of functions are introduced and discussed. 1. Introduction The concept of fuzzy sets was introduced by Zadeh in his classical paper [1]. In 1968, Chang [2] used fuzzy sets to introduce the notion of fuzzy topological spaces. ？oker [3, 4] defined the intuitionistic fuzzy topological spaces using intuitionistic fuzzy sets. Later on, Demirci and ？oker [5] defined intuitionistic fuzzy topological spaces in Kubiak？ostak’s sense as a generalization of Chang’s fuzzy topological spaces and intuitionistic fuzzy topological spaces. Mondal and Samanta [6] succeeded to make the topology itself intuitionistic. The resulting structure is given the new name “intuitionistic gradation of openness.” The name “intuitionistic” did not continue due to some doubts that were thrown about the suitability of this term. These doubts were quickly ended in 2005 by Gutiérrez García and Rodabaugh [7]. They proved that this term is unsuitable in mathematics and applications. Therefore, they replaced the word “intuitionistic” by “double” and renamed its related topologies. The notion of intuitionistic gradation of openness is given the new name “double fuzzy topological spaces.” The fuzzy type of the notion of topology can be studied in the fuzzy mathematics, which has many applications in different branches of mathematics and physics theory. For example, fuzzy topological spaces can be applied in the modeling of spatial objects such as rivers, roads, trees, and buildings. Since double fuzzy topology forms an extension of fuzzy topology and general topology, we think that our results can be applied in modern physics and GIS Problems. Jain et al. introduced totally continuous, fuzzy totally continuous, and intuitionistic fuzzy totally continuous functions in topological spaces, respectively (see [8–11]). In this paper, we introduce the notions of totally continuous, totally semicontinuous, and semitotally continuous functions in double fuzzy topological spaces and investigate some of their characterizations. Also, we study the relationships between these new classes and other classes of functions in double fuzzy topological spaces. 2. Preliminaries Throughout this paper, let be a nonempty set and let be the closed interval , and . The set of all fuzzy subsets (resp., fuzzy points) of is denoted by (resp., ). For and , a fuzzy point is defined by if
