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International Journal of Mathematical Sciences and Applications Vol. 1 No. 2 (May, 2011) Copyright © Mind Reader Publications www.ijmsa.yolasite.com GENERALISED FUZZY CONTINUOUS MAPS IN FUZZY TOPOLOGICAL SPACES Dr. Ravi Pandurangan Chaitanya Bharathi Institute of Technology, Proddatur, Kadapa District, Andhra Pradesh, India – 516362, [email protected] Abstract The study of continuity and its weaker forms constitutes an established branch of investigation in general topological spaces. Recently some researchers have tried to extend these studies to the broader framework of fuzzy topological spaces. N. Ajmal [7] proposed a theory of localization in the study of continuity and its weaker forms in fuzzy setting. Using two notions of membership of a fuzzy point to a fuzzy set, neighborhood structure of fuzzy point [8] and quasi-neighborhood structure of a fuzzy point [8], an investigation of fuzzy continuity, fuzzy - continuity, fuzzy - continuity functions, and fuzzy almost quasi continuous functions have been carried out in the present study. The purpose of this chapter is to introduce and study the concepts of generalized fuzzy continuous maps, which includes the class of fuzzy continuous maps. INTRODUCTION Let f : X Y , A I x , B I x then f(A) is a fuzzy set in Y, defined by sup A( x) / x f f ( A)( y ) 0, iff 1 ( y ) And f 1 1 ( y) , f 1 ( y) ( B) is a fuzzy set in X, defined by f 1 ( B( x)) B( f ( x)) , x X . Let f : X , Y , . Then (1) f 1 (2) f ( f B f B , for any fuzzy set B in Y , . 1 c 1 (3) A f B ) B , for any fuzzy set B in Y , . 1 ( f ( A)) , for any fuzzy set A in X , . (4) Let p x be a fuzzy point of X, A be a fuzzy set in X , and B be a fuzzy set in Y , . Then we have Ravi Pandurangan 562 (i) If f ( p x )qB , then p x qf 1 ( B) . (ii) If p x qA , then f ( p x )qf ( A) (5) Let A and B be fuzzy sets in X , and Y , respectively. Let p x be a fuzzy point in X , . Then we have (i) p x f 1 ( B ) if f ( p x ) B (ii) f ( p x ) f ( A) if p x A Definition 1.1 Let f be a mapping from a fts X , into a fts Y , . f is said to be fuzzy continuous if and only if the inverse image of any fuzzy set is a matter of . Theorem 1.2 A mapping f from a fts X , into a fts Y , is fuzzy continuous if and only if for each fuzzy point p x in X , and each fuzzy neighborhood of f p x in Y , , there is a fuzzy neighborhood of p x in X , such that f . Proof: Let f be a fuzzy continuous. Let p x be a fuzzy point in X and be a fuzzy neighborhood of f p x in Y. There is a member v of such that f p x v and so, f p x f 1 (v ) f 1 ( ) . f is fuzzy continuous and v implies f 1 v and so f 1 is fuzzy neighborhood of p x in X and f f f 1 . Conversely, let the given condition hold for f. Let and p x be any fuzzy point in f 1 . Then f p x f ( f 1 ( )) . By hypothesis there is a fuzzy neighborhood of p x such that f , i.e., f So, p x f 1 . As 1 f 1 f 1 = px : px is a fuzzy point of f 1 . . Taking union of all such relations as p x runs over f 1 , we obtain 1 1 is a fuzzy neighborhood of p x there is a member v of such that p x v . So, p x v f v f f GENERALISED FUZZY… So, f 1 563 v . Hence f is fuzzy continuous. Definition 1.3 A function f from a fts X , into a fts Y , is said to be fuzzy continuous at a fuzzy point p x in X if corresponding