Chapter 1. Introduction

Transcription

1 Chapter 1 Introduction 1.1 Fuzzy Sets and Intuitionistic Fuzzy Sets Most of our traditional tools modeling, reasoning and computing are crisp, deterministic and precise in character. By crisp we mean dichotomies, that is, yes-or-no type rather than more-or-less type. Certainty eventually indicates that we assume the structures and parameters of the model to be definitely known and that there are no doubts about their values or their occurrence. For factual model or modeling languages two major complications arise: (i) Real situations are very often not crisp and deterministic and they cannot be described precisely. (ii) The complete description of a real system often would require by far more detailed data than a human being could ever recognize, process and understand simultaneously. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition- an element either belongs or does not belong to the set. 1

2 In 1923, the philosopher Russell [58] referred to the first point when he wrote: All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life but only to an imagined celestial existence. Zadeh [72] referred to the second point when he wrote: As the complexity of a system increases, our ability to make precise and yet significant statement about its behaviour diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics. Let us consider characteristic features of real-world systems again: Real situations are very often uncertain or vague in a number of ways. Due to lack of information the future state of the system might not be known completely. This type of uncertainty (stochastic character) has long been handled appropriately by probability theory and statistics. This kolmogoroff type probability is essentially frequent and is based on set-theoretic considerations. Koopman s probability refers to the truth of statement and therefore it is based on logic. On both types of probabilistic approaches it is assumed, however, that the events ( elements of sets ) or the statements, respectively, are well defined. We shall call this type of uncertainty or vagueness stochastic uncertainty by contrast to the vagueness concerning the description of the semantic meaning of the events, phenomena or statement themselves, 2

3 which we shall call fuzziness. Fuzziness can be found in many areas of daily life, such as in Engineering (Blockley [18]), in Medicine (Vila and Delgado [69]), in Meteorology ( Cao and Chen [22] ), in Manufacturing ( Mamdani [48] ) and others. It is particularly frequent, however, the meaning of a word might even be well defined, but when using the word as a label for a set, the boundaries within which objects do or do not belong to the set become fuzzy or vague. Zadeh [72] writes: The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the fields of pattern classification and information processing. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision in the absence of sharply defined criteria of class membership rather than the presence of random variables. Fuzzy set theory provides a strict mathematical framework in which vague conceptual phenomena can be precisely and rigorously studied. Fuzziness has so far not been defined uniquely semantically and probably never will. It will mean different things, depending on the application area and the way it is measured. 3

4 In the meantime, numerous authors have contributed to this theory. Fuzzy sets are having useful and interesting applications in various fields including Probability theory, Information theory [59] and Control [61]. As an extension of classical set theory, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. The theory of intuitionistic fuzzy sets further extends both the concepts by allowing the assessment of the elements by two functions: for membership and for non-membership, which belong to the real unit interval [0, 1] and whose sum belongs to the same interval, as well. Intuitionistic fuzzy sets generalize fuzzy sets, since the indicator functions of fuzzy sets are special cases of the membership and the nonmembership functions and of intuitionistic fuzzy sets, in the case when the strict equality exists: i.e. the non-membership function fully complements the membership function to 1, not leaving room for any uncertainty. Intuitionistic fuzzy sets can be more precisely expressed. For example, the fact that the temperature of a patient changes, and other symptoms are not quite clear. There is a fair chance of the existence of a non-null hesitation part at each moment of evaluation of an unknown object. Intuitionistic fuzzy sets as a generalization of fuzzy sets can be useful in situations when description of a problem by a (fuzzy) linguistic variable, 4

5 given in terms of a membership function only, seems too rough. For example, in decision making problems, particularly in the case of medical diagnosis, sales analysis, new product marketing, financial services, etc. 1.2 Review Of Literature The concept of a fuzzy set provides a natural framework for generalizing many of the concepts of a general topology. The theory of fuzzy topological spaces was introduced and developed by Chang [24]. Since then various notions in classical topology have been extended to fuzzy topological spaces by fuzzy topologists like Azad [10], Zadeh [72], Tuna Hatice Yalvac [67,68], Brown [20], Hutton And Reilly [40], Lowen [45,46], Warren [70], Friedler [33], Gantner et. al. [34], Arya [3], Bin Shahna [16], Ceck [23], Hewitt [39] and Necla Turanli [51]. Various departures from the classical topology have also been observed in fuzzy topology. The field of Mathematical Sciences which goes under the name of topology is concerned with all questions directly or indirectly related to continuity. Fuzzy topologists have introduced and investigated many different types of fuzzy continuity. After the introduction of the concept of fuzzy sets by Zadeh [72] several researches were conducted on the generalizations of the notion of fuzzy set. The concept of intuitionistic fuzzy set was first published by Atanassov [4] and many works by the same author and his colleagues appeared in the literature [5,6,7,8]. Later topological structures in fuzzy 5

