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Asian Journal of Current Engineering and Maths 2 : 4 July – August (2013) 244 - 247 Contents lists available at www.innovativejournal.in ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS Journal homepage: http://www.innovativejournal.in/index.php/ajcem A STUDY ON FUZZY LOCALLY - CLOSED SETS D. Amsaveni, Dr. B. Amudhambigai, R. Dhivya Department of Mathematics, Sri Sarada College for Women, Salem-16, Tamil Nadu, India. ARTICLE INFO ABSTRACT Corresponding Author Dr. B. Amudhambigai Department of Mathematics, Sri Sarada College for Women, Salem16, Tamil Nadu, India. In this paper, the interrelations of fuzzy locally -regular closed sets and fuzzy locally -closed sets are studied with suitable counter examples. Also, the interrelations of fuzzy locally -regular continuous functions with other types of fuzzy locally -continuous functions are established with the required counter examples. Finally the properties and the characterizations of fuzzy locally -compact spaces and fuzzy locally -compact* spaces are discussed. Key Words: fuzzy locally -closed sets, fuzzy locally pre-closed sets, fuzzy locally regular-closed sets. ©2013, AJCEM, All Right Reserved. 1.INTRODUCTION AND PRELIMINARIES The concept of fuzzy set was introduced by zadeh [13] in his classical paper. The concept of fuzzy topological spaces was introduced and developed by Chang [5].Fuzzy sets have application in many fields such as information theory [11] and control [12]. The first step of locally closedness was done by Bourbaki [4]. Ganster and Reily used locally closed sets in [7] to define LC-continuity and LC-irresoluteness. The notions of fuzzy -closure derived from regular closed sets and fuzzy -closure of fuzzy sets in a fuzzy topological spaces were introduced by Ganguly and Saha [6] and Mukherjee and Sinha [9], respectively. Fuzzy -compactness in fuzzy topological spaces was discussed by Hanafy [8]. The concepts of fuzzy -continuity and fuzzy -compactness are studied by Seok Jong Lee and Sang Min Yin [10]. The concepts of r-fuzzy - -locally closed sets and fuzzy - -locally continuous functions were introduced by B.Amudhambigai, M.K. Uma, and E. Roja [1]. In this paper, the interrelations of fuzzy locally -regular closed sets and fuzzy locally -closed sets are studied with suitable counter examples. Also, the interrelations of fuzzy locally -regular continuous functions with other types of fuzzy locally -continuous functions are established with the required counter examples. Finally the properties and the characterizations of fuzzy locally -compact spaces and fuzzy locally -compact* spaces are discussed. 2. PRELIMINARIES Definition 2.1 A fuzzy set in a fuzzy topological space ( X, T ) is said to be a) fuzzy pre-closed if cl ( int ( ) ) . [3] b) fuzzy regular-closed if = cl ( int ( ) ). [2] Definition 2.3 [6] A fuzzy point X is said to be a fuzzy cluster point of a fuzzy set A if and only if every regular open q-neighbourhood U of in q-coincident with A. The set of all fuzzy -cluster points of A is called the fuzzy closure of A, and denoted by cl ( A ). Definition 2.4 [10] For a fuzzy subset A in a fuzzy topological space X, the fuzzy -interior is defined as follows : int ( A ) = 1 – cl ( 1 – A ). Definition 2.5 [8] A collection { Ui | i I } of fuzzy -open sets in a fuzzy topological space ( X, T ) is called a fuzzy open cover of a fuzzy set A if A holds. Definition 2.6 [8] A fuzzy topological space ( X, T ) is said to be a fuzzy -compact space if every fuzzy -open cover of ( X, T ) has a finite subcover. A fuzzy subset A of a fuzzy topological space ( X, T ) is said to be fuzzy -compact in X provided for every collection { Ui / i I } of fuzzy -open sets of X such that A , there exists a finite subset I0 of I such that A . 