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1. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 165 NEW CHARACTERIZATION OF KERNEL SET IN FUZZY TOPOLOGICAL SPACES Qays Hatem Imran Al-Muthanna University, College of Education, Department of Mathematics, Al-Muthanna, Iraq Basim Mohammed Melgat Foundation of Technical Education, Technical Institute of Al-Mussaib, Babylon, Iraq ABSTRACT In this paper we introduce a kernelled fuzzy point, boundary kernelled fuzzy point and derived kernelled fuzzy point of a subset A of X, and using these notions to define kernel set of fuzzy topological spaces. Also we introduce fuzzy topological ݇-ݎ space. The investigation enables us to present some new fuzzy separation axioms between ܶܨ and ܶܨଵ- spaces. Keywords: Fuzzy Topological Space, Kernelled Fuzzy Point, Boundary Kernelled Fuzzy Point, Derived Kernelled Fuzzy Point, Kernel Set, Weak Fuzzy Separation Axioms, ܴܨ- space, ݅ ൌ 0,1. 1. INTRODUCTION The concept of fuzzy set and fuzzy set operations were first introduced by L. A. Zadeh in 1965 [8]. After Zadeh 's introduction of fuzzy sets, Chang [3] defined and studied the notion of fuzzy topological space in 1968. In 1997, fuzzy generalized closed set (Fg - closed set) was introduced by G. Balasubramania and P. Sundaram [6]. In 1998, the notion of Fgs - closed set was defined and investigated by H. Maki el al [7]. In 2002, O. Bedre Ozbakir [11], defined the concept of fuzzy generalized strongly closed set. In 1984, fuzzy separation axioms have been introduced and investigated by A. S. Mashour and others [1]. In this paper we introduce a new characterization of kernel set through our definition kernelled fuzzy point, boundary kernelled fuzzy point and derived kernelled fuzzy point. By these notions, we obtain that the kernel of a set in fuzzy topological space ሺܺ, ܶሻ is a union of the set itself with the set of all boundary kernelled fuzzy points. In addition, it is a union of the set itself with the set of all derived kernelled fuzzy points and we give some result of ܴܨ- space by using these notions. Also in this paper we introduce fuzzy topological ݇-ݎ space iff kernel of a subset A of X is INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING & TECHNOLOGY (IJCET) ISSN 0976 – 6367(Print) ISSN 0976 – 6375(Online) Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME: www.iaeme.com/ijcet.asp Journal Impact Factor (2014): 8.5328 (Calculated by GISI) www.jifactor.com IJCET © I A E M E
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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 166 an fuzzy open set. Via this kind of fuzzy topological space, we give a new characterization of fuzzy separation axioms lying between ܶܨ and ܶܨଵ- spaces. 