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Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014 Some kinds of fuzzy connected and fuzzy continuous functions Hanan Ali Hussein Dept.of Math. College of Education for Girls Kufa University [email protected] Abstract The main aim of this paper to study new classes of fuzzy connected and fuzzy C-continuous functions , for this aim ,the nation of fuzzy open, fuzzy connected, fuzzy compact set, and fuzzy connected space introduced , and we shall study the relationship between fuzzy continuous function and fuzzy connected function . Keywords : o-connected function ,fuzz m –connected function,fuzzy weakly connected. الخالصة لددث الف اياددة اساساددن هد لددثا الدرددة لد ف اياددة لد ل افمددف هد الددف ال اله صددسة ردددسابسا الددف ال الهاد هي ردددسابسا ل ددي الهدف لس للس هفه م الهاه عدة الهف ردة رددسابسا الهاه عدة اله صدسة رددسابسا الهاه عدة الهيص صدة رددسابسا نهدس لس للدس النالادة ادم الدف ال . الها هي ردسابسا الف ال اله صسة ردسابسا .الفالة اله صسة ردسابس الفالة الها هي ردسابس: الكلمات المفتاحية Introduction With the emergence of the fundamental paper [L.A.Zadeh, 1965] by Zadeh in 1965 number of papers have appeared in literature featuring the application of fuzzy sets to pattern recognition ,decision problems ,function approximation, system theory, fuzzy logic, fuzzy algorithms, fuzzy automata, fuzzy grammars, fuzzy languages, fuzzy algebras, fuzzy topology, etc. .In this note, our interests are in the study of certain concepts in fuzzy topology. The concepts of continuity, compactness in the context of a fuzzy topological space are well known [C.V.Negoita and D.A.Raleescu,1975].In this paper the concept of fuzzy connected function is introduced and some kinds of fuzzy continuous function are studied. Throughout this paper, simply by X and Y we shall denote fuzzy topological spaces ( X , T1 ) and (Y , T2 ) and f : X Y will mean that f is a fuzzy function from ( X , T1 ) to (Y , T2 ) . This paper includes three sections, in the first section, we recall the concepts of fuzzy sets and some properties , in the second section, we have dealt with the concepts fuzzy connected ,fuzzy O-connected, fuzzy M- connected and fuzzy L-connected functions, and studied the relationship between their functions . Finally ,in the third section we have discussed the concepts fuzzy Ccontinuous, fuzzy weakly continuous and fuzzy almost continuous, and proved that every fuzzy continuous function is fuzzy connected functions . 1.Prilimeries 1.1 Definition [S.Dang,A.Behra and S.Nanda, 1994] A fuzzy point x in X is a fuzzy set defined as follows : ify x x ( y) yx 0 Where 0 1; is called its value and x is support of x . 1.2 Definition[M.H.Rashid and D.M.Ali,2008] A fuzzy point x is said to belong to a fuzzy set A in X ( denoted by: x A ) if and only if A(x ) ,for some x X . 2333 Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014 1.3 Definition[S.Dang,A.Behra and S.Nanda, 1994] Let A and B are fuzzy sets in X . Then A B if and only if x B for all x A . 1.4 Definition[D.H.Foster,1979] Let A and B are fuzzy sets in X ,then : 1) A B if and only if A( x) B( x) x X . 2) A B if and only if A( x) B( x) x X . 3) Z A B if and only if Z ( x) min{ A( x), B( x)} x X . 4) D A B if and only if D( x) max{ A( x), B( x)} x X . 5) E Ac ( The complement of A )if and only if E ( x) 1 A( x) x X . 1.5 Theorem[A.B.Saeid,2006] Let X and Y be two fuzzy topological spaces and let f : X Y be a function ,let { Ai }iI , {B j } jJ be families of fuzzy sets in X and Y respectively ,then : 1) f ( Ai ) f ( Ai ) . iI iI 2) f ( Ai ) f ( Ai ) . iI iI 1 3) f ( B j ) f 1 ( B j ) . jJ 4) f 1 jJ ( Bj ) f jJ jJ 1 (B j ) . 