Intuitionistic Supra Gradation of Openness

Applied Mathematics & Information Sciences 2(3) (2008), 291-307 An International Journal c 2008 Dixie W Publishing Corporation, U. S. A. Intuitionistic Supra Gradation of Openness A. M. Zahran 1, S. E. Abbas 2 and E. El-Sanousy 2 1 Department of Mathematics, Faculty of Science, Azhar University, Assuit, Eygpt Email Address: amzahran2000@yahoo.com 2 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Eygpt Email Addresses: sabbas73@yahoo.com; elsanowsy@yahoo.com Received August 23, 2007; Revised January 12, 2008 In this paper, we have used the intuitionistic supra gradation of openness that was created from an intuitionistic fuzzy bitopological spaces to introduce and study the concepts of continuity, some kinds of separation axioms and compactness. Keywords: Intuitionistic supra gradation of openness, IF P -continuous mapping, IF P -separation axioms, IF P -compact. 1 Introduction and Preliminaries Kubiak [10] and Šostak [15] introduced the fundamental concept of a fuzzy topological structure, as an extension of both crisp topology and fuzzy topology [3], in the sense that not only the objects are fuzzified, but also the axiomatics. In [16,17] Šostak gave some rules and showed how such an extension can be realized. Chattopadhyay et al. [4] have redefined the same concept under the name gradation of openness. A general approach to the study of topological type structures on fuzzy powersets was developed in [7-11]. As a generalization of fuzzy sets, the notion of intuitionistic fuzzy sets was introduced by Atanassov [2]. By using intuitionistic fuzzy sets, Çoker and his colleague [5,6] defined the topology of intuitionistic fuzzy sets. Recently, Samanta and Mondal [14] introduced the notion of intuitionistic gradation of openness of fuzzy sets, where to each fuzzy subsets there is a definite grade of openness and there is a grade of nonopenness. Thus, the concept of intuitionistic gradation of openness is a generalization of the concept of gradation of openness and the topology of intuitionistic fuzzy sets. In this paper, we have used the intuitionistic supra gradation of openness that was created from an intuitionistic fuzzy bitopological spaces to introduce and study the concepts of continuity, some kinds of separation axioms and compactness.

292 A. M. Zahran et al. Throughout this paper, let X be a nonempty set, I = [0, 1], I 0 = (0, 1], I 1 = [0, 1). For α I, α(x) = α for each x X. The set of all fuzzy subsets of X are denoted by I X. For x X and t I 0 a fuzzy point is defined by x t (y) = { t, if y = x 0, if y x. x t λ iff t λ(x). We denote a fuzzy set λ which is quasi-coincident with a fuzzy set µ by λqµ, if there exists x X such that λ(x) + µ(x) > 1. Otherwise by λqµ. Definition 1.1. [1,14] An intuitionistic supra gradation of openness (ISGO, for short) on X is an ordered pair (τ, τ ) of mappings from I X to I such that (ISGO1) τ(λ) + τ (λ) 1, λ I X. (ISGO2) τ(0) = τ(1) = 1, τ (0) = τ (1) = 0. (ISGO3) τ( i λ i) i τ(λ i) and τ ( i λ i) i τ (λ i ), λ i I X, i. The triplet (X, τ, τ ) is called an intuitionistic supra fuzzy topological space (isfts, for short). An ISGO (τ, τ ) is called an intuitionistic gradation of openness (IGO, for short) on X iff (IT) τ(λ 1 λ 2 ) τ(λ 1 ) τ(λ 2 ) and τ (λ 1 λ 2 ) τ (λ 1 ) τ (λ 2 ), λ 1, λ 2 I X. The triplet (X, τ, τ ) is called an intuitionistic fuzzy topological space (ifts, for short). τ and τ may be interpreted as gradation of opennes and gradation of nonopenness, respectively. The (X, (τ, τ ), (ν, ν )) is called an intuitionistic fuzzy bitopological space (ifbts, for short) where (τ, τ ) and (ν, ν ) are IGO s on X. Definition 1.2. [1] A map C : I X I 0 I 1 I X is called an intuitionistic supra fuzzy closure operator on X if for λ, µ I X and r I 0, s I 1, it satisfies the following conditions: (C1) C(0, r, s) = 0. (C2) λ C(λ, r, s). (C3) C(λ, r, s) C(µ, r, s) C(λ µ, r, s). (C4) C(λ, r 1, s 1 ) C(λ, r 1, s 2 ) if r 1 r 2 and s 1 s 2. (C5) C(C(λ, r, s), r, s) = C(λ, r, s). The pair (X, C) is called an intuitionistic supra fuzzy closure space. The intuitionistic supra fuzzy closure space (X, C) is called the intuitionistic fuzzy closure space iff (C) C(λ, r, s) C(µ, r, s) = C(λ µ, r, s). Theorem 1.1. [1] Let (X, τ, τ ) be an isfts. Then λ I X, r I 0, s I 1 we define an operator C τ,τ : I X I 0 I 1 I X as follows: C τ,τ (λ, r, s) = {µ I X : λ µ, τ(1 µ) r, τ (1 µ) s}.