2. PRELIMINARIES Fuzzy sets theory, introduced by Lotfi. A. Zada in 1965 [8], is the extension of classical set theory by allowing the membership of elements to range from 0 to 1. Let X be the universe of a classical set of objects. Membership in a classical subset A of X is often viewed as a characteristic function µA from X into {0, 1}, where µA()ݔ = for any ∈ݔ X. {0,1} is called a valuation set (see [13]). If the valuation set is allowed to be the real interval [0,1], A is called a fuzzy set in X. µA()ݔ (or simply A())ݔ is the membership value (or degree of membership) of ݔ in A. Clearly, A is a subset of X that has no sharp boundary. A fuzzy set A in X can be represented by the set of pairs: A ={( ݔ , A(,))ݔ ∈ݔ X}. Let A : X → [0,1] be a fuzzy set. If A()ݔ = 1, for each ∈ݔ X, we denote it by 1X and if A()ݔ = 0, for each ∈ݔ X, we denote it by 0X . That is, by 0X and 1X , we mean the constant fuzzy sets taking the values 0 and 1 on X, respectively [2]. Let Ι =[0,1]. The set of all fuzzy sets in X, denoted by ΙX [10]. Definition 2.1:[4] Let A be a fuzzy set of a set X. The support of A is the elements ݔ whose membership value is greater than 0, i.e., supp(A) = { ∈ݔ X : A()ݔ > 0}. Definition 2.2:[5] Let A and B be any two fuzzy sets in X. Then we define A ∨ B : X → [0,1] as follows: (A ∨ B)()ݔ = max {A(,)ݔ B(.})ݔ Also, we define A ∧ B : X → [0,1] as follows: (A ∧ B)()ݔ = min {A(,)ݔ B(.})ݔ By A ∨ B (A ∧ B), we mean the union (intersection) between two fuzzy sets A and B of X. Definition 2.3:[4] Let A be any fuzzy set in a set X. The complement of A, is denoted by 1X − A or Ac and defined as follows: Ac ()ݔ = 1 − A(,)ݔ for each ∈ݔ X. Remark 2.4: From definition (2.2) and definition (2.3) , we have, if A,B ∈ ΙX , then A ∨ B, A ∧ B and 1X − A ∈ ΙX . Definition 2.5:[9] A fuzzy point ݔఒ in a set X is a fuzzy set defined as follows: ݔఒሺݕሻ = Where 0 < λ ≤ 1. Now, supp(ݔఒ) = { ݕ : ݔఒሺݕሻ > 0}, but λ if ݕ ൌ ݔ 0 otherwise , 1 , for ∈ݔ A 0 , for ∉ݔ A ( see [4] )
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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 167 ݔఒሺݕሻ = supp(ݔఒ) = ݔ , so the value at ݔ is λ, and call the point ݔ its support of fuzzy point ݔఒ and λ is the height of ݔఒ. That is, ݔఒ has the membership degree 0 for all ∈ݕ X except one, say ∈ݔ X. Definition 2.6:[3] A fuzzy topology on a set X is a family T of fuzzy sets in X which satisfies the following conditions: (i) 0X , 1X ∈ T, (ii) If A , B ∈ T , then A ∧ B ∈ T, (iii) If {Ai : i ∈ J } is a family in T, then ∨ Ai ∈ T. T is called a fuzzy topology for X and the pair ሺܺ, ܶሻ(or simply X) is a fuzzy topological space or fts for short. Every element of T is called T - fuzzy open set (fuzzy open set, for short ). A fuzzy set is T - fuzzy closed (or simply fuzzy closed), if its complement is fuzzy open set. As ordinary topologies, the indiscrete fuzzy topology on X contains only 0X and 1X (i.e., φ, X) , while the discrete fuzzy topology on X contains all fuzzy sets in X. Example 2.7: Let X = [−1,1], and let B1 , B2 and B3 are fuzzy sets in X defined as follows: B1()ݔ = B2()ݔ = B3()ݔ = Let T = { 0X , B1 , B2 , B3 , B1∨B3 , 1X }, then T is a fuzzy topology on X, and ሺܺ, ܶሻ is a fts. Example 2.8: Let X = [0,1] and T = {0X, K , 1X }. Then T is a fuzzy topology on X, where