6 topological spaces [24] is generalized to intuitionistic fuzzy topological spaces by Coker in [30], and then the concept of intuitionistic set is introduced by Coker in [31]. Later this concept was generalized to intuitionistic L-fuzzy sets by Atanassov And Stoeva [6]. This concept is the discrete form of intuitionistic fuzzy set, and it is one of several ways of introducing vagueness in mathematical objects. Levine [43], Mashhour et. al.[49] and Njastad [52] introduced semi-open sets, preopen sets and α -sets respectively. The concept of β -open sets in topological spaces was introduced by Abd. El-Monsef et al [1]. The concept of intuitionistic fuzzy g-closed sets was introduced by Thakur and Rekha Chaturvedi [64]. Thangaraj and Balasubramanian [65] introduced and studied the concepts of fuzzy basically disconnected spaces. The concepts of fuzzy Hausdorff topological spaces was first introduced and studied by Rekha Srivastava et.al.[54]. The concepts of fuzzy S- closed spaces was studied by Coker [27]. Fuzzy topologists have introduced and investigated many different generalizations of fuzzy continuous functions. The concept of fuzzy G δ set was first introduced and studied by Balasubramanian [13]. The concepts of connectedness and disconnectedness in fuzzy topological spaces have been studied by Balasubramanian [12,14]. Roja, Uma and Balasubramanian [55,56,57] studied the concepts of continuity, connectedness and disconnectedness by using fuzzy G δ sets. The concept of semi θ continuity in an intuitionistic fuzzy topological spaces was introduced and studied by Hanafy, Abd El-Aziz and Salman [38]. 6

7 Lowen et. al.[47] introduced and developed the concept of fuzzy con- vergence spaces. Geetha [35] established the concept of F-semigroup compactifications. Sostak [60] introduced the concepts of L-fuzzy quasi uniformity. He also studied several properties of L-fuzzy quasi uni- form space and the topology generated by L-fuzzy quasi uniformity. Cocker [28,29,30] introduced and studied various concepts like continuity, compactness, connectedness and disconnectedness in intuitionistic fuzzy topological spaces.the concept of fuzzy faintly continuous functions and fuzzy faintly α -continuous functions are introduced and studied by Anjan Mukherjee [9]. The concepts of bicompact extension of topological spaces was introduced and developed by Aleksandrov [2]. The concepts of extensions of topologies were introduced and developed by Borges [19]. Balasubramanian [11] fuzzified the concept of extensions of topological spaces. Roja, Uma and Balasubramanian [55] extended the concept of extensions of fuzzy topological spaces to fuzzy bitopological spaces. The concept of closure spaces was introduced and studied by Cech [23]. The notion of fuzzy closure spaces was first introduced and developed by Mashhour and Ghanim [50]. Chawalit Boonpok [26] introduced the concept of biclosure spaces. Tapi and Navalakhe [62,63] have introduced the concept of fuzzy biclosure spaces. The generalisation of fuzzy continuous functions was studied by Balasubramanian and Sundaram [14]. The concepts of g-closed sets was introduced by Levine [44]. The extension principle of closure operator is studied by Biacino [17]. 7

