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Item Compactness and Connectedness Concept in Intuitionistic Fuzzy Topological Spaces(University of Rajshahi, 2020-12) Mahbub, Md. Aman; Hossain, Md. Sahadat; Hossain, Mohd. AltabThe fundamental concept of a fuzzy set and fuzzy set operations was first introduced by L. A. Zadeh (Zadeh, 1965) in 1965 and it provides a natural foundation for treating mathematically the fuzzy phenomena, which exists pervasively in our real world and for building new branches of fuzzy mathematics. This also provides a natural frame work for generalizing various branches of mathematics such as fuzzy topology, fuzzy group, fuzzy rings, fuzzy vector spaces, fuzzy number, fuzzy system, fuzzy function, fuzzy relation, fuzzy logic and fuzzy computation. The concepts of fuzzy topology was introduced by C. L. Chang (Chang, 1968) in 1968 based on fuzzy set. Ming and Ming (Pao-Ming & Ying-Ming, 1980) (Pao-Ming & Ying-Ming, 1980), Khedr (Khedr et al., 2001), Hutton (Hutton, 1975), Azad (Azad, 1981), Ali (Ali, 1992) (Ali et al., 1990), Lowen (Lowen, 1976) etc. discussed various properties of fuzzy topology using fuzzy sets and fuzzy topology. Fuzzy compactness occupies a very important place in fuzzy topological spaces and so does some of its forms. Fuzzy compactness first discussed by C. L. Chang [C. L. Chang, Fuzzy Topological Spaces, J. Math. Anal. Appl., 24(1968), 182–190], T. E. Gantner et al. [T. E. Gantner, R. C. Steinlage and R. H. Warren, Compactness in Fuzzy Topological Spaces, J. Math. Anal. Appl., 62(1978), 547–562] introduced 𝛼 -compactness, A. D. Concilio and G. Gerla [A. D. Concilio and G. Gerla, Almost Compactness in Fuzzy Topological Spaces, Fuzzy Sets and Systems, 13(1984), 187–192] discussed almost compact spaces and M. N. Mukherjee and A. Bhattacharyya [M. N. Mukherjee and A. Bhattacharyya, 𝛼 -Almost Compactness for Crisp Subsets in a Fuzzy Topological Spaces, J. Fuzzy Math, 11(1) (2003), 105–113] discussed almost 𝛼 -compact spaces. After two decades, in 1983, Atanassov (K. T. Atanassov, “Intuitionistic Fuzzy Sets,” VII ITKR`s Session, (V, Sgurev, Ed.), Sofia (1983), Bulgaria) introduced the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets which looks more accurately to uncertainty quantification and provides the opportunity to precisely model the problem based on the existing knowledge and observations. An intuitionistic fuzzy set (A-IFS), developed by Atanassov (K. T. Atanassov, “Intuitionistic Fuzzy Sets,” Theory and Applications, Springer-Verlag (1999), Heidelberg, New York & K. T. Atanassov, “Intuitionistic Fuzzy Sets,” Fuzzy Sets and Systems (1986), vol. 20, 87 – 96) is a powerful tool to deal with vagueness. A prominent characteristic of A-IFS is that it assigns to each element a membership degree and a non-membership degree, and thus, A-IFS constitutes an extension of Zadeh’s fuzzy set. He added a new component (which determines the degree of non-membership) in the definition of fuzzy set. The fuzzy sets give the degree of membership of an element in a given set (and the non-membership degree equals one minus the degree of membership), while intuitionistic fuzzy sets give both a degree of membership and a degree of non-membership which are more-or-less independent from each other, the only requirement is that the sum of these two degrees is not greater than 1. In the last few years various concepts in fuzzy sets were extended to intuitionistic fuzzy sets. Intuitionistic fuzzy sets have been applied in a wide variety of fields including computer science, engineering, mathematics, medicine, chemistry and economics (K. P. Huber and M. R. Berthold, “Application of Fuzzy Graphs for Metamodeling”, Proceedings of the 2002 IEEE Conference, 640 644). In 1997, Coker [D. Coker, “An Introduction to Intuitionistic Fuzzy Topological Space,” Fuzzy Sets and Systems (1997), vol. 88, 81 –89] introduced the concept of intuitionistic fuzzy topological spaces. S. Bayhan and D. Coker, “On Fuzzy Separation Axioms in Intuitionistic Fuzzy Topological Space,” BUSEFAL (1996), vol. 67, 77 –87, D. Coker and A. Es. Hyder, “On Fuzzy Compactness in Intuitionistic Fuzzy Topological Spaces,” The Journal of Fuzzy Mathematics (1995), vol. 3, no. 4, 899 –909, S. Ozcag and D. Coker, “On Connectedness in Intuitionistic Fuzzy Special Topological Spaces,” Int. J. Math. Math. Sciences (1998), vol. 21, no. 1, 33 –40] gave some other concepts of intuitionistic fuzzy topological spaces, such as fuzzy continuity, fuzzy compactness, fuzzy connectedness, fuzzy Hausdorff space and separation axioms in intuitionistic fuzzy topological spaces. After this, many concepts in fuzzy topological spaces are being extended to intuitionistic fuzzy topological spaces. Recently many