Intuitionistic Supra Gradation of Openness 293 Then (X, C τ,τ ) is an intuitionistic supra fuzzy closure space. The mapping I τ,τ : I X I 0 I 1 I X defined by I τ,τ (λ, r, s) = {µ I X : µ λ, τ(µ) r, τ (µ) s} is an intuitionistic supra fuzzy interior space. And I τ,τ (1 λ, r, s) = 1 C τ,τ (λ, r, s). Theorem 1.2. [1] Let (X, C) be an intuitionistic (intuitionistic supra) fuzzy closure space. Define the mappings τ c, τ c : I X I on X by τ c (λ) = {r I 0 : C(1 λ, r, s) = 1 λ}, τ c (λ) = {s I 1 : C(1 λ, r, s) = 1 λ}. Then, (1) (τ c, τ c ) is an IGO s (ISGO s) on X, (2) C τc,τ C C. Theorem 1.3. [1] Let (X, (τ 1, τ 1 ), (τ 2, τ 2 )) be an isfbts. We define the mappings C 12, I 12 : I X I 0 I 1 I X as follows: C 12 (λ, r, s) = C τ1,τ1 (λ, r, s) C τ 2,τ2 (λ, r, s), I 12 (λ, r, s) = I τ1,τ1 (λ, r, s) I τ 2,τ2 (λ, r, s), for all λ I X, r I 0, s I 1. Then, (1) (X, C 12 ) is an intuitionistic supra fuzzy closure space, (2) I 12 (1 λ, r, s) = 1 C 12 (λ, r, s). Corollary 1.1. [1] Let (X, C 12 ) be an intuitionistic supra fuzzy closure space. Then, the mappings τ C12, τc 12 : I X I on X defined by τ C12 (λ) = {r I 0 : C 12 (1 λ, r, s) = 1 λ} and τ C 12 (λ) = {s I 1 : C 12 (1 λ, r, s) = 1 λ} is an ISGO s on X. Theorem 1.4. [1] Let (X, (τ 1, τ 1 ), (τ 2, τ 2 )) be an ifbts. Let (X, C 12 ) be an intuitionistic supra fuzzy closure space. Define the mappings τ su, τ su : I X I on X by τ su (λ) = {τ 1 (λ 1 ) τ 2 (λ 2 ) : λ = λ 1 λ 2 }, τ su(λ) = {τ 1 (λ 1 ) τ 2 (λ 2 ) : λ = λ 1 λ 2 }, where and are taken over all families {λ 1, λ 2 : λ = λ 1 λ 2 }. Then, (1) (τ su, τ su) = (τ c12, τ c 12 ) is the coarsest ISGO on X which is finer than both of (τ 1, τ 1 ) and (τ 2, τ 2 ). (2) C 12 = C τsu,τ su = C τ c12,τ c 12.

294 A. M. Zahran et al. Definition 1.3. [13, 14] Let f : (X, (τ 1, τ1 ), (τ 2, τ2 )) (Y, (ν 1, ν1), (ν 2, ν2)) be a mapping. Then f is called (1) IF P -continuous iff τ i (f 1 (µ)) ν i (µ) and τi (f 1 (µ)) νi (µ) µ IY, i = 1, 2; (2) IF P -open iff τ i (λ) ν i (f(λ)) and τi (λ) ν i (f(λ)) λ IX, i = 1, 2; (3) IF P -closed iff τ i (1 λ) ν i (1 f(λ)) and τi (1 λ) ν i (1 f(λ)) λ I X, i = 1, 2; (4) IF P -weakly open iff τ i (λ) r and τi (λ) s = ν i(f(λ)) r and νi (f(λ)) s λ I X, i = 1, 2; (5) IF P -weakly closed iff τ i (1 λ) r and τi (1 λ) s = ν i(1 f(λ)) r and νi (1 f(λ)) s λ IX, i = 1, 2. 2 IF P -Continuous Mapping Definition 2.1. Let f : (X, (τ 1, τ 1 ), (τ 2, τ 2 )) (Y, (ν 1, ν 1), (ν 2, ν 2)) be a mapping. Then f is called IF P -continuous (resp. IF P -open, IF P -closed) iff f : (X, τ su, τ su) (Y, ν su, ν su) is IF -continuous (resp. IF -open, IF -closed). Theorem 2.1. Every IF P -continuous (resp. IF P -open, IF P -closed) is IF P - continuous (resp. IF P -open, IF P -closed). Proof. Let f : (X, (τ 1, τ 1 ), (τ 2, τ 2 )) (Y, (ν 1, ν 1 ), (ν 2, ν 2 )) be an IF P -continuous mapping and (X, τ su, τ su), (Y, ν su, ν su) their associated isfts. Suppose that there exists µ I Y and s 0 I 1 such that τ su(f 1 (µ)) s 0 ν su(µ). There exist µ 1, µ 2 I Y with µ = µ 1 µ 2 such that ν su(µ) = ν 1(µ 1 ) ν 2(µ 2 ) s 0. Then ν 1(µ 1 ) s 0 and ν 2 (µ 2 ) s 0. By IF P -continuity, we have τ 1 (f 1 (µ 1 )) ν 1(µ 1 ) s 0 and τ 2 (f 1 (µ 2 )) ν 2 (µ 2 ) s 0. This implies that τ 1 (f 1 (µ 1 )) τ 2 (f 1 (µ 2 )) s 0, and so τ su(f 1 (µ)) s 0. It is contradiction. Hence τ su(f 1 (µ)) ν su(µ), µ I Y. By the same way, we can prove τ su (f 1 (µ)) ν su (µ), µ I Y. So, f is IF P - continuous. The other parts can be proved in a similar manner. Example 2.1. Let X = {a, b, c}. Define ρ 1, ρ 2, µ 1, µ 2 I X as follows ρ 1 (a) = 0.3, ρ 1 (b) = 0.5, ρ 1 (c) = 0.4, ρ 2 (a) = 0.2, ρ 2 (b) = 0.3, ρ 2 (c) = 0.5, µ 1 (a) = 0.3, µ 1 (b) = 0.5, µ 1 (c) = 0.2, µ 2 (a) = 0.5, µ 2 (b) = 0.4, µ 2 (c) = 0.3.

