تئوری مجموعه های راف برای فضاهای توپولوژیک

The topology induced by binary relations is used to generalize the basic rough set concepts. The suggested topological structure opens up the way for applying rich amount of topological facts and methods in the process of granular computing, in particular, the notion of topological membership functions is introduced that integrates the concept of rough and fuzzy sets.

The concept of topological structures [3] and their generalizations are one of the most powerful notions in system analysis. Many works have appeared recently for example in structural analysis [4], in chemistry [17], and physics [1]. The purpose of the present work is to put a starting point for the applications of abstract topological theory into fuzzy set theory, granular computing and rough set analysis. Fuzzy set theory appeared for the first time in 1965, in famous paper by Zadeh [19]. Since then a lot of fuzzy mathematics have been developed and applied to uncertainty reasoning. In this theory, concepts like fuzzy set, fuzzy subset, and fuzzy equality (between two fuzzy sets) are usually depend on the concept of numerical grades of membership. On the other hand, rough set theory, introduced by Pawlak in 1982 [13], is a mathematical tool that supports also the uncertainty reasoning but qualitatively. Their relationships have been studied in [11,12,14,18]. In this paper, we will integrate these ideas in terms of concepts in topology. Topology is a branch of mathematics, whose concepts exist not only in almost all branches of mathematics, but also in many real life applications. We believe topologica

In this work, we generalize rough set theory in the frameworks of topological spaces. We believe such generalized rough set theory will be useful in digital topology [16] as well as biomathematics [17]. Our approach in essence is to topologize information tables (also known as information systems). Our theory connects rough sets, topological spaces, fuzzy sets, and neighborhood systems (binary relations, pretopology). This theory brings in all these techniques to information analysis and knowledge processing. We believe that topological structure is the appropriate umbrella
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