Weakly continuous functions on mixed fuzzy topological spaces

Transcription

1 cta Scentarum ISSN prnted: ISSN on-lne: Do: 0405/actasctechnolv3664 Weakly contnuous functons on mxed fuzzy topologcal spaces Bnod Chandra Trpathy and Gautam Chandra Ray Mathematcal Scences Dvson, Insttute of dvanced Study n Scence and Technology, Pashm Boragaon, Guwahat, 78035, ssam, Inda Department of Mathematcs, Central Insttute of Technology, Kokrajhar, ssam, Inda uthor for correspondence E-mal: trpathybc@yahoocom BSTRCT The notons of contnuty was generalzed n the fuzzy settng by Chang (968 Later on zad (98 ntroduced some weaker form of fuzzy contnuty lke fuzzy almost contnuty, fuzzy sem-contnuty and fuzzy weak contnuty These are natural generalzaton of the correspondng weaker forms of contnuty n topologcal spaces Recently rya and Sngal (00a and b ntroduce another weaker form of fuzzy contnuty, namely fuzzy sub contnuty as a natural generalzaton of subweak contnuty ntroduced by Rose (984 In ths paper we ntroduce fuzzy weak contnuty n mxed fuzzy topologcal space Keywords: fuzzy weak contnuty, fuzzy pont, mxed fuzzy topologcal space, fuzzy subspace Funções contínuas fracas sobre espaços topológcos dfusos msturados RESUMO s noções de contnudade foram generalzados no ambente dfuso de Chang (968 Mas tarde, zad (98 apresentou formas mas fracas de contnudade dfusa, como contnudade quase dfusa, sem-contnudade dfusa e contnudade dfusa fraca São generalzações naturas das formas correspondentes de contnudades mas fracas em espaços topológcos Recentemente, rya e Sngal (00a e b apresentaram uma outra forma mas fraca de contnudade dfusa, ou seja, contnudade subfraca dfusa como uma generalzação natural da contnudade sub-fraca de Rose (984 presenta-se nesse trabalho a contnudade fraca dfusa no espaço topológco dfuso msto Palavras-chave: contnudade fraca dfusa, ponto dfuso, espaço topológco dfuso msto, sub-espaço dfuso Introducton The noton of topologcal space has been generalzed n many ways The noton of btopologcal space and mxed topologcal space has been ntroduced and nvestgated n the resent past Btopologcal spaces have recently been studed by Ganguly and Sngha (984, Trpathy and Sarma (0b, 0 and others Mxed topology les n the theory of strct topology of the spaces of contnuous functons on locally compact spaces The concept of mxed topology s very old Mxed topology s a technque of mxng two topologes on a set to get a thrd topology on that set The works on mxed topology s due to Cooper (97, Buck (95, Das and Bashya (995, Trpathy and Ray (0, Wweger (96 and many others In 965 L Zadeh ntroduced the concept of fuzzy sets Snce then the noton of fuzzness has been appled for the study n all the branches of scence and technology It has been appled for studyng dfferent classes of sequences of fuzzy numbers by Trpathy and Baruah (00, Trpathy and Borgohan (008, 0, Trpathy and Dutta (00, Trpathy and Sarma (0a, Trpathy et al (0, and many workers on sequence spaces n the recent years The noton of fuzzness has been appled n topology and the noton of fuzzy topologcal spaces has ntroduced and nvestgated by many researches on topologcal spaces Dfferent propertes of fuzzy topologcal spaces have been nvestgated by rya and Sngal (00a and b, Chang (968, Das and Bashya (995, Ganster et al (005, Ganguly and Sngha (984, Ghanm et al (984, Katsaras and Lu (977, Petrcevc (998, Srvastava et al (98, Warren (978, Wong (974a and b and many others Recently mxed fuzzy topologcal spaces have been nvestgated from dfferent aspect by Das and Bashya (995 and others Prelmnares Let X be a non-empty set and I, the unt nterval [0, ] fuzzy set n X s characterzed by a functon : X I, where s called the membershp functon of (x represents the membershp grade of x n The empty fuzzy set s defned by ( t 0 for all tx lso X can be regarded as a fuzzy set n tself defned by X for all tx Further, an ordnary subset of X can also be regarded as a fuzzy set n X f ts cta Scentarum Technology Marngá, v 36, n, p , pr-june, 04