K : X → [0,1] defined as: K()ݔ = ݔ2 , for all ∈ݔ X, and ሺܺ, ܶሻ is a fts. Definition 2.9:[3] Let A be any fuzzy set in a fts X. The interior of A is the union of all fuzzy open sets contained in A, denoted by int(A). That is, int(A) = ∨{B : B is fuzzy open set , B ≤ A}. Definition 2.10:[3] Let A be any fuzzy set in a fts X. The closure of A is the intersection of all fuzzy closed sets containing A, denoted by cl(A). That is, cl(A) = ∧{B : B is fuzzy closed set, B ≥ A}. λ if ݕ ൌ ݔ 0 otherwise , and 0 < λ ≤ 1 . Then, i∈ J 1 , if −1 ≤ ݔ < 0 0 , if 0 ≤ ݔ ≤ 1 , 0 , if −1 ≤ ݔ < 0 1 , if 0 ≤ ݔ ≤ 1 , 0 , if −1 ≤ ݔ < 0 1/5 , if 0 ≤ ݔ ≤ 1 .
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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 168 The most important properties of the closure and interior of fuzzy sets are listed in the following proposition. Proposition 2.11:[12] If A is any fuzzy set in X then: (i) A is a fuzzy open (closed) set if and only if A = int(A) ( A = cl(A)), (ii) cl(1X − A) = 1X − int(A), (iii) int(1X − A) = 1X − cl(A). Proposition 2.12:[13] Let A,B be two fuzzy sets in a fts X. Then: (i) int(A) ≤ A, int(int(A)) = int(A), (ii) int(A) ≤ int(B), whenever A ≤ B, (iii) int(A∧B) = int(A) ∧ int(B), int(A∨B) ≥ int(A) ∨ int(B), (iv) A ≤ cl(A), cl(cl(A)) = cl(A), (v) cl(A) ≤ cl(B), whenever A ≤ B, (vi) cl(A∧B) ≤ cl(A) ∧ cl(B), cl(A∨B) = cl(A) ∨ cl(B). Definition 2.13:[1] Let ሺܺ, ܶሻ be a fuzzy topological space. Then ܺ is called: (i) fuzzy ܶ- space (ܶܨ- space, for short) iff for each pair of distinct fuzzy points in ܺ, there exists an fuzzy open set in ܺ containing one and not the other . (ii) fuzzy ܶଵ- space (ܶܨଵ- space, for short) iff for each pair of distinct fuzzy points ݔఒ and ݕఈ of ܺ, there exists an fuzzy open sets ,ܩ ܪ containing ݔఒ and ݕఈ respectively such that ݕఈ ב ܩ and ݔఒ ב .ܪ (iii) fuzzy ܶଶ- space (ܶܨଶ- space, for short) iff for each pair of distinct fuzzy points ݔఒ and ݕఈ of ܺ, there exist disjoint fuzzy open sets ,ܩ ܪ in ܺ such that ݔఒ א ܩ and ݕఈ א .ܪ (iv) fuzzy regular space iff for each fuzzy closed set ܣ and for each ݔఒ ב ,ܣ there exist disjoint fuzzy open sets ,ܩ ܪ such that ݔఒ א ܩ and ܣ .ܪ (v) fuzzy normal space iff for each pair of disjoint fuzzy closed sets ܣ and ,ܤ there exist disjoint fuzzy open sets ܩ and ܪ such that ܣ ܩ and ܤ .ܪ 3. KERNEL SET IN FUZZY TOPOLOGICAL SPACES Definition 3.1: The intersection of all fuzzy open subsets of a fuzzy topological space ሺܺ, ܶሻ containing ܣ is called the kernel of ܣ (briefly ݇݁ݎሺܣሻ ), this means that ݇݁ݎሺܣሻ ൌ ר ሼܩ א ܶ: ܣ ܩሽ. Definition 3.2: In a fuzzy topological space ሺܺ, ܶሻ, a set ܣ is said to be weakly ultra separated from ܤ if there exists a fuzzy open set ܩ such that ܣ ܩ and ܩ ר ܤ ൌ 0 or ܣ ר ݈ܿሺܤሻ ൌ 0. Remark 3.3: By definition (3.2), we have the following: For every two distinct fuzzy points ݔఒ and ݕఈ of ܺ, ݇݁ݎሼݔఒሽ ൌ ሼ ݕఈ: ሼݔఒሽ is not weakly ultra separated from ሼ ݕఈሽ ሽ. Definition 3.4: A fuzzy topological space ሺܺ, ܶሻ is called fuzzy ܴ- space (ܴܨ- space, for short) if for each fuzzy open set ܷ and ݔఒ א ܷ then ݈ܿሼݔఒሽ ܷ. Definition 3.5: A fuzzy topological space ሺܺ, ܶሻ is called fuzzy ܴଵ- space (ܴܨଵ- space, for short) if for each two distinct fuzzy points ݔఒ and ݕఈ of ܺ with ݈ܿሼݔఒሽ ് ݈ܿሼ ݕఈሽ, there exist disjoint fuzzy open sets ܷ, ܸ such that ݈ܿሼݔఒሽ ܷ and ݈ܿሼݔఒሽ ܸ.