to any fuzzy neighborhood of f( p x ) in Y , there is a fuzzy neighborhood in X , such that f . Proposition 1.4 A mapping f from a fts X , into a fts Y , is fuzzy continuous if and only if f is fuzzy continuous at every fuzzy point p x in X. Proof: Since a fuzzy set is the union of all its fuzzy points the Proposition follows directly from the Theorem 1.2. Proposition 1.5 If a fuzzy point p x in a fts X , belongs to the closure of a fuzzy set in X, then every fuzzy neighborhood of p x intersects . Proof: Let p x Cl . If then every fuzzy neighborhood of p x intersects . Let p x Cl . Let be an open fuzzy neighborhood of p x . If possible, let 0 . Let the support of be F. Then F support of = . Let us define a fuzzy set v in X as follows: 0 v x cl x mincl ( x),1 ( x) for those values of F for which x 1 for x F for those values of F for which x 1 Thus obviously, v . But v cl is a closed set and v cl ( ) , which implies a contradiction. So 0 must be hold. The converse of the above Proposition is not true. Definition 1.6 Ravi Pandurangan 564 A fuzzy point p of a fuzzy set in a fuzzy topological space X ,X is said to be a cluster point of if every fuzzy neighborhood containing a fuzzy point having same support as p has non-null intersection with . Proposition 1.7 Every fuzzy point of the closure of a fuzzy set in a fts is a cluster point of the fuzzy set. PRODUCT FUZZY TOPOLOGY Let X 1 ,X 1 and X 2 ,X 2 be two fuzzy topological spaces. Let X X 1 X 2 be the Cartesian product of X 1 and X 2 and pi be the projection of X onto X i . The product fuzzy space of X 1 and X 2 is a set X ( X X 1 X 2 ) equipped with the fuzzy topology generated by the family p 1 1 1 ( ), p2 ( ) : X 1 and X 2 . The family B : X 1 and X 2 from a base for the product topology X on X. Result 1.8 If is a fuzzy set in a fts X and is a fuzzy set in a fts Y, then Cl Cl Cl ( ) and Int Int Int ( ) . Definition 1.9 A fuzzy space X is product related to another fuzzy space Y if for any fuzzy set v of X and of Y whenever c v and c implies c 1 1 c v , where X and Y , there exists 1 X and 1 Y such that c v or c and 1 1 1 1 c 1 1 c c c Result 1.10 If X and Y are fuzzy topological spaces such that x is product related to Y, then for a fuzzy set of X and a fuzzy set of Y, (a) Cl Cl Cl (b) Int Int Int GENERALISED FUZZY… GENERALISED FUZZY CONTINUOUS MAPS IN FUZZY TOPOLOGICAL SPACES Levine [9] introduced the concept of generalized closed sets of a topological space and a class of topological spaces called T1 / 2 - spaces. Dunhamm and Levine further studied some properties of generalized closed sets and T1 / 4 - spaces. Maki and Umehara studied the characterizations of T1 / 2 - spaces and T - spaces by using generalized new sets. Balachandran, Sundran and Maki studies generalized continuous maps in topological spaces. Zadeh [1] introduced fuzzy sets. Chang [10] and Lowen [4] introduced the concept of fuzzy topological spaces and studied many properties of fuzzy topological spaces. The purpose of this chapter is to introduce and study the concepts of generalized fuzzy continuous maps, which includes the class of fuzzy continuous maps. Based on the notion of generalized closed set introduced and studied by Norman Levine [9], we give the following definition: Definition 1.11 Let ( X , ) be a fuzzy topological