1.6 Theorem[S.M.Al-Khafaji,2010] Let X and Y be two fuzzy topological spaces and let f : X Y be a function, then the following statements are holds : 1) if B1 B2 ,then f 1 ( B1 ) f 1 ( B2 ) , B1 , B2 are fuzzy sets in Y . 2) if A1 A2 ,then f ( A1 ) f ( A2 ) , A1 , A2 are fuzzy sets in X . 3) A f 1 ( f ( A)) , A is fuzzy set in X . 4) if f : X Y is an injective function ,then f 1 ( f ( A)) A . 5) f ( f 1 ( B)) B , B is fuzzy set in Y . 6) if f : X Y is an serjective function, then f ( f 1 ( B)) B . 7) if A is a fuzzy set in X ,and B is fuzzy set in Y ,then f ( A) B if and only if A f 1 ( B) . 1.7 Definition[C.L.Change,1968] Let A be a fuzzy set in X ,then : 1) The union of all fuzzy open sets contained in A is called the fuzzy interior of A and denoted by A0 . i.e; A0 {B : B A, B T } . 2) The intersection of all fuzzy closed sets containing A is called the fuzzy closure of A ,and denoted by A .i.e; A {B : A B, B c T } . 1.8 Remarks[C.L.Change,1968] 1) The interior of a fuzzy set A is the largest fuzzy open set contained in A , and trivially a fuzzy set is fuzzy open set if and only if A A0 . 2)The closure of a fuzzy set A is the smallest fuzzy closed set containing A , and trivially a fuzzy set A is fuzzy closed if and only if A A . 1.9 Theorem[X.Tang,2004] Let A and B are two fuzzy sets in X ,then : 1) 0 X 0 X ,1X 1X . 2334 Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014 2) A0 A A . 3) if A B, then A B . 4) if A B, then A0 B 0 . 1.10 Definition[S.Carlson,2005] Let f be a function from X to Y ,and let B be a fuzzy set in Y ,then the inverse image of B under f is the fuzzy set f 1 ( B) in X with membership function defined by the rule : f 1 ( B)( x) B( f ( x)) for x X .(i.e; f 1 ( B) B f ) For a fuzzy set A in X ,the image of A under f is the fuzzy set f ( A) in Y with membership function f ( A)( y ) , y Y defined by : f ( A)( y ) sup A( x ) xf -1 ( y ) 0 if f 1 ( y ) is not empty otherewise 1 where f ( y) {x : f ( x) y} . 1.11 Definition[D.H.Foster,1979] The union (respectively intersection ) of the fuzzy sets Ai is defined by : iI ( Ai )( x) sup{ Ai ( x) : i I } iI ( Ai )( x) inf{ Ai ( x) : i I } iI 11.2 Definition[K.S,Raja Sethupathy and S.Lakshmivarahan, 1977] A fuzzy topological space X is said to be connected if X can not be represented as the union of two non-empty disjoint open fuzzy sets on X ,otherwise X is called disconnected space . 1.13 Remark[K.S,Raja Sethupathy and S.Lakshmivarahan, 1977] If 1X A B where A B 0 X , A and B are non-empty fuzzy open sets of X , then they are complements to each other ,and hence both are fuzzy open and fuzzy closed set . 1.14 Definition[A.M.Zahran,2000] A family of fuzzy sets is called a cover of a fuzzy set A if and only if A {Bi : Bi } ,and it is called a fuzzy open cover if each member Bi is a fuzzy open sets .A sub cover of is a sub family of which is also a cover of A . 1.15 Definition[D.H.Foster,1979] Let A be a fuzzy set in a fuzzy topological space X .Then A is said to be a fuzzy compact set if for every fuzzy open cover of A has a finite sub cover of A . Let A X , then X is called a fuzzy compact space ,that is Ai T for every i I and A 1 X ,then there are finitely many indices i1, i2 ,..., in I such that iI i A 1X . ijI ij 2. Some kinds of fuzzy connected functions . 2.1 Definition[K.S,Raja Sethupathy and S.Lakshmivarahan, 1977] A function f : X Y is said to be fuzzy connected if and only if f (W ) is fuzzy connected set in Y for each W is fuzzy connected set in X ,otherwise f is called fuzzy disconnected function . 