2 33 Trpathy and Ray membershp functon s taken as usual characterstc functon of that s, for all tx and 0 for all tx- Two fuzzy sets and B are sad to be equal f B fuzzy set s sad to be contaned n a fuzzy set B, wrtten as B, f B Complement of a fuzzy set n X s a fuzzy set n X defned by c We wrte c co Unon and ntersecton of a collecton { : J} of fuzzy sets n X, to be wrtten as and respectvely, are defned as follows: μ (x sup{ I (x : J}, for all x X and for all x ( x nf{ μ X ( x : J} I Defnton fuzzy topology τ on X s a collecton of fuzzy sets n X such that, X τ; f τ, J then τ; f, Bτ then B τ The par (X, τ s called a fuzzy topologcal space (fts Members of τ are called open fuzzy set and the complement of an open fuzzy sets s called a closed fuzzy set Defnton If (X, τ s a fuzzy topologcal space, then the closure and nteror of a fuzzy set n X, denoted by cl and nt respectvely, are defned as cl = { B: B s a closed fuzzy set n X and B} The nt = {V : V s an open fuzzy n X and V } Clearly, cl (respectvely nt s the smallest (respectvely largest closed(respectvely open fuzzy set n X contanng (respectvely contaned n If there are more than one topologes on X then the closure and nteror of wth respect to a fuzzy topology τ on X wll be denoted by τ-cl and τ-nt Defnton 3 collecton Ɓ of open fuzzy sets n a fts X s sad to be an open base for X f every open fuzzy set n X s a unon of members of Ɓ Defnton 4 If s a fuzzy set n X and B s a fuzzy set n Y then, B s a fuzzy set n X Y defned by xb (x,y = mn { (x, B (y} for all xx and for all y Y Let f be a functon from X nto Y Then, for each fuzzy set B n Y, the nverse mage of B under f, wrtten as f - [B], s a fuzzy set n X defned by ( ( ( x f [ B] B f x for all xx Defnton 5 fuzzy set n a fuzzy topologcal space (X, τ s called a neghborhood of a pont xx f and only f there exsts B τ such that B and (x = B(x>0 Defnton 6 fuzzy pont x α s sad to be quas-concdent wth, denoted by x α q, f and only f α + (x > or α > ((x c Defnton 7 fuzzy set s sad to be quasconcdent wth B and s denoted by qb, f and only f there exsts a xx such that (x + B(x > Remark It s clear that f and B are quasconcdent at x both (x and B(x are not zero at x and hence and B ntersect at x Defnton 8 fuzzy set n a fts (X, τ s called a quas-neghbourhood of x λ f and only f τ such that and x λ q The famly of all Q-neghbourhood of x λ s called the system of Q-neghbourhood of x λ Intersecton of two quas-neghbourhood of x λ s a quasneghbourhood Let (X, τ and (X, τ two fuzzy topologcal spaces and let τ (τ = {I X : for every fuzzy set B n X wth qb, there exsts a τ Q-neghbourhood α, such that α qb and τ -closure, B }, Then τ (τ s a fuzzy topology on X and ths s called mxed fuzzy topology, and the space (X, τ (τ s called mxed fuzzy topologcal space Man results Theorem 3 If (X,, and (Y,, two fuzzy btopologcal spaces If f : X Y s - and - contnuous, then f s ( - ( contnuous Proof: Let be any ( open set n Y We show that f ( s an open set n X Let be any fuzzy set n X such that q f - ( (x + f - ( (x >, for some x X + f (x > where (x = (f(x q Now ( f ( x s a fuzzy set n Y and ( f ( x q Therefore by defnton of mxed topology there exsts -open set B such that closure ( f ( x qb B and Snce f s f contnuous, so we have B f ( But f ( B f B f ( Snce f s - contnuous, so we have therefore for each fuzzy set n X and -open set ( f ( x B n Y wth q B there exsts open set B such that Bq and f(b B cta Scentarum Technology Marngá, v 36, n, p , pr-june, 04