space. A fuzzy set in X is called generalized fuzzy closed (in short gfc) Cl whenever and is fuzzy open. Theorem 1.12 If 1 and 2 are gfc then 1 2 is a gfc set. Proof: Follows from the definition of gfc and the fact that Cl (1 2 ) 1 2 . However, the intersection of two gfc sets is not generalized fuzzy closed sets. Theorem 1.13 If is gfc set and Cl , then is a gfc set. Proof: Let be a fuzzy open set such that . Since , and is gfc, Cl . But Cl ( ) Cl ( ) since Cl ( ) and Cl . Hence, is a gfc set. Theorem 1.14 In a fuzzy topological space X , , F (the family of all fuzzy closed sets) 565 Ravi Pandurangan 566 Every fuzzy subset of X is a gfc set. Proof: Suppose that every fuzzy set of X is gfc. Let X F. Then since and is gfc we have Cl ( ) , but Cl ( ) and therefore Cl ( ) , i.e., , F. Also if , then F and hence 1 , i.e., F . Thus, we have seen that F . The converse is easy. Theorem 1.15 Let (X, F) be a fuzzy compact (Lindale, countably compact) space and suppose that is a gfc set of X. Then is fuzzy compact (Lindelof, countably compact). We prove the case for fuzzy compactness as the proof is similar for other cases. Let I be a family of fuzzy open sets in X such that VI . Since is a gfc set and VI is fuzzy open, it follows that V I . But is fuzzy compact and therefore it follows that Cl ( ) 1 2 ........... n for some natural number n N . Theorem 1.16 If X , is fuzzy regular and is fuzzy compact, then is a gfc set. Proof: Suppose , . As X is fuzzy regular, we can write V I , I being an index set, VI Cl ( ) . Hence, for some finite subset 0 we will have V 0 Cl (V 0 ) . This shows that Cl ( ) , i.e., is a gfc set. Definition 1.17 A fuzzy set is called generalized fuzzy open (in short gfo) 1 is gfc. We shall now prove some properties of generalized fuzzy open sets. Theorem 1.18 A fuzzy set is gfo Int whenever is fuzzy closed and . Proof: Let be a gfo set and be a fuzzy control set such that . Now 1 1 and 1 is gfc 1 1 . 1 (1 ) 1 (1 ) . But 1 (1 ) Int ( ) . Thus we find Int . That is, GENERALISED FUZZY… 567 Conversely, suppose that is a fuzzy set such that Int whenever is fuzzy closed and . We claim 1 is gfc – set. So let 1 where is fuzzy open. Now 1 1 . Hence, by assumption we must have 1 Int , i.e., 1 Int . But 1 Int Cl (1 ) ; hence, Cl (1 ) . This shows that 1 is a gfc set. Remark 1.19 i) The union of two sets is not generally gfo. ii) The intersection of any two-gfo sets is gfo. Theorem 1.20 If Int and if is gf-open, then is gf-open. Proof: Given Int , we have 1 1 1 Int Cl (1 ) . As is gf-open, 1- is gf-closed and so it follows by Theorem 1.18 that 1 is gf-closed i.e., is gfopen. Theorem 1.21 If is a gf-closed set in X and if f : X Y is f-continuous and f-closed the f( ) is gf-closed in Y. Proof: If f ( ) where is f-open in Y, then f f 1 ( ) is f-open, Cl ( ) f 1 1 ( ) . Since is gf-closed and ( ) , i.e., f ( ) . Now by assumption, f (Cl ( )) is f- closed and Cl ( f ( )) Cl ( f (Cl ( ))) f (Cl ( )) . This means f( ) is gf-closed. Definition 1.22 A map f : X Y is called generalized fuzzy continuous (in short gf-continuous) if the inverse image of every fuzzy closed set in Y is gf – closed in X. The following are the properties of gf – continuous functions. Theorem 1.23 If f : X Y is fuzzy continuous then f is gf – continuous. The converse of this Theorem is