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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 169 Remark 3.6: Each fuzzy separation axiom is defined as the conjunction of two weaker axioms: ܶܨ- space ൌ ܴܨିଵ- space and ܶܨିଵ- space ൌ ܴܨିଵ- space and ܶܨ- space, ݅ ൌ 1,2. Lemma 3.7: Let ሺܺ, ܶሻ be a fuzzy topological space then ݕఈ א ݇݁ݎሼݔఒሽ iff ݔఒ א ݈ܿሼ ݕఈሽ, for each ݔ ് ݕ א ܺ. Proof: Let ݕఈ ב ݇݁ݎሼݔఒሽ, then there exist fuzzy open set ܸ containing ݔఒ such that ݕఈ ב ܸ. Thus, ݔఒ ב ݈ܿሼ ݕఈሽ. The converse part can be proved in a similar way. Theorem 3.8: A fuzzy topological space ሺܺ, ܶሻ is ܶܨଵ- space if and only if for each ݔ ് ݕ א ܺ, ݕఈ ב ݇݁ݎሼݔఒሽ and ݔఒ ב ݇݁ݎሼ ݕఈሽ. Proof: Let ሺܺ, ܶሻ be a ܶܨଵ- space then for each ݔ ് ݕ א ܺ, there exists an fuzzy open sets ܷ, ܸ such that ݔఒ א ܷ, ݕఈ ב ܷ and ݕఈ א ܸ, ݔఒ ב ܸ. Implies ݕఈ ב ݇݁ݎሼݔఒሽ and ݔఒ ב ݇݁ݎሼ ݕఈሽ. Conversely, let ݕఈ ב ݇݁ݎሼݔఒሽ and ݔఒ ב ݇݁ݎሼ ݕఈሽ, for each ݔ ് ݕ א ܺ. Then there exists an fuzzy open sets ܷ, ܸ such that ݔఒ א ܷ, ݕఈ ב ܷ and ݕఈ א ܸ, ݔఒ ב ܸ. Thus, ሺܺ, ܶሻ is a ܶܨଵ- space. Theorem 3.9: A fuzzy topological space ሺܺ, ܶሻ is ܶܨଵ- space if and only if for each ݔ א ܺ then ݇݁ݎሼݔఒሽ ൌ ሼݔఒሽ. Proof: Let ሺܺ, ܶሻ be a ܶܨଵ- space and let ݇݁ݎሼݔఒሽ ് ሼݔఒሽ, then ݇݁ݎሼݔఒሽ contains another fuzzy point distinct from ݔఒ say ݕఈ. So ݕఈ א ݇݁ݎሼݔఒሽ. Hence by theorem (3.8), ሺܺ, ܶሻ is not a ܶܨଵ- space this is a contradiction. Thus, ݇݁ݎሼݔఒሽ ൌ ሼݔఒሽ. Conversely, let ݇݁ݎሼݔఒሽ ൌ ሼݔఒሽ, for each ݔ א ܺ and let ሺܺ, ܶሻ be not a ܶܨଵ- space. Then, by theorem (3.8) , ݕఈ א ݇݁ݎሼݔఒሽ, implies ݇݁ݎሼݔఒሽ ് ሼݔఒሽ, this is a contradiction. Thus, ሺܺ, ܶሻ is a ܶܨଵ- space. Definition 3.10: Let ሺܺ, ܶሻ be a fuzzy topological space. A fuzzy point ݔఒ is said to be kernelled fuzzy point of ܣ ܺ (Briefly ݔఒ א ݇݁ݎሺ))ܣ if and only if for each ܩ fuzzy closed set contains ݔఒ then ܩ ר ܣ ് 0. Definition 3.11: Let ሺܺ, ܶሻ be a fuzzy topological space. A fuzzy point ݔఒ is said to be boundary kernelled fuzzy point of ܣ (Briefly ݔఒ א ݇ݎௗሺ))ܣ if and only if for each fuzzy closed set ܩ contains ݔఒ then ܩ ר ܣ ് 0 and ܩ ר ܣ ് 0. Definition 3.12: Let ሺܺ, ܶሻ be a fuzzy topological space. A fuzzy point ݔఒ is said to be derived kernelled fuzzy point of ܣ (Briefly ݔఒ א ݇ݎௗሺ))ܣ if and only if for each ܩ fuzzy closed set contains ݔఒ then ܣ ר ܩ ോ ሼݔఒሽ ് 0. Definition 3.13: By definition (3.10), we have the following: For every two distinct fuzzy points ݔఒ and ݕఈ of ܺ, ݇݁ݎሼݔఒሽ = ሼݕఈ: ݔఒ א ܩ௬ഀ , ܩ௬ഀ א ܶሽ. Theorem 3.14: Let ሺܺ, ܶሻ be a fuzzy topological space and ݔ ് ݕ א ܺ. Then ݔఒ is a kernelled fuzzy point of ሼ ݕఈሽ if and only if ݕఈ is an adherent fuzzy point of ሼݔఒሽ. Proof: Let ݔఒ be a kernelled fuzzy point of ሼ ݕఈሽ. Then for every fuzzy closed set ܩ such that ݔఒ א ܩ implies ݕఈ א ,ܩ then ݕఈ א ר ሼ:ܩ ݔఒ א ܩሽ, this means ݕఈ א ݈ܿሼݔఒሽ. Thus ݕఈ is an adherent fuzzy point of ሼݔఒሽ.