3 Weakly contnuous functons on mxed fuzzy topologcal spaces 333 B f ( B f ( B f ( f(b Thus we have -open set B wth Bq and so by defnton of mxed topology, we have closure, B f ( Hence f - ( s ( -open, and so f : X Y s contnuous Ths completes the proof Theorem 3 Let (X,, and (Y,, be any two fuzzy b-topologcal spaces and f: X Y be a mappng such that f s - and - contnuous then f s ( - ( contnuous Proof Let B be any ( fuzzy open set n Y f We show that cl (f - (B ( clb, n ths case the closure s wth respect to ( We have B ( Let BqV for any fuzzy set V n Y Then there exst UqV and -open set U such that -closure cl (U B ( gan gven that f s - contnuous and U s -open fuzzy set n Y so by defnton of contnuty we have f ( B f ( clb ( lso cl (U s -closed fuzzy set n Y, so coy ( clu s -fuzzy open set n Y Hence f s contnuous By defnton of contnuty we have cl f ( co ( U f ( co ( cl ( U Y Takng complement of both sde of (3, we get Y (3 f - (U nt(f - (U (4 Now from ( we have f - (U f - (B Therefore usng ( we have f - (U f - (B (5 Further, nt(f - (U f - (U (6 Therefore usng (4 and (6 we have f - (U = nt (f - (U (7 gan from ( we have U B f ( U f ( B nt( f ( U f ( B Takng closure of both sde, we get f ( B nt( f ( U f ( U = f ( U Hence f - (B f - (clb f s ( - ( s contnuous Ths completes the proof Theorem 33 Let (X, τ, τ and (Y,, be two fuzzy b-topologcal spaces and and s fuzzy regular space If f: X Y s ( - ( contnuous, then f s contnuous Proof: Let B be any -fuzzy open set n Y We show that cl ( f ( B f ( B By hypothess we have and s fuzzy regular space, therefore ( ( B s fuzzy open set n Y Snce f: X Y s ( - ( contnuous, we have cl ( f ( B f ( B (8 (, thus the result lso we know that (8 s true for -fuzzy topology also Therefore cl ( f ( B f ( B, closure beng wth respect to topology and -fuzzy topology Hence f s contnuous Ths completes the proof Theorem 34 Let (X,, and (Y,, two fuzzy b-topologcal spaces, If f: X Y s ( - ( contnuous Then f s ( contnuous cta Scentarum Technology Marngá, v 36, n, p , pr-june, 04

4 334 Trpathy and Ray Proof: Let B be any ( open fuzzy set n Y We show that f - (B f - (B, the closure s beng wth respect to Let f: X Y be ( - ( contnuous Then f ( B f ( B closures are beng wth respect to ( and ( respectvely We know that the mxed topology ( s contaned n Snce f - (B f - (B, closure beng wth ( respect to f - (B f - (B, closure of the left hand sde beng wth respect to f s ( contnuous Ths completes the proof Theorem 35 Let (X,, and (Y,, two fuzzy b-topologcal spaces If f: X Y s contnuous and s fuzzy regular, then f s ( contnuous Proof: Let B be any fuzzy open set n Y Let f be contnuous Then by defnton of contnuty, we have f ( B f ( B closure of left hand sde s wth respect to and rght hand sde s wth respect to Further, s fuzzy regular and and ( therefore Thus closure of f - (B s wth respect to s also the closure of wth respect to Hence ( f ( B f ( B, the closure of LHS beng wth respect to Hence f s ( ( contnuous Ths completes the proof Theorem 36 Let (X,, and ( Y,, be two fuzzy b-topologcal spaces If f: X Y s ( contnuous, then f s contnuous Proof: Let f: X Y be ( contnuous Thus we have for any -fuzzy open set B n Y, we have f ( B f ( B, closure of the LHS s beng wth respect to ( We know that ( Thus, closure of f - (B wth respect to ( s same as the closure of f - (B wth respect to Thus f - (B f - (B, the closure of left hand sde s wth respect to the fuzzy topology f s contnuous Ths completes the proof Theorem 37 Let (X,, and ( Y,, two fuzzy b-topologcal spaces If f: X Y s ( contnuous, then f s contnuous Proof: Let B be -fuzzy open set n Y Let f: X Y be ( contnuous Then we have f ( B f ( B, the closure ( of left hand sde beng wth respect to ( Further we have Therefore closure of f ( B wth respect to ( s same as the closure of f ( B wth respect to Hence f - (B f - (B, the closure of left hand sde s beng wth respect to Thus f s contnuous Ths completes the proof Concluson We have ntroduced fuzzy weak contnuty n mxed fuzzy topologcal space and have nvestgated ts dfferent propertes The results