IOSR Jurnal f Mathematics (IOSR-JM e-issn: 78-578, p-issn: 39-765X Vlume, Issue 5 Ver I (Sep - Oct06, PP 8-4 wwwisrjurnalsrg A new Type f Fuzzy Functins in Fuzzy Tplgical Spaces Assist Prf Dr Munir Abdul Khalik Alkhafaji Al-Mustansiriyah University Cllege f Educatin Department f Math Abstract: In this paper, we intrduce and study sme characterizatin and sme prperties f fuzzy cntinuus functins ( Fuzzy super cntinuus functins frm fuzzy tplgical space int anther fuzzy tplgical space has its bases n the ntatin f quasi-cincidence, quasi-neighbrhd, fuzzy -clsure and - neighbrhd I Intrductin The first publicatins in fuzzy set thery by Zadeh [] shw the intentin f the authrs t generalize the classical ntin f a set and prpsitin (Statement t accmmdate fuzziness, Chang [4], Wng [9], [0] and ther applied sme basic cncepts frm general tplgy t fuzzy sets and develped a thery f fuzzy tplgical spaces in 980, pu and liu [8], intrduced the cncepts f quasi-cincidence and quasi-neighbrhd II Preliminaries We will discuss sme fundamental ntins and basic cncepts related t fuzzy tplgical space Definitin A fuzzy tplgy is a family T ~ f fuzzy sets X (X be any set f elements which satisfying the fllwing cnditins: - O, ( ie Ø and X T ~, - If A ~, B ~ T ~, then à B ~ T ~ 3- If A ~ i T ~ fr each i I (where I is the inde set, then A ~ i T ~ ii T ~ is called a fuzzy tplgy fr X, and the pair ( X, T ~ is a fuzzy tplgical space Every member f T ~ is c called T ~ - pen fuzzy set A fuzzy set C in X is a T ~ - clsed fuzzy set if and nly if its cmplement C is a T ~ - pen fuzzy set Definitin A fuzzy set A ~ in a fuzzy tplgical space ( X, T ~ is said t be quasi cincident ( q cincident, fr shrt with fuzzy set B ~, dented by A ~ qb ~, If and nly if there eist X μ Ã( + μ B ( > Definitin 3 A fuzzy set A in a fuzzy tplgical space ( X, T ~ is said t be quasi neighbrhd ( q nbd, fr thrt f a fuzzy pint ( where is the supprt and is the value f the fuzzy pint, 0 if and nly if there eists pen fuzzy set B ~ X qb A Definitin 4 Tw fuzzy sets A ~, B ~ in fuzzy tplgical space (X, T ~, A ~ B ~ if and nly if A ~ is nt q cincident with c B ~ ( cmplement f B ~ and dented by qb ~ c A ~ DOI: 09790/578-05084 wwwisrjurnalsrg 8 Page
Definitin 5 A fuzzy pint nly if each q nbd f Definitin 6 A new Type f Fuzzy Functins in Fuzzy Tplgical Spaces A ( the fuzzy clsure f fuzzy set A ~ in an fuzzy tplgical space ( X, T ~ if and is q cincident with A A fuzzy set A ~ in fuzzy tplgical space ( X, T ~ is called pen ( clsed fuzzy regularly if and nly if (A ~ A ~ ( A ~ the fuzzy interir f fuzzy set A in an fuzzy tplgical space ( X, T ~ (respectively (A ~ A ~ Definitin 7 A fuzzy pint is called a fuzzy - cluster pint f a fuzzy set A ~ in an fuzzy tplgical space ( X, T ~ if and nly if every pen fuzzy regularly q-nbd f Definitin 8 is q cincident with A ~ The set f all fuzzy -cluster pints f A ~ is called the fuzzy -cluster f A ~, t be dented by [ A ~ ] Remark A fuzzy set A ~ is fuzzy -clsed if and nly if fuzzy - pen A [A] and cmplement f a fuzzy -clsed is called Definitin 9 If X, Y are fuzzy tplgical spaces and A ~, B ~ A ~ B ~ f XY is defined as μ (Ã B (,y min(μ Ã(, μ B (y where X and y Y are fuzzy sets f X and Y respectively, then the fuzzy set Thrughut the paper, by ( X, T ~, ( Y, T ~ etc r simply by X, Y, Z, etc we shall mean fuzzy tplgical spaces Definitin 3 A mapping III Fuzzy Super Cntinuus Functins f :X Yfrm fuzzy tplgical X t an fuzzy