6.
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 170 Conversely, let ݕఈ be an adherent fuzzy point of ሼݔఒሽ. Then for every fuzzy open set ܷ such that ݕఈ א ܷ implies ݔఒ א ܷ, then ݔఒ א ר ሼܷ: ݕఈ א ܷሽ, this means ݔఒ א ݇݁ݎሼ ݕఈሽ. Thus, ݔఒ is a kernelled fuzzy point of ሼ ݕఈሽ. Theorem 3.15: Let ሺܺ, ܶሻ be a fuzzy topological space and ܣ ܺ and let ݇ݎௗሺ)ܣ be the set of all kernelled derived fuzzy points of ,ܣ then ݇݁ݎሺܣሻ ൌ ܣ ש ݇ݎௗሺ.)ܣ Proof: Let ݔఒ א ܣ ש ݇ݎௗሺ)ܣ and if ݔఒ א ݇ݎௗሺܣሻ, then for every fuzzy closed set ܩ intersects ܣ (in a fuzzy point different from ݔఒ). Therefore, ݔఒ א ݇݁ݎሼݔఒሽ. Hence, ݇ݎௗሺܣሻ ݇݁ݎሺܣሻ, it follows that ܣ ש ݇ݎௗሺܣሻ ݇݁ݎሺܣሻ. To demonstrate the reverse inclusion, we consider ݔఒ be a fuzzy point of ݇݁ݎሺܣሻ. If ݔఒ א ,ܣ then ݔఒ א ܣ ݇ݎௗሺܣሻ. Suppose that ݔఒ ב .ܣ Since ݔఒ א ݇݁ݎሺܣሻ, then for every fuzzy closed set ܩ containing ݔఒ implies ܩ ר ܣ ് 0, this means ܣ ר ܩ ോ ሼݔఒሽ ് 0. Then, ݔఒ א ݇ݎௗሺܣሻ, so that ݔఒ א ܣ ש ݇ݎௗሺܣሻ. Theorem 3.16: Let ሺܺ, ܶሻ be a fuzzy topological space and ܣ ܺ and let ݇ݎௗሺ)ܣ be the set of all kernelled boundary fuzzy points of ,ܣ then ݇݁ݎሺܣሻ ൌ ܣ ש ݇ݎௗሺ.)ܣ Proof: Let ݔఒ א ܣ ש ݇ݎௗሺܣሻ and if ݔఒ א ݇ݎௗሺܣሻ , then for every fuzzy closed set ܩ intersects ,ܣ therefore, ݔఒ א ݇݁ݎሼݔఒሽ. Hence, ݇ݎௗሺܣሻ ݇݁ݎሺܣሻ, it follows that ܣ ש ݇ݎௗሺܣሻ ݇݁ݎሺܣሻ. To demonstrate the reverse inclusion, we consider ݔఒ be a fuzzy point of ݇݁ݎሺܣሻ. If ݔఒ א ,ܣ then ݔఒ א ܣ ש ݇ݎௗሺܣሻ. Suppose that ݔఒ ב ,ܣ implies ݔఒ א ܣ . Since ݔఒ א ݇݁ݎሺܣሻ, then for every fuzzy closed set ܩ containing ݔఒ implies ܩ ר ܣ ≠ 0 and ܩ ר ܣ ് 0. Then ݔఒ א ݇ݎௗሺܣሻ, so that ݔఒ א ܣ ש ݇ݎௗሺܣሻ. Hence, ݇݁ݎሺܣሻ ܣ ש ݇ݎௗሺܣሻ. Thus, ݇݁ݎሺܣሻ ൌ ܣ ש ݇ݎௗሺܣሻ. Corollary 3.17: Every interior fuzzy point is a kernelled fuzzy point. Proof: Since ݅݊ݐሺܣሻ ܣ ݇݁ݎሺܣሻ. Thus, every interior fuzzy point is a kernelled fuzzy point. Theorem 3.18: Let ሺܺ, ܶሻ be a fuzzy topological space and ܣ is a subset of ܺ. Then ܣ is a fuzzy open set if and only if every ݔఒ kernelled fuzzy point of ܣ is an interior fuzzy point of .ܣ Proof: Let ܣ be a fuzzy open set, then ݇݁ݎሺܣሻ ൌ ܣ ൌ ݅݊ݐሺܣሻ, implies every kernelled fuzzy point is an interior fuzzy point. Conversely, let every ݔఒ kernelled fuzzy point of ܣ is an interior fuzzy point of .ܣ Then ݇݁ݎሺܣሻ ݅݊ݐሺܣሻ. Hence, ݅݊ݐሺܣሻ ܣ ݇݁ݎሺܣሻ, implies ݅݊ݐሺܣሻ = ܣ = ݇݁ݎሺܣሻ. Thus ܣ is a fuzzy open set. Corollary 3.19: A subset ܣ of ܺ is a fuzzy open set if and only if for each ݔఒ kernelled fuzzy point then ݔఒ ב ݈ܿሺܣሻ. Proof: By theorem (3.18). Theorem 3.20: A subset ܣ of ܺ is a fuzzy closed set if and only if ݇݁ݎሺܣሻ ר ݈ܿሺܣሻ ൌ 0. Proof: Let ܣ is a fuzzy closed set. Then ܣ is a fuzzy open set, implies ܣ ൌ ݇݁ݎሺܣሻ by theorem (3.18). Hence ܣ = ݈ܿሺܣሻ. Thus ݇݁ݎሺܣሻ ר ݈ܿሺܣሻ ൌ 0. Conversely, let ݇݁ݎሺܣሻ ר ݈ܿሺܣሻ ൌ 0, then for each ݔఒ א ݇݁ݎሺܣሻ, implies ݔఒ ב ݈ܿሺܣሻ, implies ݔఒ א ݁ݐݔሺܣሻ. Therefore ݔఒ א ݅݊ݐሺܣ ሻ. Hence by theorem (3.18), ܣ is a fuzzy open set. Thus ܣ is a fuzzy closed set.
7.