Intuitionistic Supra Gradation of Openness 295 We define τ 1, τ 1, τ 2, τ 2, ν 1, ν 1, ν 2, ν 2 : I X I as follows τ 1 (ρ) = τ 2 (ρ) = ν 1 (µ) = ν 2 (µ) = 1 if ρ = 0, 1 0.5 if ρ = ρ 1 1 if ρ = 0, 1 0.6 if ρ = ρ 2 1 if µ = 0, 1 0.5 if µ = µ 1 1 if µ = 0, 1 0.4 if µ = µ 2 τ1 (ρ) = τ2 (ρ) = ν1 (ρ) = ν2 (ρ) = 0 if ρ = 0, 1 0.4 if ρ = ρ 1 0 if ρ = 0, 1 0.3 if ρ = ρ 2 0 if µ = 0, 1 0.4 if µ = µ 1 0 if µ = 0, 1 0.5 if µ = µ 2 1 otherwise. The mapping f : (X, (τ 1, τ 1 ), (τ 2, τ 2 )) (X, (ν 1, ν 1), (ν 2, ν 2)) defined by f(a) = c, f(b) = a, f(c) = b, is IF P -continuous but not IF P -continuous. Theorem 2.2. Let f : (X, (τ 1, τ 1 ), (τ 2, τ 2 )) (Y, (ν 1, ν 1), (ν 2, ν 2)) be a mapping. Then the following statements are equivalent: λ I X, µ I Y, r I 0, s I 1 (1) f is IF P -continuous. (2) τ su (1 f 1 (µ)) ν su (1 µ) and τ su(1 f 1 (µ)) ν su(1 µ). (3) f(c 12 (λ, r, s)) C 12 (f(λ), r, s). (4) C 12 (f 1 (µ), r, s) f 1 (C 12 (µ, r, s)). (5) f 1 (I 12 (µ, r, s)) I 12 (f 1 (µ), r, s). Proof. (1) (2) is Obvious. (2) (3): For each λ I X, r I 0, s I 1, we have f 1 (C 12 (f(λ), r, s)) = f 1 (C νsu,νsu (f(λ), r, s)) = f 1 [ {η I Y : f(λ) η, ν su (1 η) r, ν su(1 η) s}] {f 1 (η) I X : λ f 1 (η), τ su (1 f 1 (η)) r, τ su(1 f 1 (η)) s} = C τsu,τ su (λ, r, s) = C 12(λ, r, s). Thus f(c 12 (λ, r, s)) C 12 (f(λ), r, s). (3) (4): For each µ I Y, r I 0, s I 1, put λ = f 1 (µ). From (3), we have f(c 12 (f 1 (µ), r, s)) C 12 (f(f 1 (µ)), r, s) C 12 (µ, r, s),

296 A. M. Zahran et al. which implies that C 12 (f 1 (µ), r, s) f 1 (f(c 12 (f 1 (µ), r, s))) f 1 (C 12 (µ, r, s)). (4) (5): For each µ I Y, r I 0, s I 1, we have C 12 (f 1 (1 µ), r, s)) f 1 (C 12 (1 µ, r, s)), which implies that 1 f 1 (C 12 (1 µ, r, s)) 1 C 12 (f 1 (1 µ), r, s), f 1 [1 C 12 (1 µ, r, s)] 1 C 12 (f 1 (1 µ), r, s). By Theorem 1.3 (2), we have f 1 (I 12 (1 µ, r, s)) I 12 (1 f 1 (1 µ), r, s) = I 12 (f 1 (µ), r, s). (5) (1): Suppose that there exists µ I Y, r I, s I 1 such that τ su(f 1 (µ)) > s ν su(µ) and τ su (f 1 (µ)) < r ν su (µ). Then, there exist µ 1, µ 2 I Y such that ν su(µ) = ν 1(µ 1 ) ν 2(µ 2 ), ν su (µ) = ν 1 (µ 1 ) ν 2 (µ 2 ), and µ = µ 1 µ 2. This implies that ν 1(µ 1 ) s and ν 2 (µ 2 ) s. Also, ν 1 (µ 1 ) r and ν 2 (µ 2 ) r, then, I ν1,ν 1 (µ 1, r, s) = µ 1 and I ν2,ν 2 (µ 2, r, s) = µ 2. From Theorem 1.3, we have I 12 (µ, r, s) = I ν1,ν 1 (µ 1, r, s) I ν2,ν 2 (µ 2, r, s) = µ 1 µ 2 = µ. By (5), we have f 1 (µ) = I 12 (f 1 (µ), r, s) = I τsu,τ su (f 1 (µ), r, s). This implies that τ su(f 1 (µ)) s and τ su (f 1 (µ)) r, which is a contradiction. So, τ su(f 1 (µ)) ν su(µ) and τ su (f 1 (µ)) ν su (µ) µ I Y. Hence, f : (X, (τ 1, τ 1 ), (τ 2, τ 2 )) (Y, (ν 1, ν 1), (ν 2, ν 2)) is IF P -continuous. Theorem 2.3. Let f : (X, (τ 1, τ 1 ), (τ 2, τ 2 )) (Y, (ν 1, ν 1), (ν 2, ν 2)) be a mapping. Then the following statements are equivalent: λ I X, µ I Y, r I 0, s I 1 (1) f is IF P -weakly open. (2) f(i 12 (λ, r, s)) I 12 (f(λ), r, s). (3) I 12 (f 1 (µ), r, s) f 1 (I 12 (µ, r, s)).

