Some kinds of fuzzy connected and fuzzy continuous functions

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1 Journal of Babylon Unversty/Pure and Appled Scences/ No(9)/ Vol(): 4 Some knds of fuzzy connected and fuzzy contnuous functons Hanan Al Hussen Deptof Math College of Educaton for Grls Kufa Unversty Hananahussen@uokafaq Abstract The man am of ths paper to study new classes of fuzzy connected and fuzzy C-contnuous functons, for ths am,the naton of fuzzy open, fuzzy connected, fuzzy compact set, and fuzzy connected space ntroduced, and we shall study the relatonshp between fuzzy contnuous functon and fuzzy connected functon Keywords : o-connected functon,fuzz m connected functon,fuzzy weakly connected الخلاصة الهدف الا ساسي من هذا البحث هو دراسة نوع جديد من الدوال المتصلة ضبابيا والدوال المستمرة ضبابيا ولغرض هذه الدراسة تناولنا مفهوم المجموعة المفتوحة ضبابيا والمجموعة المتصلة ضبابيا والمجموعة المرصوصة ضبابيا,وكما تناولنا العلاقة 333 بين الدوال المستمرة ضبابيا والدوال المتصلة ضبابيا الكلمات المفتاحية :الدالة المتصلة ضبابيا, الدالة المستمرة ضبابيا Introducton Wth the emergence of the fundamental paper [LAZadeh, 965] by Zadeh n 965 number of papers have appeared n lterature featurng the applcaton of fuzzy sets to pattern recognton,decson problems,functon approxmaton, system theory, fuzzy logc, fuzzy algorthms, fuzzy automata, fuzzy grammars, fuzzy languages, fuzzy algebras, fuzzy topology, etc In ths note, our nterests are n the study of certan concepts n fuzzy topology The concepts of contnuty, compactness n the context of a fuzzy topologcal space are well known [CVNegota and DARaleescu,975]In ths paper the concept of fuzzy connected functon s ntroduced and some knds of fuzzy contnuous functon are studed Throughout ths paper, smply by and we shall denote fuzzy topologcal spaces (, T ) and (, T ) and f : wll mean that f s a fuzzy functon from (, T ) to (, T ) Ths paper ncludes three sectons, n the frst secton, we recall the concepts of fuzzy sets and some propertes, n the second secton, we have dealt wth the concepts fuzzy connected,fuzzy O-connected, fuzzy M- connected and fuzzy L-connected functons, and studed the relatonshp between ther functons Fnally,n the thrd secton we have dscussed the concepts fuzzy C- contnuous, fuzzy weakly contnuous and fuzzy almost contnuous, and proved that every fuzzy contnuous functon s fuzzy connected functons Prlmeres Defnton [SDang,ABehra and SNanda, 994] A fuzzy pont x n s a fuzzy set defned as follows : fy x x ( y) y x Where ; s called ts value and x s support of x Defnton[MHRashd and DMAl,8] A fuzzy pont x s sad to belong to a fuzzy set A n ( denoted by: x A ) f and only f A(x),for some x

2 Journal of Babylon Unversty/Pure and Appled Scences/ No(9)/ Vol(): 4 3 Defnton[SDang,ABehra and SNanda, 994] Let A and B are fuzzy sets n Then A B f and only f x B for all x A 4 Defnton[DHFoster,979] Let A and B are fuzzy sets n,then : ) A B f and only f A( x) B( x) x ) A B f and only f A( x) B( x) x 3) Z A B f and only f Z( x) mn{ A( x), B( x)} x 4) D A B f and only f D( x) max{ A( x), B( x)} x 334 c 5) E A ( The complement of A )f and only f E( x) A( x) x 5 Theorem[ABSaed,6] Let and be two fuzzy topologcal spaces and let f : be a functon,let{ A },{ B be famles of fuzzy sets n and respectvely,then : I j} jj ) f A ) f ( A ) ( I I ( A ) f ( A I I ) f ) 3) f ( B ) f ( B ) jj j jj 4) f ( B ) f ( B ) jj j jj j j 6 Theorem[SMAl-Khafaj,] Let and be two fuzzy topologcal spaces and let f : be a functon, then the followng statements are holds : ) f B B,then f ( B ) f ( B ), B, B are fuzzy sets n ) f A A,then f ( A ) f ( A ), A, A are fuzzy sets n 3) A f ( f ( A, A s fuzzy set n 4) f f : s an njectve functon,then f ( f ( A A 5) f ( f ( B B, B s fuzzy set n 6) f f : s an serjectve functon, then f ( f ( B B 7) f A s a fuzzy set n,and B s fuzzy set n,then f ( A) B f and only f A f ( B) 7 Defnton[CLChange,968] Let A be a fuzzy set n,then : ) The unon of all fuzzy open sets contaned n A s called the fuzzy nteror of A and denoted by A e; A { B : B A, B T} ) The ntersecton of all fuzzy closed sets contanng A s called the fuzzy closure of A,and denoted by A e; A { B : A B, B c T} 8 Remarks[CLChange,968] ) The nteror of a fuzzy set A s the largest fuzzy open set contaned n A, and trvally a fuzzy set s fuzzy open set f and only f A A )The closure of a fuzzy set A s the smallest fuzzy closed set contanng A, and trvally a fuzzy set A s fuzzy closed f and only f A A 9 Theorem[Tang,4] Let A and B are two fuzzy sets n,then : ),