fuzzy topological concepts such as fuzzy compactness [D. Coker and A. Es. Hyder, “On Fuzzy Compactness in Intuitionistic Fuzzy Topological Spaces,” The Journal of Fuzzy Mathematics (1995), vol. 3, no. 4, 899 –909 ], fuzzy connectedness [N. Turanli and D. Coker, “Fuzzy Connectedness in Intuitionistic Fuzzy Topological Spaces,” Fuzzy Sets and Systems (2000), vol. 116, no. 3, 369 –375], fuzzy separation axioms[S. Bayhan and D. Coker, “On Separation Axioms in Intuitionistic Topological Space,” Int. J. of Math. Sci. (2001), vol. 27, no. 10, 621 –630], fuzzy continuity [H. Gurcay, D. Coker and A. Es. Hayder, “On Fuzzy Continuity in Intuitionistic Fuzzy Topological Spaces,” The Journal of Mathematics of Fuzzy Mathematics (1997), vol. 5, 365 –378], fuzzy g-closed sets[S. S. Thakur and Rekha Chaturvedi, “Generalized Closed Set in Intuitionistic Fuzzy Topology,” The Journal of Fuzzy Mathematics (2008), vol. 16, no. 3, 559 –572] and fuzzy g-continuity[S. S. Thakur and Rekha Chaturvedi, “Generalized Continuity in Intuitionistic Fuzzy Topological Spaces,” Notes on Intuitionistic Fuzzy Set (2006), vol. 12 no. 1, 38 –44] have been generalized for intuitionistic fuzzy topological spaces. Deschrijver and Kerre [G. Deschrijver and E. E. Kerre, “On the Relationship between Some Extensions of Fuzzy Set Theory,” Fuzzy Sets and Systems (2003), vol. 133, 227–235], Goguen [J. Goguen, “L-fuzzy Sets,” J. Math. Anal. Applicat. (1967), vol. 18, 145– 174] established the relationships between IFSs, L-fuzzy sets, interval-valued fuzzy sets, and interval-valued IFSs. Hausdorffness in an intuitionistic fuzzy topological space has been introduced earlier by Coker[D. Coker, “An Introduction to Intuitionistic Fuzzy Topological Space,” Fuzzy Sets and Systems (1997), vol. 88, 81 –89]. Lupianez [F. G. Lupianez. “Hausdorffness in Intuitionistic Fuzzy Topological Spaces,” Mathware and Soft Computing (2003), vol. 10, 17 –22] has also defined new notions of Hausdorffness in the intuitionistic fuzzy sense and obtained some new properties in particular in convergence. Separation axioms is very impotent in any kind of topological space. Bayhan and Coker (Bayhan & Coker, 1996) introduced fuzzy separation axioms in intuitionistic fuzzy topological spaces. Singh and Srivastava (Singh & Srivastava, 2012), Yue and Fang (Yue & Fang, 2006), Bhattacharjee and Bhaumik (Bhattacharjee & Bhaumik, 2012) also studied separation axioms in intuitionistic fuzzy topological spaces. The purpose of this thesis is to suggest new definitions of compactness and connectedness axioms in intuitionistic fuzzy topological spaces. We have studied several features of these definitions and the relations among them. We have also shown ‘good extension’ properties of all these spaces. Our criteria for definitions have been preserved as much as possible the relations between the corresponding separation properties for intuitionistic fuzzy topological spaces. The materials of this thesis have been divided into six chapters. A brief scenario of which we have presented as follows: Chapter one incorporates some of the basic definitions and results of general sets, fuzzy sets, intuitionistic sets, intuitionistic fuzzy sets and topologies based on such sets. In this chapter, subspace of topological space, product space and mapping in topological spaces x has been discussed, which are to be used as references for understanding the next chapters. Most of the results are quoted from various research papers and books. Our main works start from chapter two. In this chapter, we give seven new notions of intuitionistic fuzzy compact (in short, IF-Compact) space and investigate some relationship among them. At first we show that all these notions satisfy ‘good extension’ property. Furthermore, it proves that these intuitionistic fuzzy compact spaces are hereditary and productive. Finally, we observe that all concepts are preserved under one-one, onto and continuous mapping. In chapter three, we have introduced Q-compactness in intuitionistic fuzzy compact topological spaces. Furthermore, we have established some theorems and examples of Q-compactness in intuitionistic fuzzy topological spaces and discussed different characterizations of Q-compactness. Also we have defined 𝛿−𝑄 compactness, 𝑄−𝜎 compactness and 𝛿−𝑄−𝜎 compactness in intuitionistic fuzzy topological spaces and found different properties between Q-compactness and 𝛿−𝑄 compactness, 𝑄−𝜎 compactness and 𝛿−𝑄−𝜎 compactness in intuitionistic fuzzy topological spaces. In fourth chapter, we discusses various type of compactness in intuitionistic fuzzy topological spaces. Almost compact fuzzy sets was first constructed by Concilio and Gerla which is local property. Here we give wo new possible notions of almost compactness