Intuitionistic Supra Gradation of Openness 297 Proof. (1) (2): For each λ I X and r I 0, s I 1, Since I 12 (λ, r, s) = I τsu,τsu (λ, r, s) λ, we have f(i τsu,τsu (λ, r, s)) f(λ). Also, τ su (I τsu,τsu (λ, r, s)) r and τ su(i τsu,τsu (λ, r, s) s. By (1), ν su (f(i τsu,τsu (λ, r, s))) r and ν su(f(i τsu,τsu (λ, r, s))) s. Hence f(i 12 (λ, r, s)) I 12 (f(λ), r, s). (2) (3): For each µ I Y, r I 0, s I 1, put λ = f 1 (µ). From (2), we have f(i 12 (f 1 (µ), r, s)) I 12 (f(f 1 (µ)), r, s) I 12 (µ, r, s), which implies that I 12 (f 1 (µ), r, s) f 1 (f(i 12 (f 1 (µ), r, s))) f 1 (I 12 (µ, r, s)). (3) (1): For each λ I X with τ su (λ) r, τ su(λ) s implies I 12 (λ, r, s) = λ. Put µ = f(λ), by (3), we have I 12 (λ, r, s) I 12 (f 1 (f(λ)), r, s) f 1 (I 12 (f(λ), r, s)), which implies that λ f 1 (I 12 (f(λ), r, s)) and so f(λ) I 12 (f(λ), r, s). Then ν su (f(λ)) r and ν su(f(λ)) s. Hence, f : (X, (τ 1, τ 1 ), (τ 2, τ 2 ) (Y, (ν 1, ν 1), (ν 2, ν 2)) is IF P -weakly open. Theorem 2.4. Let f : (X, (τ 1, τ 1 ), (τ 2, τ 2 )) (Y, (ν 1, ν 1), (ν 2, ν 2)) be a mapping. Then the following statements are equivalent: (1) f is IF P -weakly closed. (2) C 12 (f(λ), r, s) f(c 12 (λ, r, s)), λ I X, r I 0, s I 1. Theorem 2.5. Let f : (X, (τ 1, τ 1 ), (τ 2, τ 2 ) (Y, (ν 1, ν 1), (ν 2, ν 2)) be a bijective mapping. Then the following statements are equivalent: (1) f is IF P -weakly closed. (2) f 1 (C 12 (µ, r, s)) C 12 (f 1 (µ), r, s), µ I Y, r I 0, s I 1.

298 A. M. Zahran et al. Proof. (1) (2): Put λ = f 1 (µ), from Theorem 2.4(2) C 12 (f(f 1 (µ)), r, s) C 12 (µ, r, s) f(c 12 (f 1 (µ), r, s)). Also, since f is onto, we have f 1 (C 12 (µ, r, s)) f 1 (f(c 12 (f 1 (µ), r, s))) = C 12 (f 1 (µ), r, s). (2) (1): Put µ = f(λ). Since f is injective, f 1 (C 12 (f(λ), r, s)) C 12 (f 1 (f(λ)), r, s) = C 12 (λ, r, s). Since f is onto, C 12 (f(λ), r, s) f(c 12 (λ, r, s)). 3 Some Types of Separation Axioms Definition 3.1. For i, j {1, 2}, i j, an ifbts (X, (τ 1, τ1 ), (τ 2, τ2 )) is called (1) IF P R 0 iff x t qc τi,τi (y m, r, s) implies that y m qc τj,τj (x t, r, s) for any x t y m. (2) IF P R 1 iff x t qc τi,τi (y m, r, s) implies that there exist λ, µ I X with τ i (λ) r, τi (λ) s and τ j(µ) r, τj (µ) s such that x t λ, y m µ and λ q µ. (3) IF P R 2 iff x t q ρ = C τi,τi (ρ, r, s) implies that there exist λ, µ IX with τ i (λ) r, τi (λ) s and τ j(µ) r, τj (µ) s such that x t λ, ρ µ and λ qµ. (4) IF P R 3 iff η = C τi,τi (η, r, s) qρ = C τ j,τj (ρ, r, s) implies that there exist λ, µ I X with τ i (λ) r, τi (λ) s and τ j(µ) r, τj (µ) s such that η λ, ρ µ and λ q µ. (5) IF P T 0 iff x t q y m implies that there exist λ I X such that τ i (λ) r, τi (λ) s and x t λ, y m q µ or y m λ, x t q µ. (6) IF P T 1 iff x t q y m implies that there exist λ I X such that for i = 1 or 2 τ i (λ) r, τi (λ) s, x t λ and y m q λ. (7) IF P T 2 iff x t q y m implies that there exist λ, µ I X with τ i (λ) r, τi (λ) s and τ j (µ) r, τj (µ) s such that x t λ, y m µ and λ q µ. (8) IF P T 2 1 iff x t q y 2 m implies that there exist λ, µ I X with τ i (λ) r, τi (λ) s and τ j (µ) r, τj (µ) s such that x t λ, y m µ and C τj,τj (λ, r, s) q C τ i,τi (µ, r, s). (9) IF P T 3 iff it is IF P R 2 and IF P T 1. (10) IF P T 4 iff it is IF P R 3 and IF P T 1. (11) IF P R i iff its associated isfts (X, τ su, τsu) is IF R i, i = 0, 1, 2. (12) IF P T i iff its associated isfts (X, τ su, τsu) is IF T i, i = 0, 1, 2, 2 1 2, 3, 4. In this definition if i = j we have the definition of IF R 0, IF R 1, IF R 2, IF R 3, IF T 0, IF T 1, IF T 2, IF T 2 1, IF T 3 and IF T 2 4, respectively.