3 Journal of Babylon Unversty/Pure and Appled Scences/ No(9)/ Vol(): 4 ) A A A 3) f A B, then A B 4) f A B, then A B Defnton[SCarlson,5] Let f be a functon from to,and let B be a fuzzy set n,then the nverse mage of B under f s the fuzzy set f ( B ) n wth membershp functon defned by the rule : f ( B)( x) B( f ( x for x (e; f ( B) B f ) For a fuzzy set A n,the mage of A under f s the fuzzy set f (A) n wth membershp functon f ( A)( y), y defned by : f ( A )( y ) sup A ( x ) - x f ( y ) f f ( y ) s not otherewse empty where f ( y) { x : f ( x) y} Defnton[DHFoster,979] The unon (respectvely ntersecton ) of the fuzzy sets ( A )( x) sup{ A ( x) : I} I ( A )( x) nf{ A ( x) : I} I A I s defned by : Defnton[KS,Raja Sethupathy and SLakshmvarahan, 977] A fuzzy topologcal space s sad to be connected f can not be represented as the unon of two non-empty dsjont open fuzzy sets on,otherwse s called dsconnected space 3 Remark[KS,Raja Sethupathy and SLakshmvarahan, 977] If A B where A B, A and B are non-empty fuzzy open sets of, then they are complements to each other,and hence both are fuzzy open and fuzzy closed set 4 Defnton[AMZahran,] A famly of fuzzy sets s called a cover of a fuzzy set A f and only f A { B : B },and t s called a fuzzy open cover f each member B s a fuzzy open sets A sub cover of s a sub famly of whch s also a cover of A 5 Defnton[DHFoster,979] Let A be a fuzzy set n a fuzzy topologcal space Then A s sad to be a fuzzy compact set f for every fuzzy open cover of A has a fnte sub cover of A Let A, then s called a fuzzy compact space,that s A T for every I and A I,then there are fntely many ndces,,, n I such that A j I j Some knds of fuzzy connected functons Defnton[KS,Raja Sethupathy and SLakshmvarahan, 977] A functon f : s sad to be fuzzy connected f and only f f (W ) s fuzzy connected set n for each W s fuzzy connected set n,otherwse f s called fuzzy dsconnected functon 335

4 Journal of Babylon Unversty/Pure and Appled Scences/ No(9)/ Vol(): 4 Defnton A functon f : s sad to be fuzzy O-connected functon f and only f f (W ) s fuzzy connected set n for each W s fuzzy open and fuzzy connected set n 3 Proposton Every fuzzy connected functon s fuzzy O-connected functon Let f : be a fuzzy connected functon, and let W be a fuzzy open and fuzzy connected set n Snce W s connected set n, and snce f s connected functon Then f (W ) s connected set n Hence f s fuzzy O-connected functon The converse of the above proposton s not true n general as shown n the followng example 4 Example Let { a, b, c}, { x, y, z} and T {, }, T {,, B, B} be two fuzzy topologes on and respectvely, such that B : [,] defned by B ( x), B ( y) 5 B ( z) and let B : [,] defned be B ( x), B ( y) 3 B ( z) 3 And let f : be a functon defned by f ( a) x, f ( b) y, f ( c) z Then f s fuzzy O-connected, because, are all fuzzy open and fuzzy connected sets n, such that f ( ) s fuzzy connected set n And f ( ) s fuzzy connected set n But f s fuzzy dsconnected because a fuzzy set A : [,] whch s defned by A( a), A(b) 5 A(c) 3s fuzzy connected set,but f ( A) { x, y5, z 3} s fuzzy dsconnected,snce f ( A) B B and f (A) B and f (A) B 5 Defnton A functon f : s sad to be fuzzy L-connected functon f and only f f (W ) s fuzzy connected set n for each W s fuzzy closed and fuzzy connected set n 6 Proposton Every fuzzy connected functon s fuzzy L-connected functon Let f : be a fuzzy connected functon, and let W be a fuzzy closed and fuzzy connected set n Snce f s fuzzy connected functon and W s fuzzy connected set Then f (W ) s fuzzy connected set n Then f s fuzzy L-connected functon The converse of the above proposton s not true n general as the followng example 7 Example Let { a, b}, { x, y} and T {, }, T {,, C, C} be two fuzzy topologes on and respectvely, such that C : [,] defned by 336