8 1.3 Outline of the Thesis This section presents a chapterwise summary of results obtained on intuitionistic fuzzy regular G δ sets, intuitionistic fuzzy regular G δ continuous functions, intuitionistic fuzzy regular semi (resp., pre, α and β ) open sets, intuitionistic fuzzy regular G δ compact (resp., connected and normal) spaces, intuitionistic fuzzy regular G δ T 1/2 spaces, intuitionis- tic fuzzy convergence spaces, intuitionistic fuzzy convergence topological spaces, intuitionistic fuzzy convergence topological semigroups, intuitionistic fuzzy Bohr lim compactifications, intuitionistic fuzzy lim semi group compactifications, intuitionistic fuzzy quasi uniform topological spaces, intuitionistic fuzzy quasi uniform regular G δ sets, intuitionistic fuzzy quasi uniform faintly regular G δ continuous functions, intuitionistic fuzzy quasi uniform regular G δ compact spaces, simple extension of intuitionistic fuzzy quasi uniform bitopological spaces, pairwise intuitionistic fuzzy quasi uniform connected spaces, pairwise intuitionistic fuzzy quasi uniform disconnected spaces, pairwise intuitionistic fuzzy quasi uniform regular G δ basically disconnected spaces, pairwise intuitionistic fuzzy quasi uniform extremally disconnected spaces, pairwise intuitionistic fuzzy quasi uniform T i (i = 0, 1 and 2) spaces, intuitionistic fuzzy closure spaces, intuitionistic fuzzy biclosure spaces, intuitionistic fuzzy C -open (resp., closed) sets and intuitionistic fuzzy G C Hausdorff (resp., normal and regular) biclosure spaces. In chapter 2, the concept of intuitionistic fuzzy regular G δ sets is 8

9 introduced. Some interesting properties are studied. Interesting interrelations are established. Counter examples are given wherever necessary. Also the concept of intuitionistic fuzzy regular G δ continuous functions is introduced. Interesting properties are studied. The concept of intuitionistic fuzzy regular G δ compact (resp., connected and normal) spaces is introduced and studied. In chapter 3, the concept of intuitionistic fuzzy convergence topological spaces is introduced by using the limit function (in short, lim). The concept of intuitionistic fuzzy convergence topological semigroups is introduced and studied. Besides by providing several propositions, intuitionistic fuzzy Bohr lim compactifications and intuitionistic fuzzy lim semigroup compactifications are established. The main motive of chapter 4 is to introduce the concept of intuitionistic fuzzy quasi uniform topological spaces and to discuss the interesting properties of intuitionistic fuzzy quasi uniform faintly regular G δ continuous functions. Some interesting interrelations are established with suitable examples. Further, the concept of intuitionistic fuzzy quasi uniform regular G δ compact spaces is introduced. In this connection, interrelations are studied. Chapter 5 deals with the extension of intuitionistic fuzzy quasi uniform bitopological spaces. Some interesting characterization of pairwise intuitionistic fuzzy quasi uniform connected spaces are established. Properties of pairwise intuitionistic fuzzy quasi uniform compact (resp., 9

10 Lindelof ) spaces are studied. In this connection, separation axioms are established. In chapter 6, the concept of intuitionistic fuzzy closure spaces is introduced. Some interesting properties are studied. The concept of intuitionistic fuzzy biclosure spaces is introduced. Several properties are established. The concept of intuitionistic fuzzy G C Hausdorff (resp., normal and regular) spaces are introduced and studied. 1.4 Basic Concepts In this scetion, some basic concepts of topology, intuitionistic topology, fuzzy topology and intuitionistic fuzzy topology have been recalled. Also, related results, propositions and important theorems are collected from various research articles. Throughout this thesis X be a non empty set, I = [0, 1] and ζ X is the set of all intuitionistic fuzzy sets. Definition [24] Let X be a non-empty set and I be the unit interval. A fuzzy set in X is an element of the set I X of all functions from X to I. Definition [24] Let (X, T ) be any fuzzy topological space. The characteristic function 10

11 of a subset A of X is denoted by χ A and defined as 0 if x ƒ A χ A (x) = 1 if x ƒ A Definition [4] Let X be a non empty fixed set and I the closed interval [ 0, 1 ]. An intuitionistic fuzzy set ( IFS ) A is an object of the following form A = {(x, µ A (x), γ A (x)) : x X } where the functions µ A : X I and γ A : X I denote the degree of membership ( namely µ A (x) ) and the degree of non membership ( namely γ A (x) ) for each element x X to the set A respectively and 0 µ A (x) + γ A (x) 1 for each x X. Obviously, every fuzzy set A on a nonempty set X is an IFS of the following form A = {(x, µ A (x), 1 µ A (x)) : x X }. For the sake of simplicity, we shall use the symbol A = (x, µ A (x), γ A (x)) for the intuitionistic fuzzy set A = {(x, µ A (x), γ A (x) : x X )}. For a given non empty set X, denote the family of all intuitionistic fuzzy sets in X by the symbol ζ X. Definition [4] Let A and B be the intuitionistic fuzzy sets of the form A = {(x, µ A (x), γ A (x)) : x X } and B = {(x, µ B (x), γ B (x)) : x X }. Then (i) A B if and only if µ A (x) µ B (x) and γ A (x) γ B (x). (ii) A = {(x, γ A (x), µ A (x)) : x X }. (iii) A B = {(x, µ A (x) µ B (x), γ A (x) γ B (x)) : x X }. 11