2335 Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014 2.2 Definition A function f : X Y is said to be fuzzy O-connected function if and only if f (W ) is fuzzy connected set in Y for each W is fuzzy open and fuzzy connected set in X. 2.3 Proposition Every fuzzy connected function is fuzzy O-connected function . Proof:Let f : X Y be a fuzzy connected function , and let W be a fuzzy open and fuzzy connected set in X . Since W is connected set in X , and since f is connected function . Then f (W ) is connected set in Y . Hence f is fuzzy O-connected function .□ The converse of the above proposition is not true in general as shown in the following example . 2.4 Example Let X {a, b, c} , Y {x, y, z} and T1 {0 X ,1X },T2 {0Y ,1Y , B1 , B2 } be two fuzzy X and Y respectively, such that B1 : Y [0,1] defined by topologies on and let defined B1 ( x) 0.1 , B1 ( y) 0.5 B1 ( z) 0.2 B2 : Y [0,1] be B2 ( x) 0.2 , B2 ( y) 0.3 B2 ( z) 0.3 . And let f : X Y be a function defined by f (a) x , f (b) y , f (c) z Then f is fuzzy O-connected, because 0 X ,1X are all fuzzy open and fuzzy connected sets in X , such that f (0 X ) 0Y is fuzzy connected set in Y . And f (1X ) 1Y is fuzzy connected set in Y . But f is fuzzy disconnected because a fuzzy set A : X [0,1] which is defined by A(a) 0.2 , A(b) 0.5 A(c) 0.3 is fuzzy connected set ,but f ( A) {x0.2 , y0.5 , z 0.3 } is fuzzy disconnected ,since f ( A) B1 B2 and f ( A) B1 and f ( A) B2 . 2.5 Definition A function f : X Y is said to be fuzzy L-connected function if and only if f (W ) is fuzzy connected set in Y for each W is fuzzy closed and fuzzy connected set in X . 2.6 Proposition Every fuzzy connected function is fuzzy L-connected function . Proof:Let f : X Y be a fuzzy connected function , and let W be a fuzzy closed and fuzzy connected set in X . Since f is fuzzy connected function and W is fuzzy connected set . Then f (W ) is fuzzy connected set in Y . Then f is fuzzy L-connected function . □ The converse of the above proposition is not true in general as the following example . 2.7 Example Let X {a, b} , Y {x, y} and T1 {0 X ,1X },T2 {0Y ,1Y , C1 , C2 } be two fuzzy topologies on X and Y respectively, such that C1 : Y [0,1] defined by 2336 Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014 and let C2 : Y [0,1] defined by C1 ( x) 0.4 , C1 ( y) 0.9 C2 ( x) 0.6 , C2 ( y) 0.7 . And let f : X Y be a function defined by f (a) x , f (b) y Then f is fuzzy L-connected, because 0 X ,1X are all fuzzy closed and fuzzy connected sets in X , such that f (0 X ) 0Y is fuzzy connected set in Y . And f (1X ) 1Y is fuzzy connected set in Y . But f is fuzzy disconnected because a fuzzy set A : X [0,1] which is defined by A(a) 0.6 , A(b) 0.9 is fuzzy connected set ,but f ( A) is fuzzy disconnected ,since f ( A) C1 C2 and f ( A) C1 and f ( A) C2 . 2.8 Remarks 1) not every fuzzy L-connected function is fuzzy O-connected , and the converse is true . 2) not every fuzzy L-connected and fuzzy O-connected function is fuzzy connected . 2.9 Definition A function f : X Y is said to be fuzzy M-connected function if and only if f (W ) is fuzzy connected set in Y for each W is fuzzy compact and fuzzy connected set in X . 2.10 Proposition Every fuzzy connected function is fuzzy M-connected function . Proof:Let f : X Y be a fuzzy connected function , and let W be a fuzzy compact and fuzzy connected set in X . Since f is fuzzy connected function . Then f (W ) is fuzzy connected set in Y . Then f is fuzzy M-connected function . 