tplgical space Y is said t be fuzzy super f X if and nly if fr every q nbd U f f(, there is a q nbd V f cntinuus at a fuzzy pint ( equivalently, we can take U,V t be pen fuzzy set f is said t be fuzzy super f(v U cntinuus n X if and nly if it is fuzzy super cntinuus at each fuzzy pint f X Definitin 3 A functin f : X Y frm an fuzzy tplgical space X t an fuzzy tplgical space Y is said t be fuzzy cntinuus at a fuzzy pint f X if and nly if fr every q nbd U f f( there is a q nbd V f f is called fuzzy cntinuus n X if and nly if f is fuzzy cntinuus at each fuzzy f (V U pint f X DOI: 09790/578-05084 wwwisrjurnalsrg 9 Page
Remark 3 A new Type f Fuzzy Functins in Fuzzy Tplgical Spaces If a mapping f : (X, T ~ (Y, F, frm an fuzzy tplgical space (X, T ~ t an fuzzy tplgical space (Y, F is fuzzy super cntinuus at a fuzzy pint fuzzy cntinuus at Therem 3, but the cnverse is false f X, then f is Fr a mapping f : (X, T ~ (Y, F, frm an fuzzy tplgical space (X, T ~ t an fuzzy tplgical space (Y, F, the fllwing are equivalent: (a f is a fuzzy super cntinuus (b f([ A ~ ] f(a ~, fr every fuzzy set A ~ in X (c (A ~ f (A ~ [f ], fr every fuzzy set A ~ in Y (d Fr every clsed fuzzy set A ~ in Y, f (A ~ is fuzzy -clsed in X (e Fr every pen fuzzy set A ~ in Y, f ( A ~ is fuzzy -pen in X (f Fr every pen fuzzy pint f X and fr each pen fuzzy q-nbd M f f( q-nbd N ~ f f(n M Prf (a (b : Let [A ~ ] and U be any pen fuzzy q-nbd f y f( By (a, there is pen fuzzy q-nbd N ~ f f(n U Nw, if [A ~ ] then (N ~ q A ~ s that f(n ~ q f(a ~ and hence Uq f(a Therefre f( (f(a ~ and it fllws that - f (f(a Thus [A ~ f (f(a ~ - ] (b (c : By (b, if and hence f([ A ~ ] f(a ~ - f([f (A] f[f (A] A [f (A ~ ] f (A ~ - - Then it fllws that (c (d : Immediate (d (e : Clear (e (f : N ~ f (M ~ serves the purpse, where y=f(, there is a fuzzy - pen (f (a : Fr any fuzzy pint and pen fuzzy q-nbd V f f(, there is a fuzzy - pen q-nbd N ~ f f(n V Then N ~ G ~ ( say is fuzzy -clsed in X and G ~ S the eist a pen fuzzy regularly q-nbd M f M is nt q-cincident with G DOI: 09790/578-05084 wwwisrjurnalsrg 0 Page
q M q M G N Nw, Frm which it fllws that A new Type f Fuzzy Functins in Fuzzy Tplgical Spaces f M f(n V and thus f is fuzzy super cntinuus Therem 3 A functin f :X Y frm fuzzy is a fuzzy super cntinuus if and nly if fr any fuzzy pint fr each q-nbd M f f(, there q q-nbd N ~ f Lemma 3 Let :X Y fllwing are equivalent: f(n M f X and f be a mapping frm an fuzzy tplgical space X t an fuzzy tplgical space Y the (a Fr any fuzzy pint f(n M (b Fr any fuzzy pint f (N M Prf : it is immediate f X and fr any q-nbd M f f(, there is a q-nbd N ~ f f X and fr any q-nbd M f f(, there is a q-nbd N ~ f Crllary 3 If ne f the cnditins f Lemma 3 - hlds then f is fuzzy super cntinuus That the cnverse f the abve crllary is false is shwn by the fllwing eample Eample 3 Let X be a nn-empty set and Cnsider T ~ {0,,A ~ }, a X Where μ Ã (a /3 and μ Ã (=0, fr f :(X,T ~ (X,T ~ a ( X, We cnsider the fuzzy pint a f X, where Then A ~ is pen fuzzy q-nbd f f(a, But f (U A, fr U A r 5 6 and cnsider the identity map Hence cnditin (a f lemma 3 fails But if clearly fuzzy super cntinuus Lemma 3 f : A X and f : A Y be any functins f :A X Y be defined by f(a (f(a,f (a, fr aa where X Y Let A, X, Y be fuzzy tplgical spaces and Let the prduct fuzzy tplgy is prvided with (a If B,U,U are fuzzy sets in A, X and Y respectively f(b U U then f (B U and f (B U DOI: 09790/578-05084 wwwisrjurnalsrg Page
