International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Vol...
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  1. 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
  2. 2. 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] )
  3. 3. 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 .
  4. 4. 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 ൑ ܸ.
  5. 5. 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 ൌ ‫ܴܨ‬௜ିଵ- space and ‫ܶܨ‬௜ିଵ- space ൌ ‫ܴܨ‬௜ିଵ- 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 ݇‫ݎ‬௕ௗሺ‫))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 ݇‫ݎ‬ௗ௥ሺ‫))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. 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 ݇‫ݎ‬ௗ௥ሺ‫)ܣ‬ be the set of all kernelled derived fuzzy points of ‫,ܣ‬ then ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫.)ܣ‬ Proof: Let ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫)ܣ‬ and if ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ, then for every fuzzy closed set ‫ܩ‬ intersects ‫ܣ‬ (in a fuzzy point different from ‫ݔ‬ఒ). Therefore, ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ. Hence, ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ, it follows that ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. To demonstrate the reverse inclusion, we consider ‫ݔ‬ఒ be a fuzzy point of ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. If ‫ݔ‬ఒ ‫א‬ ‫,ܣ‬ then ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ൑ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ. 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 ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ, so that ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ. Theorem 3.16: Let ሺܺ, ܶሻ be a fuzzy topological space and ‫ܣ‬ ൑ ܺ and let ݇‫ݎ‬௕ௗሺ‫)ܣ‬ be the set of all kernelled boundary fuzzy points of ‫,ܣ‬ then ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫.)ܣ‬ Proof: Let ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ and if ‫ݔ‬ఒ ‫א‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ , then for every fuzzy closed set ‫ܩ‬ intersects ‫,ܣ‬ therefore, ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ. Hence, ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ, it follows that ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ ൑ ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. To demonstrate the reverse inclusion, we consider ‫ݔ‬ఒ be a fuzzy point of ݇݁‫ݎ‬ሺ‫ܣ‬ሻ. If ‫ݔ‬ఒ ‫א‬ ‫,ܣ‬ then ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ. 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 ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ, so that ‫ݔ‬ఒ ‫א‬ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ. Hence, ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൑ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬௕ௗሺ‫ܣ‬ሻ. Thus, ݇݁‫ݎ‬ሺ‫ܣ‬ሻ ൌ ‫ܣ‬ ‫ש‬ ݇‫ݎ‬ௗ௥ሺ‫ܣ‬ሻ. 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 ݈ܿሺ‫ܣ‬௖ሻ. Proof: By theorem (3.18). Theorem 3.20: A subset ‫ܣ‬ of ܺ is a fuzzy closed set if and only if ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ ‫ר‬ ݈ܿሺ‫ܣ‬ሻ ൌ 0௑. Proof: Let ‫ܣ‬ is a fuzzy closed set. Then ‫ܣ‬௖ is a fuzzy open set, implies ‫ܣ‬௖ ൌ ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ by theorem (3.18). Hence ‫ܣ‬ = ݈ܿሺ‫ܣ‬ሻ. Thus ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ ‫ר‬ ݈ܿሺ‫ܣ‬ሻ ൌ 0௑. Conversely, let ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ ‫ר‬ ݈ܿሺ‫ܣ‬ሻ ൌ 0௑, then for each ‫ݔ‬ఒ ‫א‬ ݇݁‫ݎ‬ሺ‫ܣ‬௖ሻ, 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. 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 ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ 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 ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ = ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ െ ሼ‫ݔ‬ఒሽ. Thus, ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ 0௑. Conversely, let ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ 0௑. By theorem (3.14), ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ ൌ ሼ‫ݔ‬ఒሽ ‫ש‬ ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ, 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 ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ is a fuzzy open set.
  8. 8. 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 ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ ݇݁‫ݎ‬ሼ‫ݔ‬ఒሽ െ ሼ‫ݔ‬ఒሽ, implies ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ ൌ 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, ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ is a fuzzy open set, therefore, there exist two cases: (i) ‫ݕ‬ఈ ‫א‬ ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ is a fuzzy open set. Since ‫ݔ‬ఒ ‫ב‬ ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ. Thus ሺܺ, ܶሻ is a ‫ܶܨ‬଴- space. (ii) ‫ݕ‬ఈ ‫ב‬ ݇‫ݎ‬ௗ௥ሼ‫ݔ‬ఒሽ, 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.
  9. 9. 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 ‫ܴܨ‬௜ିଵ- 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 ‫ܴܨ‬௜ିଵ- 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 ‫ܴܨ‬௜ିଵ- space and ‫ܶܨ‬ே-space, ݅ ൌ 1,2. Proof: It follows from theorem (4.9) and remark (3.6).
  10. 10. 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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