3. ON FUZZY LOCALLY -REGULAR CLOSED SETS Definition 3.1 Let ( X, T ) be a fuzzy topological space. Any X I is called a fuzzy locally -closed set ( briefly, fl-cls ) if = , where is fuzzy -open and is a fuzzy closed set. Definition 3.2 Let ( X, T ) be a fuzzy topological space. Any X I is called a fuzzy locally pre-closed set (briefly, fl pre-cls) if = , where is fuzzy -open and is a fuzzy pre-closed set. Definition 3.3 Let ( X, T ) be a fuzzy topological space. Any X I is called a fuzzy locally regular-closed set (briefly, fl-reg cls) if = , where is fuzzy -open and is a fuzzy regular closed set. Proposition 3.1 Every fuzzy locally -regular closed set is fuzzy locally -closed. Remark 3.1 The converse of the above Proposition 3.1 need not be true. Example 3.1 Every fuzzy locally -closed set need not be fuzzy locally -regular closed. 244 Amudhambigai et.al/A Study on Fuzzy Locally - Closed Sets X Let X = { a, b } and 1, 2, 3 I be defined as, 1( a ) = 0.4, 1( b ) = 0.3; 2( a ) = 0.5, 2( b ) = 0.7; 3( a ) = 0.45, 3( b ) = 0.65. Define the fuzzy topology T as T = { 0, 1, 1, 2, 3 }. Clearly, (X,T) is a fuzzy topological space. For any fuzzy open set 1 and fuzzy closed set 1 - 3, 1 ( 1 - 3 ) = (1 - 3) is fuzzy locally -closed. But, is not fuzzy locally regular closed. Proposition 3.2 Every fuzzy locally -closed set is fuzzy locally pre-closed. Proposition 3.3 Every fuzzy locally -regular closed set is fuzzy locally pre-closed. Remark 3.2 The converse of the above Proposition 3.2 and Proposition 3.3 need not be true. Example 3.2 Every fuzzy locally pre-closed set need not be fuzzy locally - closed and fuzzy locally -regular closed. of all fuzzy -open sets containing That is, f - int () = { µ : µ is fuzzy -open and µ ≤ }. Remark 4.1 Let ( X, T ) be any fuzzy topological space and let be any fuzzy set in ( X, T ). Then, (a) f - cl ( ) = 1− f - int ( ) (b) f - int ( ) = 1− f - cl ( ). Proposition 4.1 Let ( X, T ) and ( Y, S ) be any two fuzzy topological spaces and then for any function f : ( X, T ) ( Y, S ), the following statements are equivalent: (a) f is fuzzy locally -continuous. X (b) For every I , f ( fl -cl ( ) ) cl ( f ( ) ) Y -1 Y -1 (c) For every I , f ( cl ( ) ) fl -cl f -1 ( ) ) (d) For every I , f ( int ( ) ) fl -int f -1 ( ) ) Proposition 4.2 Every fuzzy locally -regular continuous function is fuzzy locally -continuous. Let X = { a, b }, and 1, 2, 3 I be defined as, Remark 4.2 The converse of the above Proposition 4.1 need not be true. ) = 0.5, 2( b ) = 0.7; 3( a ) = 0.4, 3( b ) = 0.6. Let IX be Example 4.1 Every fuzzy locally -continuous function need not be fuzzy locally -regular continuous. X 1( a ) = 0.4, 1( b ) = 0.3; 2( a defined as ( a ) = 0.6 and ( b ) = 0.4. Then, is fuzzy preclosed. Thus, 1 = is fuzzy locally pre-closed. But, is not fuzzy locally -regular closed and hence not fuzzy locally -closed. Since it is not an intersection of any fuzzy -open set with fuzzy regular closed set. Remark 3.3From the above discussions the following implications hold. Let X = { a, b }. Define the fuzzy topology T as T = { 0, 1, 1, X 2, 3 }, where 1, 2, 3 I be defined as follows, 1( a ) = 0.4, 1( b ) = 0.3; 2( a ) = 0.5, 2( b ) = 0.7; 3( a ) = 0.45, 3( b ) = 0.65. Define S = { 0, 1, } and f : ( X, T ) ( Y, S ) as f ( a ) = b ; f ( b ) = a. -1 f (1 – ) = ( 0.55, 0.35 ) = 1 - 3 is fuzzy locally -closed but not fuzzy locally -regular closed. Therefore, every fuzzy locally -continuous function need not be fuzzy locally -regular continuous function. Proposition 4.3 Every fuzzy locally -continuous function is fuzzy locally pre-continuous function. 