5 Journal of Babylon Unversty/Pure and Appled Scences/ No(9)/ Vol(): 4 C ( x) 4, C( y) 9 and let C : [,] defned by C ( x) 6, C ( y) 7 And let f : be a functon defned by f ( a) x, f ( b) y Then f s fuzzy L-connected, because, are all fuzzy closed and fuzzy connected sets n, such that f ( ) s fuzzy connected set n And f ( ) s fuzzy connected set n But f s fuzzy dsconnected because a fuzzy set A : [,] whch s defned by A ( a) 6, A(b) 9 s fuzzy connected set,but f (A) s fuzzy dsconnected,snce f ( A) C C and f (A) C and f (A) C 8 Remarks ) not every fuzzy L-connected functon s fuzzy O-connected, and the converse s true ) not every fuzzy L-connected and fuzzy O-connected functon s fuzzy connected 9 Defnton A functon f : s sad to be fuzzy M-connected functon f and only f f (W ) s fuzzy connected set n for each W s fuzzy compact and fuzzy connected set n Proposton Every fuzzy connected functon s fuzzy M-connected functon Let f : be a fuzzy connected functon, and let W be a fuzzy compact and fuzzy connected set n Snce f s fuzzy connected functon Then f (W ) s fuzzy connected set n Then f s fuzzy M-connected functon 3 Some knds of fuzzy contnuous functons 3 Defnton A fuzzy set A of a fuzzy topologcal space s sad to be a fuzzy contnuum f and only f A s fuzzy connected and fuzzy compact set 3 Example A sngleton set n any fnte fuzzy dscreet topologcal space s fuzzy contnuum 33 Defnton[Güner,Erdal,7] A functon f : s sad to be fuzzy contnuous functon f and only f f ( W ) s fuzzy open (closed ) set n for each W s fuzzy open ( closed ) set n,otherwse f s called fuzzy dscontnuous 34 Defnton A functon f : s sad to be fuzzy contnuum functon f and only f f (W ) s fuzzy contnuum set n for each W s fuzzy contnuum set n 35 Proposton Every fuzzy contnuous functon s fuzzy contnuum functon Let f : be a fuzzy contnuous functon, and suppose that f s not fuzzy contnuum functon Then,there s fuzzy contnuum set W n,and f (W ) s not be fuzzy connected set 337

6 Journal of Babylon Unversty/Pure and Appled Scences/ No(9)/ Vol(): 4 Then there s two non-empty fuzzy open sets A, A n such that f ( W ) A A and A A Snce f s fuzzy contnuous functon,then f ( A ), f ( A ) are fuzzy open sets n Then W f A A ) f ( A ) f ( A ), and ( ( A ) f ( A A ) f ( f ( A ) f ) Then W s fuzzy dsconnected set, and ths contradcton Then f s fuzzy contnuum functon 36 Defnton A functon f : s sad to be fuzzy C-contnuous functon f and only f f ( W ) s fuzzy open (closed ) set n for each W s fuzzy open ( closed ) and fuzzy compact set n 37 Proposton Every fuzzy contnuous functon s fuzzy C-contnuum functon Let f : be a fuzzy contnuous functon, and let W s fuzzy closed and fuzzy compact set n Snce W s fuzzy closed and f s fuzzy contnuous functon, then f ( W ) s fuzzy closed set n Then f s fuzzy C-contnuous functon 38 Example Let { x, x}, { a, a} and T {, }, T {, } be two fuzzy topologes on and respectvely, and let f : defned by f ( x ) a, f ( x a ), Snce are fuzzy closed and fuzzy compact sets n such that f ( ) s fuzzy closed set n,and f ( ) s fuzzy closed set n Then f s fuzzy C-contnuous functon 39 Defnton A functon f : s sad to be fuzzy weakly contnuous f and only f for all fuzzy pont x n,and for all fuzzy open set V n contans f x ) there exst open set U n contan x such that f ( U ) V 3 proposton A functon f : s fuzzy weakly contnuous f and only f for each fuzzy open set V n,such that f ( V ) ( f ( V Let x be a fuzzy pont n and V be a fuzzy open set n contans f ( x ) Snce V be a fuzzy open set n,then Snce Snce f ( x ) V,then x f ( V ) ( f ( V s fuzzy open set n, then f f ( V ) ( f ( V Snce x f ( V ),then x f ( V ), and so x ( V ) ( f ( V ( f ( V ( 338