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 171 Theorem 3.21: A fuzzy topological space ሺܺ, ܶሻ is ܴܨ- space if and only if every adherent fuzzy point of ሼݔఒሽ is a kernelled fuzzy point of ሼݔఒሽ. Proof: Let ሺܺ, ܶሻ be an ܴܨ- space. Then, for each ݔ א ܺ, ݇݁ݎሼݔఒሽ = ݈ܿሼݔఒሽ by lemma (3.7). Thus, every adherent fuzzy point of ሼݔఒሽ is a kernelled fuzzy point of ሼݔఒሽ. Conversely, let every adherent fuzzy point of ሼݔఒሽ is a kernelled fuzzy point of ሼݔఒሽ and let ܷ ܺ and ݔఒ א ܷ. Then ݈ܿሼݔఒሽ ݇݁ݎሼݔఒሽ for each ݔ א ܺ. Since ݇݁ݎሼݔఒሽ ൌ ר ሼܷ: ܷ א ܶ, ݔఒ א ܷሽ, implies ݈ܿሼݔఒሽ ܷ, for each ܷ fuzzy open set contains ݔఒ. Thus, ሺܺ, ܶሻ is an ܴܨ- space. Theorem 3.22: A fuzzy topological space ሺܺ, ܶሻ is ܶܨ- space if and only if for each ݔ ് ݕ א ܺ, either ݔఒ is not kernelled fuzzy point of ሼ ݕఈሽ or ݕఈ is not kernelled fuzzy point of ሼݔఒሽ. Proof: Let ሺܺ, ܶሻ be an ܶܨ- space. Then for each ݔ ് ݕ א ܺ there exists a fuzzy open set ܷ such that ݔఒ א ܷ, ݕఈ ב ܷ (say), implies ݕఈ א ܷ . Hence ܷ is a fuzzy closed, then ݕఈ is not kernelled fuzzy point of ሼݔఒሽ. Thus either ݔఒ is not kernelled fuzzy point of ሼ ݕఈሽ or ݕఈ is not kernelled fuzzy point of ሼݔఒሽ. Conversely, let for each ݔ ് ݕ א ܺ, either ݔఒ is not kernelled fuzzy point of ሼ ݕఈሽ or ݕఈ is not kernelled fuzzy point of ሼݔఒሽ. Then there exist a fuzzy closed set ܩ such that ݔఒ א ܩ , ܩ ר ሼ ݕఈሽ ൌ 0 or ݕఈ א ܩ , ܩ ר ሼݔఒሽ ൌ 0, implies ݔఒ ב ܩ , ݕఈ א ܩ or ݔఒ א ܩ , ݕఈ ב ܩ . Hence ܩ is a fuzzy open set. Thus, ሺܺ, ܶሻ is a ܶܨ- space. Theorem 3.23: A fuzzy topological space ሺܺ, ܶሻ is ܶܨଵ- space if and only if ݇ݎௗሼݔఒሽ ൌ 0, for each ݔ א ܺ. Proof: Let ሺܺ, ܶሻ be an ܶܨଵ- space. Then for each ݔ א ܺ, ݇݁ݎሼݔఒሽ = ሼݔఒሽ by theorem (3.9). Since ݇ݎௗሼݔఒሽ = ݇݁ݎሼݔఒሽ െ ሼݔఒሽ. Thus, ݇ݎௗሼݔఒሽ ൌ 0. Conversely, let ݇ݎௗሼݔఒሽ ൌ 0. By theorem (3.14), ݇݁ݎሼݔఒሽ ൌ ሼݔఒሽ ש ݇ݎௗሼݔఒሽ, implies ݇݁ݎሼݔఒሽ ൌ ሼݔఒሽ. Hence, by theorem (3.9), ሺܺ, ܶሻ is a ܶܨଵ- space. Theorem 3.24: A fuzzy topological space ሺܺ, ܶሻ is ܶܨଵ- space if and only if for each ݔ ് ݕ א ܺ, ݔఒ is not kernelled fuzzy point of ሼ ݕఈሽ and ݕఈ is not kernelled fuzzy point of ሼݔఒሽ. Proof: Let ሺܺ, ܶሻ be a ܶܨଵ- space. Then for each ݔ ് ݕ א ܺ there exist fuzzy open sets ܷ, ܸ such that ݔఒ א ܷ, ݕఈ ב ܷ and ݕఈ א ܸ, ݔఒ ב ܸ, implies ݔఒ א ܸ , ሼ ݕఈሽ ר ܸ ൌ 0 and ݕఈ א ܷ , ሼݔఒሽ ר ܷ ൌ 0. Hence ܷ and ܸ are fuzzy closed sets. Thus ݔఒ is not kernelled fuzzy point of ሼ ݕఈሽ and ݕఈ is not kernelled fuzzy point of ሼݔఒሽ. Conversely, let for each ݔ ് ݕ א ܺ, ݔఒ is not kernelled fuzzy point of ሼ ݕఈሽ and ݕఈ is not kernelled fuzzy point of ሼݔఒሽ. Then, there exist fuzzy closed sets ܩଵ, ܩଶ such that ݔఒ א ܩଵ , ܩଵ ר ሼ ݕఈሽ ൌ 0 and ݕఈ א ܩଶ, ܩଶ ר ሼݔఒሽ ൌ 0, implies ݔఒ א ܩଶ , ݕఈ ב ܩଶ and ݕఈ א ܩଵ , ݔఒ ב ܩଵ . Hence ܩଵ and ܩଶ are fuzzy open sets. Thus, ሺܺ, ܶሻ is ܶܨଵ- space. 4. kr- Spaces in Fuzzy Topological Spaces: Definition 4.1: A fuzzy topological space ሺܺ, ܶሻ is said to be a ݇-ݎ space if and only if for each subset ܣ of ܺ then, ݇݁ݎሺܣሻ is a fuzzy open set. Definition 4.2: A fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is called fuzzy ܶ- space (ܶܨ- space, for short) if and only if for each ݔఒ א ܺ, then ݇ݎௗሼݔఒሽ is a fuzzy open set.