of ths artcle can be appled for futher nvestgatons and applcatons n studyng dfferent propertes weak contnuty of functons n mxed fuzzy topologcal spaces References RY, S P; SINGL, N Fuzzy sub contnuous functons Mathmatcs Student, v 70, n -4, p 3-40, 00a RY, S P; SINGL, N Fuzzy sub α-contnuous functons Mathematcs Student, v 70, n -4, p 4-46, 00b ZD, K K On fuzzy sem-contnuty, fuzzy almost contnuty and fuzzy contnuty Journal of Mathematcal nalyss and pplcatons, v 8, n, p 4-3, 98 BUCK, R C Operator algebras and dual spaces Proceedngs of the mercan Mathematcal Socety, v, 3, n 5, p , 95 cta Scentarum Technology Marngá, v 36, n, p , pr-june, 04

5 Weakly contnuous functons on mxed fuzzy topologcal spaces 335 CHNG, C L Fuzzy topologcal spaces Journal of Mathematcal nalyss and pplcatons, v 4, n, p 8-90, 968 COOPER, J B The strct topology and spaces wth mxed topologes Proceedngs of the mercan Mathematcal Socety, v, 30, n 3, p , 97 DS, N R; BISHY, P C Mxed fuzzy topologcal spaces The Journal of Fuzzy Mathematcs, v 3, n 4, p , 995 GNGULY, S; SINGH, D Mxed topology for a Btopologcal spaces Bulletn of Calcutta Mathematcal Socety, v 76, n 3, p , 984 GNSTER, M; GEORGIOU, D N; JFRI, S; MOSOKHO, S On some applcaton of fuzzy ponts ppled General Topology, v 6, n, p 9-33, 005 GHNIM, M H; KERRE, E E; MSHHOUR, S Separaton axoms, subspaces and sums n fuzzy topology Journal of Mathematcal nalyss and pplcatons, v 0, n, p 89-0, 984 KTSRS, K; LIU, D B Fuzzy vector spaces and fuzzy topologcal vector spaces Journal of Mathematcal nalyss and pplcatons, v 58, n, p 35-46, 977 PETRICEVIC, Z On s-closed and extremally dsconnected fuzzy topologcal spaces Matematички Bechик, v 50, n /, p 37-45, 998 ROSE, D Weak contnuty and almost contnuty Internatonal Journal of Mathematcs and Mathematcal Scences, v 7, n, p 3-38, 984 SRIVSTV, R; LL, S N; SRIVSTV, K Fuzzy Hausdorff topologcal spaces Journal of Mathematcal nalyss and pplcatons, v 8, n, p , 98 TRIPTHY, B C; BRUH, Lacunary statstcally convergent and lacunary strongly convergent generalzed dfference sequences of fuzzy real numbers Kyungpook Mathematcal Journal, v 50, n 4, p , 00 TRIPTHY, B C; BORGOGIN, S Some classes of dfference sequence spaces of fuzzy real numbers defned by Orlcz functon dvances n Fuzzy Systems, v 0, n, p -6, 0 TRIPTHY, B C; BORGOGIN, S The sequence space m(m,, n m, p F Mathematcal Modellng and nalyss, v 3, n 4, p , 008 TRIPTHY, B C; DUTT, J Bounded varaton double sequence space of fuzzy real numbers Computers and Mathematcs wth pplcatons, v 59, n, p , 00 TRIPTHY, B C; RY, G C On mxed fuzzy topologcal spaces and countablty Soft Computng, v 6, n 0, p , 0 TRIPTHY, B C; SRM, B Double sequence spaces of fuzzy numbers defned by Orlcz functon cta Mathematca Scenta, v 3B, n, p 34-40, 0a TRIPTHY, B C; SRM, D J On b-locally open sets n btopologcal spaces Kyungpook Mathematcal Journal, v 5, n 4, p , 0b TRIPTHY, B C; SRM D J On parwse b locally open and parwse b locally closed functons n btopologcal spaces Tamkang Journal of Mathematcs, v 43, n 4, p , 0 TRIPTHY, B C; SEN, M; NTH, S I-convergence n probablstc n-normed space Soft Computng, v 6, n 6, p 0-07, 0 WRREN, R H Neghbourhood bases and contnuty n fuzzy topologcal spaces Rocky Mountan Journal of Mathematcs, v 8, n 3, p , 978 WIWEGER, Lnear spaces wth mxed topology Studa Mathematca, v 0, n, p 47-68, 96 WONG, C K Fuzzy topology: product and quotent theorems Journal of Mathematcal nalyss and pplcatons, v 45, n, p 5-5, 974a WONG, C K Fuzzy ponts and local propertes of fuzzy topology Journal of Mathematcal nalyss and pplcatons, v 46, n, p36-38, 974b Receved on March 5, 0 ccepted on May 8, 0 Lcense nformaton: Ths s an open-access artcle dstrbuted under the terms of the Creatve Commons ttrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted cta Scentarum Technology Marngá, v 36, n, p , pr-june, 04