3. Some kinds of fuzzy continuous functions 3.1 Definition A fuzzy set A of a fuzzy topological space is said to be a fuzzy continuum if and only if A is fuzzy connected and fuzzy compact set . 3.2 Example A singleton set in any finite fuzzy discreet topological space is fuzzy continuum . 3.3 Definition[Güner,Erdal,2007] A function f : X Y is said to be fuzzy continuous function if and only if f 1 (W ) is fuzzy open (closed ) set in X for each W is fuzzy open ( closed ) set in Y ,otherwise f is called fuzzy discontinuous. 3.4 Definition A function f : X Y is said to be fuzzy continuum function if and only if f (W ) is fuzzy continuum set in Y for each W is fuzzy continuum set in X . 3.5 Proposition Every fuzzy continuous function is fuzzy continuum function . Proof:Let f : X Y be a fuzzy continuous function , and suppose that f is not fuzzy continuum function . Then ,there is fuzzy continuum set W in X ,and f (W ) is not be fuzzy connected set . 2337 Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014 Then there is two non-empty fuzzy open sets A1 , A2 in Y such that f (W ) A1 A2 and A1 A2 0Y . Since f is fuzzy continuous function ,then f 1 ( A1 ) , f 1 ( A2 ) are fuzzy open sets in X. Then W f 1 ( A1 A2 ) f 1 ( A1 ) f 1 ( A2 ) , and f 1 ( A1 ) f 1 ( A2 ) f 1 ( A1 A2 ) f 1 (0Y ) 0 X . Then W is fuzzy disconnected set , and this contradiction . Then f is fuzzy continuum function . □ 3.6 Definition A function f : X Y is said to be fuzzy C-continuous function if and only if 1 f (W ) is fuzzy open (closed ) set in X for each W is fuzzy open ( closed ) and fuzzy compact set in Y . 3.7 Proposition Every fuzzy continuous function is fuzzy C-continuum function . Proof:Let f : X Y be a fuzzy continuous function , and let W is fuzzy closed and fuzzy compact set in Y . Since W is fuzzy closed and f is fuzzy continuous function , then f 1 (W ) is fuzzy closed set in X . Then f is fuzzy C-continuous function . □ 3.8 Example Let X {x1 , x2 } , Y {a1 , a2 } and T1 {0 X ,1X },T2 {0Y ,1Y } be two fuzzy topologies on X and Y respectively, and let f : X Y defined by f ( x1 ) a1 , f ( x2 ) a2 . Since 0Y ,1Y are fuzzy closed and fuzzy compact sets in Y such that f 1 (0Y ) 0 X is fuzzy closed set in X ,and f 1 (1Y ) 1X is fuzzy closed set in X . Then f is fuzzy C-continuous function . 3.9 Definition A function f : X Y is said to be fuzzy weakly continuous if and only if for all fuzzy point x in X ,and for all fuzzy open set V in Y contains f ( x ) there exist open set U in X contain x such that f (U ) V . 3.10 proposition A function f : X Y is fuzzy weakly continuous if and only if for each fuzzy open set V in Y ,such that f 1 (V ) ( f 1 (V )) 0 . Proof: Let x be a fuzzy point in X and V be a fuzzy open set in Y contains f ( x ) . Since V be a fuzzy open set in Y ,then f Since f ( x ) V ,then x f 1 (V ) 1 (V ) ( f 1 (V )) 0 Since ( f 1 (V )) 0 is fuzzy open set in X , then f 1 (V ) ( f 1 (V )) 0 Since x f 1 (V ) ,then x f 1 (V ) , and so x ( f 1 (V )) 0 2338 Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014 Let U ( f 1 (V )) 0 ,then U is a fuzzy open set in 1 X contains x such that 1 f (U ) f (( f (V )) ) f ( f (V )) V . See ( theorem 1.6(5) ) Then f is fuzzy weakly continuous function . Let V be a fuzzy open set in Y ,and let x f 1 (V ) , then f ( x ) V Since f is fuzzy weakly continuous function, then there exists fuzzy open set U contains x such that f (U ) V . 