Intuitionistic Supra Gradation of Openness 299 Theorem 3.1. Let (X, (τ 1, τ 1 ), (τ 2, τ 2 )) be an ifbts. Then we have (1) IF P R i IF P R i, i = 0, 1, 2, 3. (2) IF P T i IF P T i, i = 0, 1, 2, 2 1 2, 3. (3) IF P T i IF P T i, i = 0, 1. (4) IF P R 2 IF P R 1 IF P R 0. (5) IF P T 4 IF P T 3 IF P T 2 1 2 IF P T 2 IF P T 1 IF P T 0. Proof. (1) Let (X, (τ 1, τ1 ), (τ 2, τ2 )) be an IF P R 0 and let x t qc τsu,τsu (y m, r, s). From Theorem 1.4(2), we have x t qc 12 (y m, r, s). Also, by Theorem 1.3, we have x t q[c τ1,τ1 (y m, r, s) C τ2,τ2 (y m, r, s)]. Then, x t 1 [C τ1,τ1 (y m, r, s) C τ2,τ2 (y m, r, s)] = [1 C τ1,τ1 (y m, r, s)] [1 C τ2,τ2 (y m, r, s)], this implies that x t 1 C τ1,τ1 (y m, r, s) or x t 1 C τ2,τ2 (y m, r, s). Therefore, x t qc τ1,τ1 (y m, r, s) or x t qc τ2,τ2 (y m, r, s). Since (X, (τ 1, τ1 ), (τ 2, τ2 )) is IF P R 0, we have y m qc τ1,τ1 (x t, r, s) or y m qc τ2,τ2 (x t, r, s) this implies that y m q[c τ1,τ1 (x t, r, s) C τ2,τ2 (x t, r, s)] = C 12 (x t, r, s) = C τsu,τsu (x t, r, s), so, (X, (τ 1, τ1 ), (τ 2, τ2 )) is IF P R 0. (2) Let (X, (τ 1, τ1 ), (τ 2, τ2 )) be an IF P T 2 1 and x t q y 2 m. Then there exist λ, µ I X with τ i (λ) r, τi (λ) s and τ j(µ) r, τj (µ) s for i, j {1, 2}, i j such that x t λ, y m µ and C τj,τj (λ, r, s)qc τ i,τi (µ, r, s). Since C τ su,τsu C τ i,τi for i = 1, 2 we have, C τsu,τsu (λ, r, s)q C τ su,τsu (µ, r, s). Then (X, (τ 1, τ1 ), (τ 2, τ2 )) is IF P T 2 1. 2 (3) Let (X, (τ 1, τ1 ), (τ 2, τ2 )) be an IF P T 1 and x t qy m. Then there exists λ I X such that x t λ, τ su (λ) r, τsu(λ) s and y m qλ. Since, τ su (λ) r, τsu(λ) s there exist λ 1, λ 2 I X such that τ su (λ) = τ 1 (λ 1 ) τ 2 (λ 2 ), τsu(λ) = τ1 (λ 1 ) τ2 (λ 2 ) and λ = λ 1 λ 2, then τ 1 (λ 1 ) r, τ 2 (λ 2 ) r and τ1 (λ 1 ) s, τ2 (λ 2 ) s. And x t λ implies that x t λ 1 or x t λ 2. Also, y m q λ implies that y m q λ 1 and y m q λ 2. Thus (x t λ 1, τ 1 (λ) r, τ1 (λ) s and y m q λ 1 ) or (x t λ 2, τ 2 (λ) r, τ2 (λ) s and y m q λ 2 ). Hence, (X, (τ 1, τ1 ), (τ 2, τ2 )) is IF P T 1. (4) and (5) obvious from the definition. Other parts are similarly proved. Lemma 3.1. Let (X, (τ 1, τ 1 ), (τ 2, τ 2 )) be an ifbts. Then (1) If (X, τ 1, τ 1 ) or (X, τ 2, τ 2 ) is IF T i, then (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is IF P T i, i = 0, 1, 2, 2.5, 3. (2) If (X, τ 1, τ 1 ) or (X, τ 2, τ 2 )) is IF R i, then (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is IF P R i, i = 0, 1, 2. Proof. (2) Let (X, τ 1, τ 1 ) or (X, τ 2, τ 2 ) be an IF R 0. For any two fuzzy points x t y m such that x t q C τsu,τ su (y m, r, s) this implies that x t q [C τ1,τ 1 (y m, r, s) C τ2,τ 2 (y m, r, s)] implies x t q C τ1,τ 1 (y m, r, s) or x t q C τ2,τ 2 (y m, r, s). Then y m q C τ1,τ 1 (x t, r, s) or y m q C τ2,τ 2 (x t, r, s) this implies that y m q [C τ1,τ 1 (x t, r, s) C τ2,τ 2 (x t, r, s)] = C 12 (x t, r, s) = C τsu,τ su (x t, r, s). This implies that (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is IF P R 0.

300 A. M. Zahran et al. Example 3.1. Let X = {a, b}. Define τ i, τ i τ 1 (λ) = : I X I, i {1, 2, 3,..., 12} as follows: 1 if λ = 0, 1 0.5 if λ {a α b 0.5, a 0.5 b α }, α (0, 1) {0.5} 0.5 if λ = α, α (0, 1) τ1 (λ) = 0 if λ = 0, 1 0.5 if λ {a α b 0.5, a 0.5 b α }, α (0, 1) {0.5} 0.5 if λ = α, α (0, 1) 1 if λ = 0, 1 τ 2 (λ) = 0.5 if λ = 0.4 0 if λ = 0, 1 τ2 (λ) = 0.5 if λ = 0.4, 0.5 τ 3 (λ) = 1 if λ = 0, 1 0.5 if 0 λ < 0.5 0.4 if 0.5 < λ 1 τ3 (λ) = 0 if λ = 0, 1 0.5 if 0 λ < 0.5 0.6 if 0.5 < λ 1 0.6 if λ = 0.5 1 if λ = 0, 1 τ 4 (λ) = 0.4 if λ = 0.5 0 if λ = 0, 1 τ4 (λ) = 0.6 if λ = 0.5 τ 5 (λ) = 1 if λ = 0, 1 0.5 if λ {a 1, b 1, a 0.4, b 0.4 } 0.6 if λ {0.4, a 0.4 b 1, a 1 b 0.4 } τ5 (λ) = 0 if λ = 0, 1 0.5 if λ {a 1, b 1, a 0.4, b 0.4, a 0.6, b 0.6, a 0.4 b 0.6, a 0.6 b 0.4 } 0.4 if λ {0.4, 0.6, a 0.4 b 1, a 1 b 0.4, a 0.6 b 1, a 1 b 0.6 }