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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 172 Theorem 4.3: In fuzzy topological ݇-ݎ space ሺܺ, ܶሻ, every ܶܨଵ-space is a ܶܨ- space. Proof: Let ሺܺ, ܶሻ be a ܶܨଵ- space. Then, for each ݔఒ א ܺ, ݇݁ݎሼݔఒሽ ൌ ሼݔఒሽ by theorem (3.9). As ݇ݎௗሼݔఒሽ ൌ ݇݁ݎሼݔఒሽ െ ሼݔఒሽ, implies ݇ݎௗሼݔఒሽ ൌ 0. Thus, ሺܺ, ܶሻ is a ܶܨ- space. Theorem 4.4: In fuzzy topological ݇-ݎ space ሺܺ, ܶሻ, every ܶܨ- space is a ܶܨ- space. Proof: Let ሺܺ, ܶሻ be a ܶܨ- space and let ݔ ് ݕ א ܺ. Then, ݇ݎௗሼݔఒሽ is a fuzzy open set, therefore, there exist two cases: (i) ݕఈ א ݇ݎௗሼݔఒሽ is a fuzzy open set. Since ݔఒ ב ݇ݎௗሼݔఒሽ. Thus ሺܺ, ܶሻ is a ܶܨ- space. (ii) ݕఈ ב ݇ݎௗሼݔఒሽ, implies ݕఈ ב ݇݁ݎሼݔఒሽ. But ݇݁ݎሼݔఒሽ is a fuzzy open set. Thus ሺܺ, ܶሻ is a ܶܨ- space. Definition 4.5: A fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is said to be fuzzy ܶ- space (ܶܨ- space, for short) if and only if for each ݔ ് ݕ א ܺ, ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ is degenerated (empty or singleton fuzzy set). Theorem 4.6: In fuzzy topological ݇-ݎ space ሺܺ, ܶሻ, every ܶܨଵ- space is ܶܨ- space. Proof: Let ሺܺ, ܶሻ be a ܶܨଵ- space. Then for each ݔ ് ݕ א ܺ, ݇݁ݎሼݔఒሽ ൌ ሼݔఒሽ and ݇݁ݎሼ ݕఈሽ ൌ ሼ ݕఈሽ by theorem (3.9), implies ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ ൌ 0. Thus ሺܺ, ܶሻ is a ܶܨ- space. Theorem 4.7: In fuzzy topological ݇-ݎ space ሺܺ, ܶሻ, every ܶܨ- space is a ܶܨ- space. Proof: Let ሺܺ, ܶሻ be a ܶܨ- space. Then for each ݔ ് ݕ א ܺ, ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ is degenerated (empty or singleton fuzzy set). Therefore there exist three cases: (i) ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ ൌ 0, implies ሺܺ, ܶሻ is a ܶܨ- space. (ii) ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ ൌ ሼݔఒሽ ݎ ሼ ݕఈሽ, implies ݕఈ ב ݇݁ݎሼݔሽ or ݔఒ ב ݇݁ݎሼ ݕఈሽ implies ሺܺ, ܶሻ is a ܶܨ- space. (iii) ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ ൌ ൛ݖఉൟ , ݖ ് ݔ ് ݕ , ݖ א ܺ, implies ݕఈ ב ݇݁ݎሼݔఒሽ and ݔఒ ב ݇݁ݎ ሼ ݕఈሽ, implies ሺܺ, ܶሻ is a ܶܨ- space. Definition 4.8: A fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is said to be a fuzzy ܶே- space (ܶܨே- space, for short) if and only if for each ݔ ് ݕ א ܺ, ݇݁ݎ ሼݔఒሽ ר ݇݁ݎ ሼ ݕఈሽ is empty or {ݔఒ} or { ݕఈ}. Theorem 4.9: In fuzzy topological ݇-ݎ space ሺܺ, ܶሻ, every ܶܨଵ- space is ܶܨே- space. Proof: Let ሺܺ, ܶሻ be a ܶܨே- space. Then for each ݔ ് ݕ א ܺ, ݇݁ݎሼݔఒሽ ൌ ሼݔఒሽ and ݇݁ݎሼ ݕఈሽ ൌ ሼ ݕఈሽ by theorem (3.9), implies ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ ൌ 0. Thus ሺܺ, ܶሻ is a ܶܨே- space. Theorem 4.10: In fuzzy topological ݇-ݎ space ሺܺ, ܶሻ, every ܶܨே- space is a ܶܨ- space. Proof: Let ሺܺ, ܶሻ be a ܶܨே- space. Then for each ݔ ് ݕ א ܺ, ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ is degenerated (empty or singleton fuzzy set). Therefore there exist two cases: (i) ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ ൌ 0 , implies ሺܺ, ܶሻ is a ܶܨ- space. (ii) ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ ൌ ሼݔఒሽ or ሼ ݕఈሽ, implies ݕఈ ב ݇݁ݎሼݔఒሽ or ݔఒ ב ݇݁ݎሼ ݕఈሽ, implies ሺܺ, ܶሻ is a ܶܨ- space.
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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 173 Theorem 4.11: A fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is ܶܨଶ- space iff for each ݔ ് ݕ א ܺ, then ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ ൌ 0. Proof: Let a fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is ܶܨଶ- space. Then for each ݔ ് ݕ א ܺ there exist disjoint fuzzy open sets ܷ, ܸ such that ݔఒ א ܷ, and ݕఈ א ܸ. Hence ݇݁ݎሼݔఒሽ ܷ and ݇݁ݎሼ ݕఈሽ ܸ. Thus ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ = 0. Conversely, let for each ݔ ് ݕ א ܺ, ݇݁ݎሼݔఒሽ ר ݇݁ݎሼ ݕఈሽ ൌ 0. Since ሺܺ, ܶሻ is a fuzzy topological ݇-ݎ space, this means kernel is a fuzzy open set. Thus ሺܺ, ܶሻ is ܶܨଶ- space. Theorem 4.12: A fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is fuzzy regular space iff for each ܩ fuzzy closed set and ݔఒ ב ,ܩ then ݇݁ݎሺܩሻ ר ݇݁ݎሼݔఒሽ ൌ 0. Proof: By the same way of proof of theorem (4.11). Theorem 4.13: A fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is fuzzy normal space iff for each disjoint fuzzy closed sets ,ܩ ,ܪ then ݇݁ݎሺܩሻ ר ݇݁ݎሺܪሻ ൌ 0. Proof: By the same way of proof of theorem (4.11). Theorem 4.14: A fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is ܶܨଵ- space iff it is ܴܨ- space and ܶܨ- space. Proof: It follows from theorem (4.3) and remark (3.6). Theorem 4.15: A fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is ܶܨଵ- space iff it is ܴܨ- space and ܶܨ- space. Proof: It follows from theorem (4.6) and remark (3.6). Theorem 4.16: A fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is ܶܨଵ- space if and only if it is ܴܨ- space and ܶܨே- space. Proof: It follows from theorem (4.9) and remark (3.6). Theorem 4.17: A fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is ܶܨ- space if and only if it is ܴܨିଵ- space and ܶܨ-space, ݅ ൌ 1,2. Proof: It follows from theorem (4.3) and remark (3.6). Theorem 4.18: A fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is ܶܨ- space if and only if it is ܴܨିଵ- space and ܶܨ-space, ݅ ൌ 1,2. Proof: It follows from theorem (4.6) and remark (3.6). Theorem 4.19: A fuzzy topological ݇-ݎ space ሺܺ, ܶሻ is ܶܨ- space if and only if it is ܴܨିଵ- space and ܶܨே-space, ݅ ൌ 1,2. Proof: It follows from theorem (4.9) and remark (3.6).