12 (iv) A B = {(x, µ A (x) µ B (x), γ A (x) γ B (x)) : x X }. (v) A = B if and only if A B and B A. (vi) [ ]A = {(x, µ A (x), 1 µ A (x)) : x X }. (vii) ( )A = {(x, 1 γ A (x), γ A (x)) : x X }. Definition [30] The intuitionistic fuzzy sets 0 and 1 are defined by 0 = {(x, 0, 1) : x X } and 1 = {(x, 1, 0) : x X } Definition [30] Let {A i /i J } be an arbitrary family of intuitionistic fuzzy sets in X. Then (i) T A i = {(x, V µ A (x), W γ A (x)) : x X }. (ii) S A i = {(x, W µ A (x), V γ A (x)) : x X }. Corollary [30] Let A, B and C be intuitionistic fuzzy sets in X. Then (i) A B and C D A C B D and A C B D. (ii) A B and A C A B C. (iii) A C and B C A B C. (iv) A B and B C A B C. (v) A B and B C A C. 12

13 (vi) A B = A B. (vii) A B = A B. (viii) A B B A. (ix) A = A. (x) 1 = 0. (xi) 0 = 1. Definition [30] Let X and Y be two non empty sets and f : X Y be a function. (i) If B = {(y, µ B (y), γ B (y)) : y Y } is an intuitionistic fuzzy set in Y, then the inverse image of B under f is an IFS defined by f 1 (B) = {(x, f 1 (µ B )(x), f 1 (γ B )(x)) : x X }. (ii) If A = {(x, λ A (x), ϑ A (x)) : x X } is an intuitionistic fuzzy set in X. Then the image of A under f is an IFS defined by f (A) = {(y, f (λ A )(y), (1 f (1 ϑ A ))(y)) : y Y } where 1 supx f 1 (y)λ A (x) f (y) = 0 f (λ A )(y) = 0 otherwise 1 infx f 1 (y)ϑ A (y) f (y) = 0 (1 f (1 ϑ A ))(y) = 0 otherwise Definition [30] Let A, A i (i J ) be IFSs in X. B, B j (j K) IFSs in Y and f : X Y be a function. Then 13

14 (i) A 1 A 2 f (A 1 ) f (A 2 ). (ii) B 1 B 2 f 1 (B 1 ) f 1 (B 2 ). (iii) A f 1 (f(a)) {If f is injective, then A = f 1 (f(a))}. (iv) f (f 1 (B)) B {If f is surjective, then A = f (f 1 (A))}. (v) f 1 ( S B j ) = S f 1 (B j ). (vi) f 1 ( T B j ) = T f 1 (B j ). (vii) f ( S A j ) = S f (A j ). (viii) f ( T A j ) T f (A j ). {If f is injective, then f ( T A j ) = T f (A j )}. (ix) f 1 (1 ) = 1. (x) f 1 (0 ) = 0. (xi) f (1 ) = 1, if f is surjective. (xii) f (0 ) = 0. (xiii) f (A) f (A), if f is surjective. (xiv) f 1 (B) f 1 (B). Definition [30] An intuitionistic fuzzy topology ( IFT ) in Coker s sense on a non empty set X is a family τ of IFSs in X satisfying the following axioms. (i) 0, 1 τ. 14

15 (ii) G 1 G 2 τ for any G 1, G 2 τ. (iii) G i τ for arbitrary family {G i /i I } τ. In this case the pair (X, τ ) is called an intuitionistic fuzzy topological space (IFTS for short) and any IFS in τ is known as an intuitionistic fuzzy open set (IFOS for short) in X. The complement A of an IFOS A in X is called an intuitionistic fuzzy closed set (IFCS for short) in X. Definition [30] Let (X, τ ) be an IFTS and A = (x, µ A, γ A ) be an IFS in X. Then the fuzzy interior and fuzzy closure of A are defined by int A = {G / G is an IFOS in X and G A}. cl A = {G / G is an IFCS in X and G A}. Proposition [30] For any IFS A in ( X, τ ) we have (i) cl(a) = int(a). (ii) int(a) = cl(a). Definition [30] Let (X, τ ) be an IFTS. cl(a) is an IFCS and int(a) is an IFOS in X. Then (i) A is an IFCS in X iff cl(a) = A. 15