not true. Ravi Pandurangan 568 Theorem 1.24 Let f : X , 1 Y , 2 be a map. Then the following are equivalent: a) f is gf – continuous. b) The inverse image of each fuzzy set in Y is gf – open in X. Based on Dunhaam’s introduction of generalized closure operator, we define the generalized fuzzy closure operator Cl* for any fuzzy set in X , as follows: Cl * ( ) and is gf closed . Theorem 1.25 Let f : X Y be gf – continuous. Then f [Cl * ( X )] Cl[ f ( )] where is any fuzzy set in X. Proof: Now Cl f ( ) is a fuzzy closed set in Y. As f is gf – continuous, f – closed in X. f 1 (Cl f ( )) 1 (Cl ( )) is gf and Cl * ( ) f 1 [Cl f ( )] . Hence, f [Cl * ( X )] Cl[ f ( )] . The converse of this Theorem is not true. Definition 1.26 A fuzzy topological space X , is said to be fuzzy T1 / 2 if every generalized fuzzy closed set in x is a fuzzy closed in X. Theorem 1.27 Let f : X Y and Let g : Y Z be mappings and Y be fuzzy T1 / 2 . If f and g are gf – continuous, then the composition g f is gf – continuous. This Theorem is not valid if Y is not fuzzy T1 / 2 . FUZZY - CONTINUOUS MAPPINGS Definition 1.28 A fuzzy point p x is a fuzzy - cluster point of a fuzzy set in a fts X if and only if every fuzzy regularly open set containing a fuzzy point q x (having same support as p) has non-null intersection with . Definition 1.29 GENERALISED FUZZY… 569 Let be a fuzzy subset of a fts X. Let be fuzzy subset of X satisfying the following conditions: (a) every fuzzy point p x in is a fuzzy - cluster point of , (b) if v is a fuzzy set of X such that v , then there is a fuzzy point p x in v which is not a - cluster point of . Any fuzzy subset of X having same support as is defined to be a - closure of . We denote a - cluster point of a fuzzy set in a fts by . Thus a property related with the notation will always imply that the property holds for all - closures of . Definition 1.30 A fuzzy set of a fts X is said to be fuzzy - closed if is equal to one of its - closures. Complement of a fuzzy - closed set is said to be fuzzy - open. Remark 1.31 - Closure of a fuzzy set in a fts is not unique. However, it is unique up to its support. Any fuzzy regularly open set of a fts X is a fuzzy open set of X. Therefore, the set of all fuzzy regularly open sets a subset of the set of all fuzzy open sets of a fts. So, if a fuzzy point p x of a fuzzy set in a fts X is a fuzzy cluster point , then p x is also a fuzzy - cluster point of . A fuzzy regularly closed set is not always a fuzzy - closed set. Definition 1.32 A mapping f from a fts X , into a fts Y , is said to be a fuzzy - continuous at a fuzzy point Px in X, if for each fuzzy open neighborhood of f( Px ), there exists an open fuzzy neighborhood of Px such that f Int Cl Int Cl . Theorem 1.33 Let f be a mapping form a fts X , into a fts Y , . Then the following two conditions are equivalent: (a) f is fuzzy - continuous. Ravi Pandurangan 570 (b) For each fuzzy point Px in X and each fuzzy regular open set containing f( Px ), there exists a fuzzy regular open set containing Px such that f . Proof: a b : Let be a fuzzy point in X and be a fuzzy regularly open set containing f( Px ). Since every fuzzy regularly open set is a fuzzy set, it follows from (a) that there exists a fuzzy open neighborhood 1 of Px such that, f Int Cl 1 Int Cl . Taking