Intuitionistic Supra Gradation of Openness 301 1 if λ = 0, 1 τ 6 (λ) = 0.6 if λ = 0.7 τ6 (λ) = 0 if λ = 0, 1 0.4 if λ = 0.7 0.6 if λ = 0.3 τ 7 (λ) = τ7 (λ) = 1 if λ = 0, 1 0.5 if λ {a α, b α }, α (0, 1) 0.6 if λ = α, α (0, 1) 0 if λ = 0, 1 0.5 if λ {a α, b α, a α b 1, a 1 b α }, α (0, 1) 0.4 if λ = α, α (0, 1) 1 if λ = 0, 1 τ 8 (λ) = 0.5 if λ = 0.6 1 if λ = 0, 1 τ 9 (λ) = 0.5 if 0 λ < 1 1 if λ = 0, 1 τ 10 (λ) = 0.4 if 0.5 < λ < 1 τ8 (λ) = 0 if λ = 0, 1 0.5 if λ = 0.6 0.6 if λ = 0.4 0 if λ = 0, 1 τ9 (λ) = 0.5 if 0 λ < 1 0 if λ = 0, 1 τ10(λ) = 0.6 if 0.5 < λ < 1 1 if λ = 0, 1 τ 11 (λ) = 0.5 if λ {α, a α b 1, a 1 b α }, α (0, 1) τ11(λ) = 0 if λ = 0, 1 0.5 if λ {α, 1}, {1, α}, α (0, 1) 0.4 if λ = α, α (0, 1)

302 A. M. Zahran et al. 1 if λ = 0, 1 τ 12 (λ) = 0.5 if λ = 0.3 τ12(λ) = 0 if λ = 0, 1 0.5 if λ = 0.3 0.6 if λ = 0.4 1 otherwise. (1) For 0 < r 0.5, 0.5 s < 1, (X, (τ 1, τ1 ), (τ 2, τ2 )) is IF P R 1, but it is neither IF P R 1 nor IF P R 0. (2) For 0 < r 0.4, 0.6 s < 1, (X, (τ 3, τ3 ), (τ 4, τ4 )) is IF P R 2, but it is not IF P R 2. (3) For 0 < r 0.4, 0.6 s < 1, (X, (τ 5, τ5 ), (τ 6, τ6 )) is IF P T 2, but it is not IF P T 2. (4) For 0 < r 0.4, 0.6 s < 1, (X, (τ 7, τ7 ), (τ 8, τ8 )) is IF P T 2 1, but it is not 2 IF P T 2 1. 2 (5) For 0 < r 0.4, 0.6 s < 1, (X, (τ 9, τ9 ), (τ 10, τ10)) is IF P T 3, but it is not IF P T 3. (6) For 0 < r 0.4, 0.6 s < 1, (X, (τ 11, τ11), (τ 12, τ12)) is IF P R 0, but it is not IF P R 1. Lemma 3.2. [12] Let (X, (τ 1, τ 1 ), (τ 2, τ 2 )) be an ifbts. For r I 0, s I 1, we have (1) For λ I X with τ su (λ) r, τ su(λ) s, λqµ iff λqc 12 (µ, r, s), µ I X. (2) x t qc 12 (λ, r, s) iff λqµ for all µ I X with τ su (µ) r, τ su(µ) s and x t µ. Theorem 3.2. Let (X, (τ 1, τ1 ), (τ 2, τ2 )) be an ifbts. Then, λ I X, r I 0, s I 1, the following statements are equivalent: (1) (X, (τ 1, τ1 ), (τ 2, τ2 )) is IF P R 0. (2) C 12 (x t, r, s) λ with τ su (λ) r, τsu(λ) s, x t λ. (3) If x t q λ = C τsu,τsu (λ, r, s), there exists µ IX with τ su (µ) r, τsu(µ) s such that x t q µ and λ µ. (4) If x t q λ = C τsu,τsu (λ, r, s) then, C τ su,τsu (x t, r, s) q λ = C τsu,τsu (λ, r, s). Proof. (1) (2): Let y m q C 12 (x t, r, s). By Theorem 1.3, we have y m q C τsu,τsu (x t, r, s). Using (1), we obtain x t q C τsu,τsu (y m, r, s), i.e. x t q C 12 (y m, r, s). Using Lemma 3.2(2), we find that y m q µ µ I X with τ su (µ) r, τsu(µ) s and x t µ. Then, we have C 12 (x t, r, s) µ. (2) (1): If y m q C τsu,τsu (x t, r, s), we have y m 1 C τsu,τsu (x t, r, s). By (2) and the fact τ su (1 C τsu,τsu (x t, r, s)) r, τsu(1 C τsu,τsu (x t, r, s)) s, we get C 12 (y m, r, s) 1 C τsu,τ su (x t, r, s) 1 x t. Thus, x t q C 12 (y m, r, s) = C τsu,τ su (y m, r, s). Hence, (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is IF P R 0.