16 (ii) A is an IFOS in X iff int(a) = A. Proposition [30] Let (X, τ ) be an IFTS and A, B be IFSs in X. Then the following properties hold. (i) int(a) A. (ii) A cl(a). (iii) A B int(a) int(b). (iv) A B cl(a) cl(b). (v) int(int(a)) = int(a). (vi) cl(cl(a)) = cl(a). (vii) int(a B) = int(a) int(b). (viii) cl(a B) = cl(a) cl(b). (ix) int(1 ) = 1. (x) cl(0 ) = 0. Definition [30] A function f : X Y from an IFTS X into an IFTS Y is called an intuitionistic fuzzy continuous if f 1 (B) is an IFOS in X, for each IFOS in Y. 16

17 Definition [30] Let (X, τ ) and (Y, Φ) be two IFTSs and let f : X Y be a function. Then f is said to be fuzzy open iff the image of each IFS in τ is an IFS in Φ. Definition [9] Let f : (X, T ) (Y, S) be a function. The graph g : X X Y of f is defined by g(x) = (x, f (x)) x X. Definition [37] Let X, Y be non empty sets and U = {(x, λ U (x), ϑ U (x)) : x X }, V = {(y, λ V (y), ϑ V (y)) : y Y } IFSs of X and Y respectively. Then U V is an IFS of X Y defined by: (U V ) = (x, y), min(µ U (x), µ V (y)), max(γ U (x), γ V (y))). Definition [36,42] Let A be an IFS of an IFTS X. Then A is called an (i) intuitionistic fuzzy regular open set (IFROS) if A = int(cl(a)). (ii) intuitionistic fuzzy semi open set (IFSOS) if A cl(int(a)). (iii) intuitionistic fuzzy β open set (IF β OS) if A cl(int(cl(a))). (iv) intuitionistic fuzzy preopen set (IFPOS) if A int(cl(a)). (v) intuitionistic fuzzy α open set (IF α OS) if A int(cl(int(a))). 17

18 Definition [36,42] Let A be an IFS in IFTS.Then (i) βint(a) = {G / G is an IF β OS in X and G A} is called an intuitionistic fuzzy β -interior of A. (ii) βcl(a) = {G / G is an IF β CS in X and G A} is called an intuitionistic fuzzy β -closure of A. (iii) int s (A) = {G / G is an IFSOS in X and G A} is called an intuitionistic fuzzy semi interior of A. (iv) cl s (A) = {G / G is an IFCS in X and G A} is called an intuitionistic fuzzy semi closure of A. Definition [56] A fuzzy topological space ( X, T ) is said to be fuzzy β T 1/2 space if every gf β - closed set in ( X,T) is fuzzy closd in (X, T ). Definition [13] Let ( X, T ) be a fuzzy topological space and λ be a fuzzy set in X. λ is called fuzzy G δ set if λ = V i= λ i where each λ i T. i=1 The complement of a fuzzy G δ is a fuzzy F σ set. Definition [38] A function f : (X, Ψ) (Y, Φ) is said to be intuitionistic fuzzy semi θ - continuous if for each IFP c(a, b) in X and V N sq (f (c(a, b))), there exists U N θq (c(a, b)) such that f (U) V. 18

19 Definition [66] Let S be a non empty set and be a binary operation on S. The algebraic system (S, ) is said to be a semigroup if the operation is associative. Definition [66] Let (S, ) be a semigroup and T S. If the set T is closed under the operation, then (T, ) is said to be a subsemigroup of (S, ). Definition [66] Let (S, ) and (Y, ) be two algebraic systems of the same type in the sense that both and x are binary operations. A function g : X Y is said to be a homomorphism from (S, ) and (Y, ) if for any x 1, x 2 X, g(x 1 x 2 ) = g(x 1 ) x g(x 2 ). Definition [66] Let g be a homomorphism from (X, ) to (Y, ). If g : X Y is one to one and onto then g is said to be an isomorphism. Definition [66] A semigroup is a non empty set S together with an associative multiplication (x, y) xy from S S into S. If S has a Hausdorff topology such that (x, y) xy is continuous with the product topology on S S, then S is said to be topological semigroup. 19

