On Smooth Bitopological Spaces

Abstract

Fuzzy sets are a generalization of conventional set theory that were introduced by Zadeh [91] in 1965 as a mathematical way to represent uncertainty and vagueness and to provide formalized tools for dealing with the imprecision intrinsic to many problems in everyday life. The notion of a fuzzy set has caused great interest among both `pure' and applied mathematicians. It has also raised enthusiasm among some engineers, biologists, psychologists, economists, and experts in other areas, who use (or at least try to use) mathematical ideas and methods in their research. After the discovery of the fuzzy sets, much attention has been paid to generalize the basic concepts of classical topology in fuzzy setting and thus a modern theory of fuzzy topology is developed. Chang [13] in 1968 introduced the notion of fuzzy topology when he replaced sets by fuzzy sets in the ordinary de nition of topology. Since then, a lot of contributions to the development of fuzzy topology have been published (cf.[24, 25, 28, 58, 59, 60, 64]). In 1980, Pu and Liu [65, 66] introduced the concept of quasi-coincidence relation which done further research on Chang's concept of fuzzy topology. However, Chang de nition for fuzzy topology was criticized by some authors, that his notion did not really describe fuzziness with respect to the concept of openness of a fuzzy set, which seems to be a drawback in the process of fuzzi cation of the concept of topological spaces. In the light of this di culty, many mathematicians try to make a fuzzy treatment for this structures. Independently by Kubiak [47] and Sostak [84] in 1985 introduced the fundamental concept of a `fuzzy topological structure', as an extension of both crisp topology and Chang's fuzzy topology, according to which a fuzzy topology on a set X is a mapping on the power set IX (i.e., a mapping : IX 􀀀! I where I = [0; 1] v is the closed unite interval) satisfying certain axioms. With respect to this mapping each fuzzy set 2 IX is open with a suitable degree ( ), i.e., each fuzzy subset has a degree of openness, in the sense that not only the object were fuzzi ed, but also the axiomatics. Sostak gave some rules and showed how such an extension can be realized. Every fuzzy topology in Sostak sense is a fuzzy topology in Chang sense if : IX 􀀀! f0; 1g. Subsequently, Badard [8] in 1986, introduced the concept of `smooth topological space'. In 1992, Chattopadhyay et al. [14] and Chattopadhyay and Samanta [15] in 1993, re-introduced the same concept, calling it `gradation of openess'. In 1992, Ramadan [70] and his colleagues introduced a similar de nition, namely, smooth topological space for lattice L = [0; 1]. Following Ramadan, several authors have re-introduced and further studied smooth topological space (cf.[6, 21, 43, 71, 72, 80, 85]). Thus, the terms `fuzzy topology', in Sostak's sense, `gradation of openness' and `smooth topology' are essentially referring to the same concept. In our thesis, we adopt the term smooth topology. The concept of bitopological spaces (X; 1; 2) was rst introduced by Kelly [37] in 1963, as a method of generalizing topological spaces (X; ). In 1989, Kandil [33] introduced and studied the notion of fuzzy bitopological spaces as a natural generalization of fuzzy topological spaces. In 2001, Lee et al. [53] introduced the concept of smooth bitopological spaces as a generalization of smooth topological spaces and Kandil's fuzzy bitopological spaces. Thereafter, a large number of papers have been written to generalize fuzzy topological concepts to smooth bitopological setting. Ghanim et al. [30] in 2000, introduced the notion of supra smooth topology. Abbas [1] in 2002, generated the supra smooth topological space (X; 12) from a smooth bitopological space, as an extension of generated supra fuzzy topology in the sense of Kandil et al. [35]. This induced supra smooth topological space plays an important role in the researching of the study of smooth bitopological spaces, because in this case, smooth bitopological spaces studied by one supra smooth topology 12 such that i 12; i = 1; 2 and this is the easiest.