Separation Axioms in Intuitionistic Fuzzy Topological Spaces

Abstract

In 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].