20 Definition [47] Given a set X, the pair (X, lim) is said to be a fuzzy convergence space, where lim : F(X ) I X, provided: (i) limf = inf G Pm (F)limG, F F(X ). (ii) limf c(f), F F P (X). (iii) limg limf when F G, F, G F P (X). (iv) limα1 x α1 x for 0 < α 1 and x X. Definition [71] Let (X, F ) be a fuzzy topological space, x λ, 0 < λ 1, be a fuzzy point and P a closed fuzzy set in X. Then P is called a remoted neighbourhood (R-nbd) of x λ if x ƒ P. Definition [71] A fuzzy net S is a function S : D T where D is a directed set with order relation and T the collection of all fuzzy points in X. Then for each n D, S(n) is a fuzzy point belonging to T. Let its value in (0, 1] be denoted by λ n. Thus we get a crisp net V (S) = {λ n : n D} in the half open interval (0, 1]. If V (S) converges to α (0, 1] we say V (S) is an α net. Definition [35] Let X be a semigroup and F a T 2 fuzzy topology on X. Then (X, F ) is called a fuzzy topological semi group (FTSG) if for all x, y X, 20

21 the function α : (x, y) xy of (X, F ) (X, F ) into (X, F ) is fuzzy continuous. Definition [35] Let X be a FTSG. The Bohr fuzzy compactification of X is a pair (α, Y ) such that Y is a compact FTSG. α : X Y is a F- morphism provided whenever β : X Z is an F-morphism of X into a compact FTSG Z, then there exist a unique F-morphism f : Y Z such that f α = β. Definition [35] Let X denote an FTSG. By an F-semigroup compactification of X we mean an ordered pair (α, A), where A is a compact FTSG and α : X A is a dense F-marphism. Definition [53] Let a and b be two real numbers in [0,1] satisfying the inequality a + b 1. Then the pair (a, b) is called an intuitionistic fuzzy pair. Let (a 1, b 1 ),(a 2, b 2 ) be any two intuitionistic fuzzy pairs. Then define (i) (a 1, b 1 ) (a 2, b 2 ) if and only if a 1 a 2 and b 1 b 2. (ii) (a 1, b 1 ) = (a 2, b 2 ) if and only if a 1 = a 2 and b 1 = b 2. (iii) If {(a i, b i /i J )} is a family of intuitionistic fuzzy pairs, then (a i, b i ) = ( a i, b i ) and (a i, b i ) = ( a i, b i ). (iv) The complement of an intuitionistic fuzzy pair (a, b) is the intuitionistic fuzzy pair defined by (a, b). = (b, a) 21

22 (v) 1 = (1, 0) and 0 = (0, 1). Definition [12] A fuzzy topological space X is said to be fuzzy extremally disconnected if the closure of every fuzzy open set in X is fuzzy open in X. Definition [28] (i) Let X be a non empty set. If r I 0, s I 1 are fixed real number, such that r + s 1 then the intuitionistic fuzzy set x r,s is called an intuitionistic fuzzy point (IFP for short) in X given by (r, s) if x = xp x r,s (x p ) = (0, 1) if x = xp for x p X is called the support of x r,s where r denotes the degree of membership value and s is the degree of non membership value of x r,s (ii) An intuitionistic fuzzy point x r,s is said to belong to an intuitionistic fuzzy set A if r µ A (x) and s γ A (x). (iii) An intuitionistic fuzzy singleton x r,s X is an intuitionistic fuzzy set in X taking value r (0, 1] and s [0, 1) at x and 0 elsewhere. (iv) Two intuitionistic fuzzy point or intuitionistic fuzzy singleton are said to be distinct if their support are distinct. 22

23 Definition [71] Let (X, F ) be a fuzzy topological space. A fuzzy set A in X is called N-compact if each α net, 0 < α 1, contained in A has at least a cluster point x α with value α. When A is equal to the constant fuzzy set 1 and is N-compact, we call (X, F ) an N compact fuzzy topological group. Definition [60] Let L be a completely distributive lattice and X be a set. Let D denote the family of function U : L X L X such that (i) M U (M ), for each M L X. (ii) U ( M j ) = j U (M j ) for each family {M j /j J } L X. A L-fuzzy quasi uniformity on X can now be defined as a subset U D satisfying the following axioms. (i) If U U, U V and U D, then V U. (ii) If U 1, U 2 U, then there exists V U such that V U 1 U 2. (iii) For every U U there exists V U such that V V U. A L-fuzzy quasi uniformity is called an L-fuzzy uniformity, if U U implies U 1 U (U 1 : L X L X is defined by U 1 (N ) = {M/U (M c ) N c }). The pair (X, U ) is called a Hutton L-fuzzy (quasi) uniform space. 23