Int Cl 1 = . b a : Let Px be a fuzzy point of X and be a fuzzy open neighborhood containing f( Px ). Then Int Cl is a fuzzy regular open set containing f( Px ) since Int Cl . By (b), there is a fuzzy regularly open set containing Px such that f Int Cl . Theorem 1.34 Let f be a surjection from a fts X , into a fts Y , . Then the following implications hold. (a) f is fuzzy - continuous. (b) f ([ ] ) [ f ( )] for every fuzzy set in X. (c) [ f 1 ( )] f 1 [ ] for every fuzzy set in Y. ( ) is fuzzy - closed in X. (d) For every fuzzy - closed set in Y, f 1 (e) For every fuzzy - open set in Y, f ( ) is fuzzy - open in X. 1 Proof: a b : Let p be a fuzzy point in Y such that p f [ ] then p=f(q), where q [ ] . Let f : X Y be map and A be a fuzzy subset of X. Let q be a fuzzy point of f(A) defined by q(y)=a for y y q a 0,1 and q(y)=0 for y y q y Y GENERALISED FUZZY… 571 Then a q ( y q ) f ( A)( y q ) = Choose 0 so small that a< Sup A( z ) z f 1 ( y q ) Sup A( z ) z f 1 ( y q ) But there is a z f 1 ( yq ) such that a< Sup A( z ) A( z1 ) z f 1 ( y q ) Let us define a fuzzy point p of X such that p(x)=0 for x z1 and p(x)=0 for x z1 ( x X ) . Then p A and Sup p( z ) 1 z f ( yq ) f ( p)( y ) 0 1 if f 1 ( y ) is non void otherwise So that f(p)(y)=a for y y q and f(p)(y)=0 for y y q y Y . Therefore f(p)=q. Let v be a fuzzy regularly open set containing p. Then there is a fuzzy regular open set containing q such that f ( ) v . q [ ] 0 , i.e., f ( ) 0 . But f ( ) f ( ) f ( ) 0 and f ( ) v v f ( ) 0 p [ f ( )] . a c : f ([ f 1 f 1 ( )] ) [ f ( f is a fuzzy set in X for every fuzzy set 1 c d : Let (c) [ f 1 ( )] f 1 d e : Let in Y. (d) f 1 ( ))] , so that f ([ f 1 in Y. By (b), ( )] ) [ ] . Thus [ f 1 ( )] f 1 ([ ] ) . be a fuzzy - closed set in Y. Then [ ] . ([ ] ) f 1 ( ) . So, f 1 ( ) is fuzzy - closed set in X. be a fuzzy - open set in Y. Then c is a fuzzy - closed set ( c ) is fuzzy - closed set in X. But f 1 ( c ) ( f 1 ( )) c so f 1 ( ) is fuzzy - open in X. Theorem 1.35 If f is fuzzy - continuous mapping from a fts X , into a fts Y , and g is a fuzzy - continuous mapping from a fts Y , into a fts Z , then so is g f . Proof: Ravi Pandurangan 572 It is a direct consequence of Theorem 1.33 and Theorem 1.34. Theorem 1.36 Let X 1 , X 2 and Y1 ,Y2 be fuzzy topological spaces such that Y1 is product related to Y2 and X 1 is product related to X 2 . Then the product f1 f 2 : X 1 X 2 Y1 Y2 of fuzzy - continuous mappings f 1 : X 1 Y1 and f 2 : X 2 Y2 is fuzzy - continuous. Proof: p1 , p 2 be a fuzzy point in X 1 X 2 . Let be a fuzzy open set in Let Y1 Y2 containing ( f 1 ( p1 ), f 2 ( p 2 )) . Then is the union , where A and B are indexing sets and ’s and ’s fuzzy open sets of Y1 and Y2 respectively. To obtain the result we have to show that there exists a fuzzy open set in X 1 X 2 containing ( p1 , p 2 ) such that f 1 f 2 Int Cl Int Cl i.e., Int Cl f 1 f 2 1 Int Cl ……………….