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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 165-174 © IAEME 174 Remark 4.20: The relation between fuzzy separation axioms can be representing as a matrix. Therefore, the element ܽ refers to this relation. As the following matrix representation shows: Matrix Representation The relation between fuzzy separation axioms in fuzzy topological ݇-ݎ spaces REFERENCES 1. A. S. Mashour, E, E. Kerre and M. H. Ghanim, “Separation axioms, sub spaces and sums in fuzzy topology”, J. Math Anal., 102(1984), 189-202. 2. B. Sikn, “On fuzzy FC - Compactness”, Comm. Korean Math. Soc.13 (1998), 137-150. 3. C. L. Chang, “Fuzzy topological spaces”, J. Math. Anal. Appl. 24(1968), 182-190. 4. G. J. Klir, U. S. Clair, B. Yuan, “ Fuzzy set theory ”, foundations and applications, 1997. 5. G. Balasubramanian, “Fuzzy β - open sets and fuzzy β - separation axioms”, Kybernetika 35(1999), 215-223. 6. G. Balasubramania and P. Sundaram, “On some generalizations of fuzzy continuous functions”, Fuzzy sets and Systems, 86(1) (1997), 93-100. 7. H. Maki et al., “Generalized closed sets in fuzzy topological space, I, Meetings on Topological spaces Theory and its Applications”, (1998), 23-36. 8. L. A. Zadeh, “Fuzzy sets”, Information, and control, 8 (1965), 338 -353. 9. M. Sarkar, “On fuzzy topological spaces”, J. Math. Anal. Appl. 79 (1981), 384-394. 10. M. Alimohammady and M. Roohi, “On Fuzzy φψ - Continuous Multifunction”, J. App. Math. sto. Anal. (2006), 1-7. 11. O. Berde Ozbaki, “On generalized fuzzy strongly semiclosed sets in fuzzy topological spaces”, Int. J. Math. Sci. 30 (11) (2002), 651- 657. 12. R. H. Warren, “Neighborhoods, bases and continuity in fuzzy topological spaces”, The Rocky Mountain Journal of Mathematics, Vol. 8, No.3, (1977), 459-470. 13. X. Tang, “Spatial object modeling in fuzzy topological spaces with application to lands cover change in china”, Ph. D. Dissertation, ITC Dissertation No.108, Univ. of Twente, The Nether lands, (2004). 14. Gunwanti S. Mahajan and Kanchan S. Bhagat, “Medical Image Segmentation using Enhanced K-Means and Kernelized Fuzzy C- Means”, International Journal of Electronics and Communication Engineering &Technology (IJECET), Volume 4, Issue 6, 2013, pp. 62 - 70, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472. and ܶܨ ܶܨଵ ܶܨଶ ܴܨ ܴܨଵ ܶܨ ܶܨ ܶܨே ܶܨ ܶܨ ܶܨଵ ܶܨଶ ܶܨଵ ܶܨଶ ܶܨ ܶܨ ܶܨே ܶܨଵ ܶܨଵ ܶܨଵ ܶܨଶ ܶܨଵ ܶܨଶ ܶܨଵ ܶܨଵ ܶܨଵ ܶܨଶ ܶܨଶ ܶܨଶ ܶܨଶ ܶܨଶ ܶܨଶ ܶܨଶ ܶܨଶ ܶܨଶ ܴܨ ܶܨଵ ܶܨଵ ܶܨଶ ܴܨ ܴܨଵ ܶܨଵ ܶܨଵ ܶܨଵ ܴܨଵ ܶܨଶ ܶܨଶ ܶܨଶ ܴܨଵ ܴܨଵ ܶܨଶ ܶܨଶ ܶܨଶ ܶܨ ܶܨ ܶܨଵ ܶܨଶ ܶܨଵ ܶܨଶ ܶܨ ܶܨ ܶܨ ܶܨ ܶܨ ܶܨଵ ܶܨଶ ܶܨଵ ܶܨଶ ܶܨ ܶܨ ܶܨ ܶܨே ܶܨே ܶܨଵ ܶܨଶ ܶܨଵ ܶܨଶ ܶܨ ܶܨ ܶܨே
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