Intuitionistic Supra Gradation of Openness 303 (1) (3): Let x t q λ = C τsu,τsu (λ, r, s). Since C τ su,τsu (y m, r, s) C τsu,τsu (λ, r, s), y m λ, we have x t q C τsu,τsu (y m, r, s). By (1), we have y m q C τsu,τsu (x t, r, s). Using Lemma 3.2(2), y m q C τsu,τsu (x t, r, s), there exists η I X such that x t q η, τ su (η) r, τsu(η) s and y m η. Let µ = y m λ {η : x tqη, y m η}. From the definition of ISGO, we have τ su (µ) r, τsu(µ) s. Then, x t q µ, λ µ, τ su (µ) r, τsu(µ) s. (3) (4): Let x t q λ = C τsu,τ su (λ, r, s). By (3), there exists µ IX such that x t q µ, λ µ with τ su (µ) r, τ su(µ) s. Since x t q µ, it follows that x t 1 µ, which implies that C τsu,τsu (x t, r, s) C τsu,τsu (1 µ, r, s) = 1 µ 1 λ. Hence, C τsu,τsu (x t, r, s) q λ = C τsu,τsu (λ, r, s). (4) (1): Let x t q C τsu,τsu (y m, r, s). By (4), we have C τsu,τsu (x t, r, s) q C τsu,τsu (y m, r, s) and since y m C τsu,τsu (y m, r, s), y m q C τsu,τsu (x t, r, s). Hence (X, (τ 1, τ1 ), (τ 2, τ2 )) is IF P R 0. Theorem 3.3. An ifbts (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is an IF P R 1 iff x t q C 12 (y m, r, s), there exist λ i I X for i = 1, 2 such that (1 λ 1 ) q (1 λ 2 ) and C 12 (x t, r, s) λ 2, C 12 (y m, r, s) λ 1, τ su (λ i ) r, τ su(λ i ) s. Proof. ( ) Let x t q C 12 (y m, r, s) = C τsu,τ su (y m, r, s). By IF P R 1, there exist λ i I X for i = 1, 2 with λ 1 q λ 2 such that x t λ 1, y m λ 2 and τ su (λ i ) r, τ su(λ i ) s. Since (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is an IF P R 1 implies that (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is an IF P R 0, by Theorem 3.2(4), x t q (1 λ 1 ) with τ su (λ 1 ) r, τ su(λ 1 ) s implies C 12 (x t, r, s) 1 λ 1 λ 2. Similarly, C 12 (y m, r, s) 1 λ 2 λ 1. ( ) Straightforward. Theorem 3.4. Let (X, (τ 1, τ 1 ), (τ 2, τ 2 )) be an ifbts. Then, r I 0, s I 1, the following statements are equivalent: (1) (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is IF P R 2. (2) If x t λ with τ su (λ) r, τ su(λ) s, there exist µ 1 I X with τ su (µ 1 ) r, τ su(µ 1 ) s such that x t µ 1 C τsu,τ su (µ 1, r, s) λ. (3) If x t q λ with τ su (1 λ) r, τ su(1 λ) s, there exists µ i I X with τ su (µ i ) r, τ su(µ i ) s, i = 1, 2 such that x t µ 1, λ µ 2 and C τsu,τ su (µ 1, r, s) q C τsu,τ su (µ 2, r, s).

304 A. M. Zahran et al. Proof. (1) (2): Let x t λ with τ su (λ) r, τ su(λ) s. Then x t q (1 λ). Since (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is IF P R 2, there exists µ i I X with τ su (µ i ) r, τ su(µ i ) s for i = 1, 2 such that x t µ 1, 1 λ µ 2 and µ 1 q µ 2, which implies x t µ 1 1 µ 2 λ. (2) (3): Let x t q λ with τ su (1 λ) r, τ su(1 λ) s. Then x t 1 λ. By (2), there exists µ I X with τ su (µ) r, τ su(µ) s such that x t µ C τsu,τsu (µ, r, s) 1 λ. Since τ su (µ) r, τ su(µ) s and x t µ. Again by (2), there exists µ 1 I X with τ su (µ 1 ) r, τ su(µ 1 ) s such that which implies that x t µ 1 C τsu,τsu (µ 1, r, s) µ C τsu,τsu (µ, r, s) 1 λ, λ (1 C τsu,τsu (µ, r, s)) = I τ su,τsu (1 µ, r, s) 1 µ. Put µ 2 = I τsu,τsu (1 µ, r, s). Then, C τsu,τ su (µ 2, r, s) 1 µ 1 C τsu,τ su (µ 1, r, s), that is, C τsu,τ su (µ 1, r, s) q C τsu,τ su (µ 2, r, s). (3) (1): It is trivial. 4 IF P Compactness Definition 4.1. Let (X, τ, τ ) be an ifts and µ I X, r I 0, s I 1. Then (1) The family {η j : τ(η j ) r, τ (η j ) s, j J} is called (τ, τ )-cover of µ iff for each x t µ there exists j 0 J such that x t η j0. (2) µ is C-set iff every (τ, τ )-cover of µ have a finite subcover. (3) (X, τ, τ ) is IF -compact iff λ I X such that τ(1 λ) r, τ (1 λ) s is C-set. (4) An ifbts (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is called IF P -compact iff its associated isfts (X, τ su, τ su) is IF -compact. Theorem 4.1. Let (X, (τ 1, τ 1 ), (τ 2, τ 2 )) be an ifbts. If (X, τ 1, τ 1 ) or (X, τ 2, τ 2 ) is IF - compact, then (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is IF P -compact. Proof. Suppose that (X, τ 1, τ1 ) is IF -compact, and λ I X such that τ su (1 λ) r, τsu(1 λ) s, r I 0, s I 1 and {η j : τ su (η j ) r, τsu(η j ) s, j J} be (τ su, τsu)-cover of λ. Since τ su (1 λ) r, τsu(1 λ) s, we can write λ = λ 1 λ 2, τ i (1 λ i ) r, τ i (1 λ i ) s, (i = 1, 2).

