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Abstract Since the inception of fuzzy set theory [ see Zadeh (1965) and Goguen (1967)], the efforts of many researches have been directed to develop fuzzy analogues of basic concepts of classical mathematics and to work out correspoinding theories. Topological structures on lattices of fuzzy sets were first considered in 1968 by Chang, later Chang’s ideas were developed in essentially different directions by [Goguen (1973), Wong (1973), Lowen (1976) and (1977), Warren (1978) and (1979), Hutton (1980), Rodabaugh (1980) and (1991), Kerre (1989) and (1991), etc]. It is easy to see that they have always investigated fuzzy objects with crisp methods. Fuzziness in the concept of openness of a fuzzy set has not been considered, which seems to be a drawback in the process of fuzzification of the concept of topological spaces. By the ends of eighties and beginning of nineties many mathematician remarked that the fuzziness in these extensions is not enough, since we handle with fuzzy subsets but the handling is crisp. For this reason many mathematicians try to make a fuzzy treatment for this structures, fuzzification of openness was first initiated by [H¨ohle(1980), Kubiak (1985) and ˇSostak (1985)]. In 1991, from a logical point of view, Ying (1991) studied H¨ohle’s topology and called it fuzzifying topology. This fuzzy topology is an extension of both crisp topology and Chang’s fuzzy topology and each fuzzy subset has a degree of openness, in the sense that not only the objects are fuzzified, but also the axiomatics, i.e., this topology makes a fuzzy set be open to some extend, that is to say the open property becomes fuzzy. [ˇSostak (1989a), (1989b), (1989c), (1990) and (1996)], gave some rules and showed how such an extension can be realized. In (1992), fuzzy topological spaces in ˇSostak sense Typeset by AMS-TEX 1 -2- were independently redefined by [ Ramadan (1992)], under the name of smooth topological spaces using lattices. It has been developed in many directions [Hazra et., al., (1992), Chattopadhyay and Samanta (1993), El-Gayyar et. al., (1994), H¨ohle and ˇSostak (1995), Demirci (1997), Zhang (1999) and (2002), Kotze (2003), Kubiak and ˇSostak (2004), Fang Jinming (2006)]. Attanassov (1986) introduced the idea of intuitionistic fuzzy set. ¸Coker and coworker [(1996), (1997) ] introduced the idea of the topology of intuitionistic fuzzy sets. Recently, Samanta and Mondal (2002) introduced the notion of intuitionistic gradation of openness which a generalization of both of fuzzy topological spaces [ˇSostak (1986)] and the topology of intuitionistic fuzzy sets [ ¸Coker and Demirci (1996), ¸Coker (1997)]. Our main object is to investigate more further the structure of the intuitionistic supra gradation of openness and intuitionistic gradation of openness when some information are known about their fuzzy structure. This thesis includes five chapters: Chapter I, is of introductory nature, providing the reader with results concerning, fuzzy sets, intuitionistic fuzzy sets, fuzzy topologies, intuitionistic fuzzy topologies and intuitionistic gradation of openness. Chapter II, we introduce the concept of ˘C-IF closure spaces which is a generalization of the IF-closure space introduced by [Kim and Ko (2004)]. Also, we introduced some separation axioms in ˘C-IF closure spaces. Finally, we study the notion of IF-closure systems and study many important properties of ˘C-IF closure spaces and IF-closure systems. Chapter III, we introduce and study the concept of G-closure -3- operator induced by an intuitionistic fuzzy topological space in view of the definition of Samanta and Mondal. We show that it is an IF-closure operator. Furthermore, it induces an intuitionistic gradation of openness which is finer than a given intuitionistic gradation of openness. We investigate some properties of GIF-closure operators. Also, we define (r; s)-generalized fuzzy (semi, weakly semi) closed sets in an intuitionistic fuzzy topological spaces. Moreover, We investigate some properties of GIFsemicontinuous mapping. Chapter IV, we shall give various characterizations of (r; s)- fuzzy regularity and (r; s)-fuzzy almost regularity with the help of quasi-neighbourhood [Pu and Liu (1980)], µ-closure and ±-closure operators [Kim and Park (2000)]. Also, we give some properties of (r; s)-T2 space. Chapter V, we have used the intuitionistic supra gradation of openness which created from an intutionistic fuzzy bitopological spaces to introduce and study the concepts of continuity, some kinds of separation axioms and compactness. Most of the results of this thesis either have been accepted or submitted for publications as follows: (1) Intuitionistic supra gradation of openness, Applied Mathematics and Information Sciences, Vol. 2, No. 3, (2008), 291-307. (2) Intuitionistic fuzzy G-closure operators, International Review of Fuzzy Mathematics (IRFM), Vol. 3, No. 1, (2008), 37-53. (3) Several types of intuitionistic fuzzy semiclosed sets, J. Fuzzy Mathematics (submitted) (4) Some properties of (r; s)¡T2 spaces, Journal of the Egyptian Mathematical Society (submitted) (5) Characterizations of some IF separation axioms, J. Fuzzy Mathematics (submitted) -4- (6) IF-closure systems and IF-closure operators, The 22nd International Conference on Topology and its Appications [Helwan 7-8 july (2008)], and submitted for ”Journal of the Egyptian Mathematical Society ” |