7 Journal of Babylon Unversty/Pure and Appled Scences/ No(9)/ Vol(): 4 Let U ( f ( V,then U s a fuzzy open set n contans 339 x such that f ( U ) f (( f ( V ) f ( f ( V V See ( theorem 6(5) ) Then f s fuzzy weakly contnuous functon Let V be a fuzzy open set n,and let x f ( V ), then f ( x ) V Snce f s fuzzy weakly contnuous functon, then there exsts fuzzy open set U contans x such that f ( U ) V Then U f ( V ) See ( theorem 6(7) ) Snce x U f ( V ),and snce U s fuzzy open set Then x ( f ( V Then f ( V ) ( f ( V 3 Proposton Every fuzzy weakly contnuous onto functon s fuzzy connected Let f : s fuzzy weakly contnuous onto functon Let s fuzzy connected space, and suppose that s fuzzy dsconnected space then there exst two non-empty fuzzy open sets A, A such that A A,and A A,then f ( A A ) f ( ) snce f ( ) f ( A A ) f ( A ) f ( A ) Snce A s fuzzy open set n Then by proposton ( 3) f ( A ) ( f ( A Then A s both fuzzy open and fuzzy closed, see remark ( 3) Then A A,so f ( A ) ( f ( A,but ( f ( A f ( A ) Then f ( A f ( A ) ( ( A Then f ) s fuzzy open set n Smlarty we prove that f ( A ) s fuzzy open set n Then s fuzzy dsconnected space,ths contradcts Then s fuzzy connected space Then f s fuzzy connected functon 3 Defnton A functon f : s sad to be fuzzy almost contnuous f and only f for all fuzzy pont x n,and for all fuzzy open set V n contans f x ) there exst open set U n contan x such that f ( U ) V 33 Example Let { a, b}, { x, y} and let T be dscrete fuzzy topologcal space on and T {, } be fuzzy topologcal space on If f : be a functon defned by f ( a) x, f ( b) y Then f s fuzzy almost contnuous functon 34 Proposton )Every fuzzy almost contnuous functon s fuzzy weakly contnuous ) Every fuzzy contnuous functons s fuzzy connected (

8 Journal of Babylon Unversty/Pure and Appled Scences/ No(9)/ Vol(): 4 ) Let f : s a fuzzy almost contnuous functon Let x,and let V be a fuzzy open set n contans f ( x ) Snce f s a fuzzy almost contnuous functon,then there exst fuzzy open set U n such that x U and f ( U ) V () Snce V V,then from theorem (9(3 V V () Substtute () n (),then we have f ( U) V Then f s fuzzy weakly contnuous functon ) snce every fuzzy weakly contnuous functon s fuzzy connected,see proposton (3),and snce every fuzzy almost contnuous functon s fuzzy connected,see proposton ( 34( then every fuzzy contnuous functon s fuzzy connected and the followng graph explan ths relaton Fuzzy almost contnuous functon Fuzzy contnuous functon Fuzzy weakly contnuous functon Fuzzy connected functon References Saed,6,"Fuzzy AB topologcal B-algebras ", Internatonal J of fuzzy systems,pp6-64 Zahran,,"Regularly AM open sets and good extenson on fuzzy topologcal space ",Fuzzy sets and fuzzy systems,pp Chang,968,"Fuzzy CL topologcal space ",JMath Anal App, pp8-9 Negota CV and DARalescu,975,"Applcatons of fuzzy sets to system Analyss ", JohnWely and Sons,New ork Foster,979,"Fuzzy DH topologcal groups", JMath Anal App,pp Güner,Erdal,7,"Fuzzy contractblty ", Commun Facsc, Unversty Ankara,SSeresA,Vo56,No,pp-6 Raja Sethupath KS and SLakshmvarahan,977, "connectedness n fuzzy topology ", kybernetka,vo3,no3,pp9-93 Zadeh,965,"Fuzzy sets ", LA Informaton and Control,8,3,pp

9 Journal of Babylon Unversty/Pure and Appled Scences/ No(9)/ Vol(): 4 Carlson,5, " S Fuzzy sets and topologes : early deas and obstacles ", wwwrosehulmaedu/math/semnarfels Dang,ABehra S and SNanda,994, "On fuzzy weakly sem contnuous functons ", fuzzy sets and systems,pp39-45 AL-Khafaj,, " SM On fuzzy topologcal vector spaces ", MSce,Thess, Al- Qadsya,Unversty Tang,4,"Spatal object modelng n fuzzy topologcal space ",PHD dssertaton,unversty of Twente, the Netherlands 34

Semi - - Connectedness in Bitopological Spaces

Semi - - Connectedness in Bitopological Spaces Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second صفحة 45-53 Sem - - Connectedness n Btopologcal Spaces By Qays Hatem