4. ON FUZZY LOCALLY -REGULAR CONTINUOUS FUNCTIONS Definition 4.1 Let ( X, T ) and ( Y, S ) be any two fuzzy topological spaces. A function f : ( X, T ) ( Y, S ) is said to be a fuzzy locally -regular continuous function (briefly, flY -1 X rcf) if for each fuzzy closed set I , f ( ) I is fuzzy locally -regular closed. Definition 4.2 Let ( X, T ) and ( Y, S ) be any two fuzzy topological spaces. A function f : ( X, T ) ( Y, S ) is said to be a fuzzy locally -continuous function (briefly, fl-cf)if for Y -1 X each fuzzy closed set I , f () I is fuzzy locally closed sets. Definition 4.3 Let ( X, T ) and ( Y, S ) be any two fuzzy topological spaces. A function f : ( X, T ) ( Y, S ) is said to be a fuzzy locally pre-continuous function (briefly, Y -1 X fl p-cf) if for each fuzzy closed set I , f ( ) I is fuzzy locally pre-closed. Definition 4.4 A fuzzy topological space and let be any fuzzy set in ( X, T ). Then fuzzy -closure of denoted by fcl ( ) is defined as the intersection of all fuzzy -closed sets containing . That is, f - cl ( ) = ∧ { µ : µ is a fuzzy closed set and µ ≥ }. Definition 4.5 Let ( X, T ) be any fuzzy topological space and let be any fuzzy set in ( X, T ). Then fuzzy -interior of , denoted by f - int ( ) is defined us the intersection Proposition 4.4 Every fuzzy locally -regular continuous function is fuzzy locally pre-continuous function. Remark 4.3 The converse of the above Proposition 4.2 and Proposition 4.3 need not be true. Example 4.2 Every fuzzy locally pre-continuous function need not be fuzzy locally -regular continuous and locally -continuous. Let X = { a, b }. Define the fuzzy topology T as T = { 0, 1, 1, 2, 3 }, where 1, 2, 3 IX be defined as follows : 1( a ) = 0.4, 1( b ) = 0.3; 2( a ) = 0.5 2( b ) = 0.7; 3( a ) = 0.4, Y 3( b ) = 0.6. Define S as S = {0, 1, }. Let I be defined as ( a ) = 0.6 and ( b ) = 0.4. - Define f : ( X, T ) ( Y, S ) as f( a ) = b and f( b ) = a. Then, f 1 (1 – ) = (1 - 3) is fuzzy locally pre-closed. Therefore, f is -1 fuzzy locally pre-continuous. But, f (1 – ) is not fuzzy locally -regular closed set and hence not fuzzy locally -closed set. Therefore, every fuzzy locally pre-continuous function need not be fuzzy locally -regular continuous function and fuzzy locally -continuous function. Remark 4.4 From the above discussions the following implications hold. 245 Amudhambigai et.al/A Study on Fuzzy Locally - Closed Sets Proposition 5.4 Let A and B be fuzzy subsets of a fuzzy topological space ( X, T ) such that A is fuzzy locally compact in X and B is fuzzy locally -closed in ( X,T ). Then A B is fuzzy locally -compact in X. 5. FUZZY LOCALLY -COMPACT SPACES AND FUZZY LOCALLY -COMPACT* SPACES Definition 5.1 Let ( X, T ) be a fuzzy topological space. The X collection { I : is fuzzy locally -open, i } is called the fuzzy locally -open cover of ( X, T ) if = iI 1. Definition 5.2 Any fuzzy topological space ( X, T ) is called fuzzy locally -compact if every fuzzy locally -open cover ( X, T ) has a finite subcover. Definition 5.3 A collection { Ui | i I } of fuzzy locally open sets in a fuzzy topological space ( X, T ) is called a fuzzy locally -open cover of a fuzzy set A if A holds. Definition 5.4 A fuzzy topological space ( X, T ) is said to be a fuzzy locally -compact space if every fuzzy locally open cover of ( X, T ) has a finite subcover. A fuzzy subset A of a fuzzy topological space ( X, T ) is said to be fuzzy locally -compact in X provided for every collection { Ui / i I } of fuzzy locally -open sets of X with A , there exists a finite subset I0 of I such that A . Definition 5.5 A fuzzy topological space X is said to be fuzzy locally -compact* at a fuzzy point x if there is a fuzzy locally -open subset U and a fuzzy set F which is fuzzy locally -compact in X such that x F U. If X is fuzzy locally -compact* at each of its fuzzy point, X is said to be a fuzzy locally -compact* space. Definition 5.6 A fuzzy subset A of a fuzzy topological space X is said to be fuzzy locally -compact* in X provided for each fuzzy point x in A, there is a fuzzy locally -open subset u and a fuzzy subset F which is fuzzy locally compact in X such that x F U. Definition 5.7 Let ( X, T ) and ( Y, S ) be any two fuzzy topological spaces. Any function f : ( X, T ) → ( Y, S ) is -1 X called a fuzzy locally - irresolute function if f ( ) I is fuzzy locally -open for every fuzzy locally -open set Y I . Remark : 5.1 Every fuzzy locally -open set is fuzzy locally open. Proposition 5.1 Every fuzzy locally compact space is locally -compact. Proposition 5.2 ( X, T ) is fuzzy locally -compact if and only if every family of fuzzy locally -closed subsets of ( X, T ) which has the finite intersection property has a nonempty intersection. Corollary 5.1 A fuzzy topological space ( X, T ) is fuzzy locally compact if and only if every family of fuzzy T-closed subsets of (X, T ) with the finite intersection property has a non empty intersection. Proposition 5.3 Let F be a fuzzy locally -closed subset of a fuzzy locally -compact space ( X, T ), then F is also fuzzy locally -compact in X. Proposition 5.5 Let ( X,T ) and ( Y, S ) be any two fuzzy topological space. Let f : ( X, T ) ( Y, S ) be a fuzzy locally -irresolute and surjective function. If ( X, T ) is a fuzzy locally -compact space, then ( Y, S ) is also a fuzzy locally -compact space. Proposition 5.6 Let f : ( X, T ) ( Y, S ) be a fuzzy locally continuous. If a fuzzy subset A is fuzzy locally -compact in ( X, T ), then the image f ( A ) is fuzzy locally -compact in ( Y, S ). Proposition 5.7 Let f : ( X, T ) ( Y, S ) be a fuzzy locally irresolute, fuzzy locally -open and injective mapping. If a fuzzy subset B of Y is fuzzy locally -compact in ( Y, S ), -1 then the image f ( B ) is fuzzy locally -compact in ( X, T ). Proposition 5.8 Let X be a fuzzy locally -compact* space and A a fuzzy subset of ( X, T ). If A is fuzzy locally -closed in X, then A is fuzzy locally -compact* in X. Proposition 5.9 Let a fuzzy topological space ( X, T ) be fuzzy locally -compact* and A be a fuzzy open subset of X. Then A is fuzzy locally -compact* in X. Proposition 5.10 Let ( X, T ) and ( Y, S ) be two fuzzy topological spaces and f : ( X, T ) ( Y, S ) be a fuzzy locally -irresolute, fuzzy locally -open and surjective function. If (X, T ) is fuzzy locally -compact*, then ( Y, S ) is also fuzzy locally -compact*. Proposition 5.11 Let (X, T) and (Y, S) be fuzzy topological spaces and f : (X, T) (Y, S) be a fuzzy locally -irresolute, fuzzy locally -open and injective function. If ( Y, S ) is fuzzy locally -compact*, then ( X, T ) is also fuzzy locally compact*. ACKNOWLEDGEMENT: The authors express their sincere thanks to the referees for their valuable comments regarding the improvement of the paper. REFERENCES 1. Amudhambigai.B, Uma.M.K, And Roja.E :r-Fuzzy - Locally Closed Sets And Fuzzy - -Locally Continuous Functions, Int. J. Of Mathematical Sciences And Applications, 1, Sep 2011. 2. Azad.k.k : on fuzzy semicontinuity, Fuzzy almost continuity and fuzzy weakly continuity, J.math. Appl., 82 (1981), 14-32 3. Bin Shahna.A.S : On Fuzzy Strong Semi Continuity And fuzzy Pre Continuity, Fuzzy Sets And Systems 44 (1991), 303-308. 4. Bourbaki.N: General Topology, Part I. Addison-Wesley, Reading, Mass 1996. 5. Chang.C.L: Fuzzy Topological Spaces, J.Math. Anal. Appl.,24 (1968),182-190. 6. 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