24 Definition [60] Every L-fuzzy quasi uniformity generates an L-fuzzy topology. Let (X, U ) be an L-fuzzy quasi uniform space. Then the operator I nt : L X L X defined by I nt(m ) = {N L X /U(N ) M for some U U } is the fuzzy interior operator. The corresponding L-fuzzy topology on X, τ U = {M L X /I nt(m ) = M } is called generated topology by U. Definition [28] Let A and B be intuitionistic fuzzy set in an intuitionistic fuzzy topological space (X, T ). Then A is said to be quassi coincident with B denoted by AqB if there exists an x X such that γ B (x) < µ A (x) or µ A (x) > γ B (x). Definition [28] An intuitionistic fuzzy point x r,s in X is said to be quassi coincident with an intuitionistic fuzzy set A, denoted by x r,s qa if and only if r > γ A (x) or s < µ A (x). Definition [21] A fuzzy point x r in an fts X is said to be fuzzy fc- accumulation point of a fuzzy filterbase B if for each fuzzy feebly open feeble quassi-nbd U of x r and for each B B, BqF cl(u ). Definition [30] Let ( X, T ) be an intuitionistic fuzzy topological space.then ( X, T ) is 24

25 said to be an intuitionistic fuzzy compact space if for every intuitionistic fuzzy open cover {V i /i J } of ( X, T ) there exists a finite subset J 0 of J such that S {V i /i J 0 } = 1. Definition [29] Let (X, T ) be an intuitionistic fuzzy topological space. Then (X, T ) is said to be an intuitionistic fuzzy almost compact space if for every intuitionistic fuzzy open cover {V i /i J } of ( X, T ) there exists a finite subset J 0 of J such that S {I F cl(v i )/i J 0 } = 1. Definition [30] Let (X, T ) be an intuitionistic fuzzy topological space. Then X is said to be a fuzzy (i) C 5 - disconnected space if there exists an intuitionistic fuzzy open set and intuitionistic fuzzy open set G such that G = 1 and G = 0. (ii) C 5 - connected space if it is not an intuitionistic fuzzy C 5 - disconnected space. Definition [15] A fuzzy topological space (X, τ 1, τ 2 ) is said to be pairwise fuzzy basically disconnected space if τ 1 - closure of each τ 2 fuzzy open, τ 2 fuzzy F σ is τ 2 fuzzy open and τ 2 - closure of each τ 1 fuzzy open, τ 1 fuzzy F σ is τ 1 fuzzy open. 25

26 Definition [25] A fuzzy topological space (X, τ 1, τ 2 ) is said to be pairwise fuzzy extremally disconnected space τ 1 - closure of each τ 2 fuzzy open is τ 2 fuzzy open and τ 2 - closure of each τ 1 fuzzy open is τ 1 fuzzy open. Definition [55] Let (X, τ 1, τ 2 ) be a fuzzy bitopological space and let δ ƒ τ 1 and δ ƒ τ 2. If τ 1 (δ) and τ 2 (δ) are simple extensions of the fuzzy topologies τ 1 and τ 2 respectively, then the fuzzy bitopology (τ 1 (δ), τ 2 (δ)) is called a simple extension of the fuzzy bitopology (τ 1, τ 2 ). Definition [66] Let (P, ) be a partially ordered set and A P. Any element x P is an upper bound for A if for all a A, a x. Similarly, any element x P is a lower bound for A if for all a A, x a. Definition [55] Let (X, τ 1, τ 2 ) be a fuzzy bitopological space. (X, τ 1, τ 2 ) is said to be pairwise fuzzy normal and if for every pair of fuzzy sets λ τ 1 and µ ƒ τ 2 such that λ µ, there exists τ 1 - fuzzy open set δ or τ 2 - fuzzy open set δ such that λ δ clτ i (δ) µ, for i = 1, 2. Definition [50] A function u : I X I X defined on the family I X of all fuzzy sets of X is called a fuzzy closure operator on X and the pair (X, u) is called fuzzy closure space, if the following conditions are satisfied. 26