0 Then U f 1 (V ) . See ( theorem 1.6(7) ) Since x U f 1 (V ) ,and since U is fuzzy open set Then x ( f 1 (V )) 0 Then f 1 (V ) ( f 1 (V )) 0 . □ 3.11 Proposition Every fuzzy weakly continuous onto function is fuzzy connected . Proof:Let f : X Y is fuzzy weakly continuous onto function . Let X is fuzzy connected space, and suppose that Y is fuzzy disconnected space . then there exist two non-empty fuzzy open sets A1 , A2 such that 1Y A1 A2 ,and A1 A2 0Y ,then f 1 ( A1 A2 ) f 1 (0Y ) 0 X . since 1X f 1 (1Y ) f 1 ( A1 A2 ) f 1 ( A1 ) f 1 ( A2 ) . Since A1 is fuzzy open set in Y . Then by proposition ( 3.10) f 1 ( A1 ) ( f 1 ( A1 )) 0 Then A1 is both fuzzy open and fuzzy closed, see remark ( 1.13) Then A1 A1 ,so f 1 ( A1 ) ( f 1 ( A1 )) 0 ,but ( f 1 ( A1 )) 0 f 1 ( A1 ) . Then ( f 1 ( A1 )) 0 f 1 ( A1 ) Then f 1 ( A1 ) is fuzzy open set in X . Similarity we prove that f 1 ( A2 ) is fuzzy open set in X . Then X is fuzzy disconnected space ,this contradicts Then Y is fuzzy connected space . Then f is fuzzy connected function . □ 3.12 Definition A function f : X Y is said to be fuzzy almost continuous if and only if for all fuzzy point x in X ,and for all fuzzy open set V in Y contains f ( x ) there exist _______ 0 open set U in X contain x such that f (U ) V . 3.13 Example Let X {a, b}, Y {x, y} and let T1 be discrete fuzzy topological space on X and T2 {0Y ,1Y } be fuzzy topological space on Y . If f : X Y be a function defined by f (a) x , f (b) y . Then f is fuzzy almost continuous function . 3.14 Proposition i)Every fuzzy almost continuous function is fuzzy weakly continuous . ii) Every fuzzy continuous functions is fuzzy connected . Proof:2339 Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014 1) Let f : X Y is a fuzzy almost continuous function . Let x X ,and let V be a fuzzy open set in Y contains f ( x ) . Since f is a fuzzy almost continuous function ,then there exist fuzzy open set U in _______ X such that x U and f (U ) V 0 .............(1) _______ Since V 0 V ,then from theorem (1.9(3)) V 0 V ……….(2) Substitute (2) in (1) ,then we have f (U ) V . Then f is fuzzy weakly continuous function . ii) since every fuzzy weakly continuous function is fuzzy connected ,see proposition (3.11) ,and since every fuzzy almost continuous function is fuzzy connected ,see proposition ( 3.14(i)) . then every fuzzy continuous function is fuzzy connected .□ and the following graph explain this relation . Fuzzy almost continuous function Fuzzy continuous function Fuzzy weakly continuous function Fuzzy connected function References Saeid,2006,"Fuzzy A.B topological B-algebras ", International .J. of fuzzy systems ,pp.160-164. Zahran,2000,"Regularly A.M. open sets and good extension on fuzzy topological space ",Fuzzy sets and fuzzy systems ,pp.353-359 . Chang,1968,"Fuzzy C.L. topological space ",J.Math. Anal. App1, pp.182-190. Negoita C.V. and D.A.Ralescu,1975,"Applications of fuzzy sets to system Analysis " , JohnWiely and Sons ,New York . Foster,1979,"Fuzzy D.H. topological groups", J.Math. Anal. App1,pp.549-564 . Güner ,Erdal,2007,"Fuzzy contractibility ", Commun. Facsci, University Ankara ,SSeriesA1,Vo.56,No.2,pp.11-16. Raja Sethupath K.S. and S.Lakshmivarahan,1977, "connectedness in fuzzy topology ", kybernetika,Vo.13,No.3,pp.190-193 . Zadeh,1965,"Fuzzy sets ", L.A. Information and Control ,8,3,pp.338-353 . 2340 Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014 Carlson ,2005, " S. Fuzzy sets and topologies : early ideas and obstacles ", www.rosehulma.edu/math/seminarfiels . Dang ,A.Behra S. and S.Nanda,1994, "On fuzzy weakly semi – continuous functions ", fuzzy sets and systems ,pp.239-245. AL-Khafaji ,2010, " S.M. On fuzzy topological vector spaces ", M.Sce.,Thesis, AlQadisiya ,University . 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