in intuitionistic fuzzy topological spaces are studied and investigated some of their properties. We show that these notions satisfy hereditary and productive property of intuitionistic fuzzy topological spaces. Under some conditions it is shown that image and preimage preserve intuitionistic fuzzy topological spaces. Also we give three new notions of 𝐼-compactness, 𝐶-compactness and 𝐼−𝐶-compactness in intuitionistic fuzzy topological spaces and investigate some relations between our notionss. At last we give three new notions of paracompactness and one new notion of 𝜎-compactness in intuitionistic fuzzy topological spaces and established some properties of them. In chapter five, we give some new notions of separated, connectedness and totally connectedness and one notions of 𝑇1-space in intuitionistic fuzzy topological space and investigate some relationship among them. Also we find a relation about classical topology and intuitionistic fuzzy topology. Further, we show that connectedness in intuitionistic fuzzy topological spaces are productive. In the chapter six, we have introduced (𝑟,𝑠)-connectedness in intuitionistic fuzzy topological spaces. Furthermore, we have established some theorems and examples of (𝑟,𝑠)-connectedness in intuitionistic fuzzy topological spaces and discussed different characterizations of (𝑟,𝑠)-connectedness.Item Separation Axioms in Intuitionistic Fuzzy Topological Spaces(University of Rajshahi, 2015) Ahmed, Estiaq; Hossain, Md. SahadatIn 1965, Zadeh[150] introduced to the world the term fuzzy set(FS), as a formalization of vagueness and partial truth, and represents a degree of membership for each member of the universe of discourse to a subset of it. This also provides a natural frame work for generalizing many branches of mathematics such as fuzzy rings, fuzzy vector spaces, fuzzy topology, fuzzy supra topology, fuzzy infra topology. Chang[26] introduced the concepts of fuzzy topological spaces by using fuzzy sets in 1968. Wong [144, 145], Lowen [74], Hutton[60, 61], Khedr[68], Ming and Ming [84, 85] etc. discussed various aspects of fuzzy topology using fuzzy sets. After this, there have been several generalizations of notions of fuzzy sets and fuzzy topology. In the frame work of fuzzifying topology, Shen[118] introduced T0-space, T1-space,T2-space, T3-space, T4-space separation axioms in fuzzifying topology. Khedr et. al.[68] introduced and studied the R0-space, R1-space separation axioms and found their relations with the T1-space, T2-space separation axioms respectively. After two decades, in 1983, Atanassov[13] introduced the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets which looks more accurately to uncertainty quantification and provides the opportunity to precisely model the problem based on the existing knowledge and observations. An intuitionistic fuzzy set (A-IFS), developed by Atanassov[11, 13] is a powerful tool to deal with vagueness. A prominent characteristic of A-IFS is that it assigns to each element a membership degree and a non- membership degree, and thus, A-IFS constitutes an extension of Zadeh’s fuzzy set. He added a new component (which determines the degree of non-membership) in the definition of fuzzy set. The fuzzy sets give the degree of membership of an element in a given set (and the non-membership degree equals one minus the degree of membership), while intuitionistic fuzzy sets give both a degree of membership and a degree of non-membership which are more-or-less independent from each other, the only requirement is that the sum of these two degrees is not greater than 1. In the last few years various concepts in fuzzy sets were extended to intuitionistic fuzzy sets. Intuitionistic fuzzy sets have been applied in a wide variety of fields including computer science, engineering, mathematics, medicine, chemistry and economics [58]. In 1997, Coker [29] introduced the concept of intuitionistic fuzzy topological spaces. Coker et. al.[19, 21, 32, 94] gave some other concepts of intuitionistic fuzzy topological spaces, such as fuzzy continuity, fuzzy compactness, fuzzy connectedness, fuzzy Hausdorff space and separation axioms in intuitionistic fuzzy topological spaces. After this, many concepts in fuzzy topological spaces are being extended to intuitionistic fuzzy topological spaces. Recently many fuzzy topological concepts such as fuzzy compactness[32], fuzzy connectedness[139], fuzzy separation axioms[20], fuzzy continuity[49], fuzzy g-closed sets[132] and fuzzy g-continuity[134] have been generalized for intuitionistic fuzzy topological spaces. Demirci[37]presented a Bernays-like axiomatic theory of IFSs involving five primitives and seven axioms. Bustince et. al.