(b If W ~ is pen fuzzy q-nbd f f( (f (, f ( in X Y, Then there eist pen fuzzy q-nbd U f that f( q (U V W (c If A new Type f Fuzzy Functins in Fuzzy Tplgical Spaces (f ( in X and pen fuzzy q-nbd V f ( f (A f (A U U, where A ~, A ~ are fuzzy sets in A and U,U f(v U U where V A A f and y respectively, then f (B U X, then Prf (a we prve that In fact, Let f(μ B (= Sup z f ( μ B ( Nw, zf ( f(z Let f (z y Y Then f(z (,y zf [(,y] Nw f(μ B [(,y]= Obviusly μ B (z By ( But ( Sup tf [(, y ] Sup tf [(, y ] tf [(,y ] μ B (t μ B (t Sup μ (t μ(u U [(,y ] μ (z μ [(,y ] Since ( is true fr all B B U U z f μ (z μ ( ( B U (, we have f(μ ( μ ( (f in Y such are fuzzy sets f (B U B U and hence Similarly, we can prve that f (B U (b Since W is pen fuzzy set in X Y, we have W (U V, where the U X and the V s are pen fuzzy set in Y Then bviusly U V W, fr each β and each μ We claim that fr sme β and sme μ f( q U V In nt then fr each β and each μ min(μ (f (,μ (f ( S that U β V μ (f(, fr each β and each μ, (Uβ V s are pen set in DOI: 09790/578-05084 wwwisrjurnalsrg Page
And thus But then Therefre Sup[μ (f( (U βv μ w (f( and hence f( q W, a cntradictin f( q U V, fr sme β and sme μ, min(μ (f (,μ (f ( S that U β V Then f ( qu β and f ( qv we have prved ur desired result (c Straight frward and left Therem 3 3 Let A, X, Y be fuzzy tplgical spaces and : A X, A new Type f Fuzzy Functins in Fuzzy Tplgical Spaces f f : A Y be any functins Then f :A X Y, defined by f( (f(, f (, fr all A, is fuzzy super cntinuus iff f and f are fuzzy super cntinuus Prf Let be a fuzzy pint f A and U,U respectively U U is clearly pen fuzzy q-nbd f f( (f( Then ie, f be pen fuzzy q-nbds f f ( Then by fuzzy super cntinuity f f, there is pen fuzzy q-nbd V f f(v U U By lemma 3 (a, we then have f (V U and f (V U S that f and f are fuzzy super cntinuus cnversely, Let be any fuzzy pint f A and W be any pen fuzzy q-nbd f f( and f ( in X Y in X and Y in A Then by lemma 3 (b, there eist pen fuzzy q-nbds and f( qu U W Als since f and f are fuzzy super cntinuus, there eist pen fuzzy q-nbd V and V f in A f (V U and f (V U, s that f (V f (V U U Nw by lemma 3 (c we have f(v U U, where V V V and V is bviusly pen fuzzy q-nbd Hence f is fuzzy super cntinuus functin DOI: 09790/578-05084 wwwisrjurnalsrg 3 Page U f f ( U f f ( References [] SP Arya and R Gupta, n strngly cntinuus mapping, kyungpk Math J 4( (974 3 43 [] KK Azad, n fuzzy semi cntinuity, fuzzy almst cntinuity and fuzzy weakly cntinuity, J Math Anal Appl 8 ( (98 4 3 [3] Casanvas J Trrens, J, An aimatic apprach t fuzzy cardinalities f finite fuzzy sets, fuzzy sets and systems 33 (003, n, 93-09 [4] CL Chang, Fuzzy tplgical spaces, J Math Anal Appl 4 (968 8 90 [5] B Huttn and IL Reilly, Separatin aims in fuzzy tplgical spaces, Fuzzy sets and systems 3 (980 7 4 [6] N Levine, strng cntinuity in tplgical spaces, Amer Math mnthly 67 (960 69
A new Type f Fuzzy Functins in Fuzzy Tplgical Spaces [7] S Nada, n fuzzy tplgical spaces, Fuzzy sets and systems 9(986 93 97 [8] Pa Ming Pu and Ying Ming Liu, Fuzzy tplgy, In Neighbrhd structure f fuzzy pint and Mre-smith cnvergence, J Math Anal Appl 76 (980 57 599 [9] CK Wng, Fuzzy tplgy ; prduct and qutient therems J Math Anal Appl 45 (974 [0] CK Wng, Fuzzy pints and lcal prperties f fuzzy tplgy, J Math Anal Appl 46 (974 36 38 [] LA Zadeh, Fuzzy sets, infrm, and cntrl 8 (965 338 353 DOI: 09790/578-05084 wwwisrjurnalsrg 4 Page