Intuitionistic Supra Gradation of Openness 305 Then, for every x t λ, there exists η j0 I X with τ su (η j0 ) r, τsu(η j0 ) s such that x t η j0 = η (1) η (2), for some η (i) I X with τ su (η (i) ) r, τsu(η (i) ) s, (i = 1, 2). Then, x t η (1) or x t η (2). Now, the family {η (1) i : τ 1 (η (1) i ) r, τ1 (η (1) i ) s, i } is (τ 1, τ1 )-cover of λ 1 or {η (2) i : τ 2 (η (2) i ) r, τ2 (η (2) i ) s, i } is (τ 2, τ2 )-cover of λ 2. If (X, τ 1, τ1 ) is IF -compact, then λ 1 is C-set i.e., there exists finite subset 0 of such that λ λ 1 i 0 η (1) i. Hence, λ is C-set. Consequently (X, (τ 1, τ1 ), (τ 2, τ2 )) is IF P -compact. Similarly, if (X, τ 2, τ2 ) is IF -compact, then (X, (τ 1, τ1 ), (τ 2, τ2 )) is IF P -compact. Theorem 4.2. Let (X, (τ 1, τ1 ), (τ 2, τ2 )) be an IF P T 2, x t P t(x), λ, µ I X, r I 0, s I 1. Then (1) If λ is C-set such that x t qλ, then there exist η i I X with τ su (η i ) r, τsu(η i ) s, (i = 1, 2) such that x t η 1, λ η 2 and η 1 q η 2. (2) If λ, µ are C-sets such that λ q µ, then there exist ρ i I X, τ su (ρ i ) r, τsu(ρ i ) s, (i = 1, 2) such that λ ρ 1, µ ρ 2 and ρ 1 q ρ 2. (3) If λ is C-set, then C τsu,τsu (λ, r, s) = λ. Proof. (1): Since x t q λ, then x t q y m y m λ. By IF P T 2 of (X, (τ 1, τ1 ), (τ 2, τ2 )) there exist η 1, υ I X with τ su (η 1 ) r, τsu(η 1 ) s, τ su (υ) r, τsu(υ) s such that x t η 1, y m υ and η 1 q υ. Then the family {υ i : τ su (υ i ) r, τsu(υ i ) s, i } is (τ su, τsu)-cover of λ. Since λ is C-set, there exists a finite subset 0 of 0 such that λ i 0 υ i. Put η 2 = i 0 υ i. Then τ su (η 2 ) = τ su ( υ i ) τ su (υ i ) r, i 0 i 0 τsu(η 2 ) = τsu( υ i ) τsu(υ i ) s. i 0 i 0 Since η 1 q υ i, i 0, then η 1 1 υ i, which implies that η 1 (1 υ i ) = 1 υ i = 1 η 2. i 0 i 0 Then, η 1 q η 2. (2): Let x t µ and λ q µ, then x t q λ. By (1) there exist σ, ρ 2 I X with τ su (σ) r, τsu(σ) s, τ su (ρ 2 ) r, τsu(ρ 2 ) s such that x t σ, λ ρ 2 and σ q ρ 2. Then the family {σ i : τ su (σ i ) r, τsu(σ i ) s, i } is (τ su, τsu)-cover of µ, so there exists a finite subset 0 of such that µ i 0 σ i. Put ρ 1 = i 0 ρ i, then τ su (ρ 1 ) r and τsu(ρ 1 ) s. Since ρ 2 q σ i, i 0 we have ρ 2 q ρ 1. (3): Let x t 1 λ, then x t q λ. Since λ is C-set, then by (2), there exist η 1, η 2 I X with τ su (η i ) r, τsu(η i ) s (i = 1, 2) such that x t η 1, λ η 2 and η 1 q η 2. This implies that x t η 1 1 η 2 1 λ. Thus, 1 λ = {η 1 : x t 1 λ}. So, τ su (1 λ) r, τsu(1 λ) s. Hence, C τsu,τsu (λ, r, s) = λ.

306 A. M. Zahran et al. Theorem 4.3. Let (X, (τ 1, τ 1 ), (τ 2, τ 2 )) be an IF P -compact. Then IF P T 2 IF P T 4. Proof. ( ): Since (X, (τ 1, τ1 ), (τ 2, τ2 )) is IF P T 2 it is clear that it is IF P T 1. We only need to prove that (X, (τ 1, τ1 ), (τ 2, τ2 )) is IF P R 3, Let λ 1 = C τsu,τ su (λ 1, r, s)qλ 2 = C τsu,τ su (λ 2, r, s). Then, τ su (1 λ i ) r, τ su(1 λ i ) s, (i = 1, 2). Since (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is IF P -compact, λ 1 and λ 2 are C-sets. Since λ 1 q λ 2, by Theorem 4.2(2), there exist ρ 1, ρ 2 I X, such that λ 2 ρ 2 and ρ 1 q ρ 2. Thus, (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is IF P R 3. Hence, (X, (τ 1, τ 1 ), (τ 2, τ 2 )) is IF P T 4. ( ): See Theorem 3.1(5). Theorem 4.4. Let f : (X, (τ 1, τ 1 ), (τ 2, τ 2 )) (Y, (ν 1, ν 1), (ν 2, ν 2)) be an IF P - continuous and µ I X is C-set. Then f(µ) is C-set in Y. Proof. Let {η i : i J} be (ν su, ν su)-cover of f(µ). Then, f(µ) i J η i, ν su (η i ) r, ν su(η i ) s. By IF P -continuity of f we have Also, τ su (f 1 (η i )) ν su (η i ) r, τ su(f 1 (η i )) ν su(η i ) s. µ f 1 (f(µ)) f 1 ( i J η i ) = i J f 1 (η i ). Then, the family {f 1 (η i ) : i J} is (τ su, τsu)-cover of µ. But µ is C-set, there exist a finite subset J 0 of J such that µ i J 0 f 1 (η i ), which implies that f(µ) f( f 1 (η i )) = f(f 1 (η i )) η i. i J 0 i J 0 i J 0 Hence, f(µ) is C-set in Y. Corollary 4.1. The IF P -continuous image of an IF P -compact is IF P -compact. References [1] S. E. Abbas, Intuitionistic supra fuzzy topological spaces, Chaos, Solitons and Fractals 21 (2004), 1205 1214. [2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87 96. [3] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182 190. [4] K. C. Chattopadhyay, R. N. Hazra and S. K. Samanta, Gradation of openness: fuzzy topology, Fuzzy Sets and Systems 49 ( 1992), 237 242.

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