[23] defined some intuitionistic fuzzy generators and studied the existence of the equilibrium points and dual points. They presented different characterization theorems of intuitionistic fuzzy generators and a way of constructing IFSs from a fuzzy set and the intuitionistic fuzzy generators. Mondal and Samanta [88] introduced a concept of intuitionistic gradation of openness on fuzzy subsets of a non-empty set and also defined an intuitionistic fuzzy topological space. Deschrijver and Kerre[39], Goguen[47] established the relationships between IFSs, L-fuzzy sets, interval-valued fuzzy sets, and interval-valued IFSs. Bustince and Burillo[24], Deschrijver and Kerre[38] investigated the composition of intuitionistic fuzzy relations. Park[95] defined the notion of intuitionistic fuzzy metric spaces as a natural generalization of fuzzy metric spaces. Hausdorffness in an intuitionistic fuzzy topological space has been introduced earlier by Coker[29]. Lupianez[76] has also defined new notions of Hausdorffness in the intuitionistic fuzzy sense and obtained some new properties in particular in convergence. Bayhan and D. Coker[20] introduced fuzzy separation axioms in intuitionistic fuzzy topological spaces. Yue and Fang[149], considered the separation axioms T0, T1 and T2 in an intuitionistic fuzzy (I-fuzzy) topological space. Singh and Srivastava[120, 121] studied separation axioms and also studied - and -separation axioms in intuitionistic fuzzy topological spaces. Bhattacharjee and Bhaumik[22] has discussed pre-semi separation axioms in intuitionistic fuzzy topological spaces. The purpose of this thesis is to suggest new definitions of separation axioms in intuitionistic fuzzy topological spaces. We have studied several features of these definitions and the relations among them. We have also shown ‘good extension’ properties of all these spaces. Our criteria for definitions have been preserved as much as possible the relations between the corresponding separation properties for intuitionistic fuzzy topological spaces. Our aim is to develop the theories of intuitionistic fuzzy T0-spaces, intuitionistic fuzzy T1-spaces, intuitionistic fuzzy T2-spaces, intuitionistic fuzzy separation axioms. The materials of this thesis have been divided into six chapters. A brief scenario of which we have presented as follows: In first chapter, chapter one incorporates some of the basic definitions and results of fuzzy sets, intuitionistic sets, intuitionistic fuzzy sets, fuzzy topology, intuitionistic topology, intuitionistic fuzzy topology, fuzzy mappings, intuitionistic fuzzy mappings. These results are ready references for the work in the subsequent chapter. Our work starts from the second chapter. In second chapter, we have introduced T0-properties in intuitionistic fuzzy topological spaces and added seven definitions to this list. We have established the relations among them. We have shown that all these definitions satisfy ‘good extension’ property. It is also shown that these notions are hereditary and productive. We have studied some other properties of these concepts. This chapter is based on the Article [4]. In third chapter, we have studied T1-properties in intuitionistic fuzzy topological spaces and also adjoined seven definitions to this list. We have established the relations among them. We have shown that all these definitions satisfy ‘good extension’ property. As earlier we have found that all the definitions are hereditary and productive. Also we have studied some other properties of these concepts. This chapter is based on the Article [5]. In fourth chapter, we have studied T2-properties in intuitionistic fuzzy topological spaces. Seven definitions are introduced and the relations among them are established. All these definitions satisfy ‘good extension’ property. These definitions are hereditary and productive. Several other properties of these concepts are also studied. This chapter is based on the Article [6]. In fifth chapter, we have introduced R0-properties in intuitionistic fuzzy topological spaces. Here we have added seven definitions to this list and established the relations among them. All these definitions satisfy ‘good extension’ property. We have proved that all the definitions are hereditary. Also we have studied some other properties of these concepts. This chapter is based on the Article [1]. In sixth chapter, we have studied R1-properties in intuitionistic fuzzy topological spaces. Here we have adjoined seven definitions to this list and established the relations among them. All these definitions satisfy ‘good extension’ property. We have proved that all these definitions are hereditary and projective. Also we have studied some other properties of these concepts. This chapter is based on the Article [2].