Compactness and Connectedness Concept in Intuitionistic Fuzzy Topological Spaces

Abstract

The 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.