(1) Now Cl Cl Cl ( ) So Int Cl Int Cl Int Cl Int Cl ( Int Cl ) Now f 1 f 2 1 Int Cl f 1 f 2 [ Int Cl Int Cl ] 1 [ f 1 f 2 [ f 1 f1 1 1 1 Int Cl Int Cl ] Int Cl f 2 1 ( Int Cl )] Int Cl f 1 1 2 ( Int Cl 1 )] [Where 1 A and 1 B are such that ( f ( p1 ), f ( p 2 )) 1 1 ] GENERALISED FUZZY… 573 Int Cl 1 Int Cl 1 Int cl ( 1 1 ) . Considering 1 1 = , we get (1). Theorem 1.37 Let X 1 X 2 be the product space of two fuzzy topological spaces X 1 , X 1 and X 2 , X 2 and let X , X be a fts. If f : X X 1 X 2 is fuzzy - continuous, then pi f (i 1, 2) is also fuzzy - continuous, where pi : X 1 X 2 X i (i=1, 2) is the projection of X 1 X 2 onto X i . Proof: Let p be a fuzzy point in X and be an fuzzy open set in X i containing p i of p. Since pi is fuzzy certainly f ( p) pi 1 continuous, 1 pi ( ) is fuzzy open in X1 X 2 . Since f is fuzzy - continuous, there is a fuzzy open set in and in X containing p such that 1 f ( Int (Cl )) Int pi (Cl ) So 1 pi [ f ( Int (Cl ))] pi ( Int ( pi (Cl ))) 1 pi ( pi ( Int Cl )) Int (Cl ) So, pi f ( Int (Cl )) Int (Cl ) . Definition 1.38 A mapping f from fts X , into a fts Y , is called a fuzzy almost continuous mapping if f 1 ( ) X for each fuzzy open set of Y. The definition illustrates fuzzy continuity implies fuzzy almost continuity. Theorem 1.39 For a function f form fts X , into a fts Y , , the following are true: (1) If Y is fuzzy semi-regular space and f is fuzzy - continuous, then f is fuzzy continuous. Ravi Pandurangan 574 (2) If X is fuzzy semi-regular space and f is fuzzy almost continuous then f is fuzzy - continuous. Proof: (1) Let p be a fuzzy point in X and let be an open fuzzy set in Y containing f(p). Y is fuzzy semi-regular implies , where ’s fuzzy regular open sets and f ( p ) for some and f is fuzzy - continuous implies there is a fuzzy regular open set containing p such that f ( ) . Then f is fuzzy continuous. (2) Let p X and be a fuzzy regular open set in Y containing f(p). Then f 1 ( ) is fuzzy open in X, and f X, p f 1 1 ( ) , where ’s are fuzzy regular open sets in ( ) p ( for some ) f 1 ( ) . Therefore f ( ) f ( f 1 ( )) . So, f is - continuous. REFERENCES 1. ZADEH.LA, Fuzzy Sets, Information and Control, 8, (1965), 338-353. 2. BELLMAN.R and GIERTY.M, On the analytic formation of the theory of fuzzy sets. Information Sciences 5 (1973), 149-156. 3. DIB.K.A, the Fuzzy topological spaces on a fuzzy space, Fuzzy Sets and Systems 108(1999), 103-110. 4. LOWEN.R. Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56(1976), 621-633. 5. PALANIAPPAN.N, Fuzzy Topology, Narosa Publishing House, New Delhi, 2002. 6. DIB.K.A, On Fuzzy spaces and fuzzy group theory, Inform. Sci. 80(3-4), (1994), 253-282. 7. AJMAL.N.TYAGI.K, On fuzzy almost continuous functions, Fuzzy Sets and Systems 41(1991), 221-232. 8. HU CHING-MING, Fuzzy topological spaces, Journal of Mathematical Analysis and Applications 110, (1985) 141-178. 9. LEVINE.N, Semi-open sets and semi continuity in topological spaces Amer. Math. J 22(1982), 55-60. 10. CHANG.C.L, Fuzzy topological spaces, J. Math. Anal. Appl. 24(1968), 182-190. SYMBOLS Cl - Closure Int - Interior fts - fuzzy topological space gfc - generalized fuzzy closed gfo - generalized fuzzy open GENERALISED FUZZY… FSO - fuzzy semi open f.a.o.S - fuzzy almost open (semi) f.a.o.H - fuzzy almost open (Pre semi) fsCl(A) - fuzzy semi closure of A fsInt(A) - fuzzy semi Interior of A f.w.q.c - fuzzy weakly quasi continuous f.s.w.c - fuzzy sub weakly continuous f.a.w.c - fuzzy almost weakly continuous FPO - fuzzy pro open f.w.c - fuzzy weakly continuous f.w.a.c - fuzzy weakly alpha/almost continuous f.w.c.c - fuzzy weakly completely continuous f.f.c - fuzzy faintly continuous f.s.w.c - fuzzy sub weakly continuous 575