Intuitionistic Fuzzy G δ -e-locally Continuous and Irresolute Functions

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1 Intern J Fuzzy Mathematcal rchve Vol 14, No 2, 2017, ISSN (P), (onlne) Publshed on 11 December 2017 wwwresearchmathscorg DOI http//dxdoorg/ /jmav14n2a14 Internatonal Journal o Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons G Saravanakumar 1, S Tamlselvan 2 and Vadvel 3 1 Department o Mathematcs, nnamala Unversty nnamalanagar, Taml Nadu , Inda 2 Mathematcs Secton (FET), nnamala Unversty nnamalanagar, Taml Nadu , Inda 3 Department o Mathematcs, Government rts College (utonomous) Karur, Taml Nadu , Inda Emal avmaths@gmalcom Receved 15 November 2017; accepted 7 December 2017 bstract The purpose o ths paper s to ntroduce the concepts o an ntutonstc uzzy G -e-locally neghborhood, ntutonstc uzzy G -e-locally quas neghborhood, ntutonstc uzzy G -e-locally contnuous and ntutonstc uzzy G -e-locally rresolute unctons n ntutonstc uzzy topologcal spaces lso some nterestng propertes are establshed Keywords ntutonstc uzzy G -e-locally neghborhood, ntutonstc uzzy G -elocally quas neghborhood, ntutonstc uzzy G -e-locally contnuous, ntutonstc uzzy G -e-locally rresolute uncton MS Mathematcs Subject Classcaton (2010) 5440, 5499, 03E72, 03E99 1 Introducton The concept o uzzy sets was ntroduced by Zadeh [10] and later tanassov [1] generalzed the dea to ntutonstc uzzy sets On the other hand, Coker [3] ntroduced the notons o an ntutonstc uzzy topologcal spaces, ntutonstc uzzy contnuty and some other related concepts The concept o an ntutonstc uzzy e -closed set was ntroduced by Sobana et al, [8] Ganster and Relly used locally closed sets n [5] to dene LC-contnuty and LC-rresoluteness alasubramanan [2] ntroduced and studed the concept o uzzy G set n a uzzy topologcal space In ths paper, the concepts o an ntutonstc uzzy G locally quas neghborhood, ntutonstc uzzy G locally contnuous and ntutonstc uzzy G rresolute uncton are ntroduced and studed Some nterestng propertes among contnuous uncton are dscussed 2 Prelmnares Denton 21 [1] Let X be a nonempty xed set and I be the closed nterval [0, 1] n ntutonstc uzzy set (IFS) s an object o the ollowng orm 313

2 G Saravanakumar, S Tamlselvanand Vadvel = { x,, where the mappng µ X I and γ X I denote the degree o membershp (namely µ (x ) ) and the degree o nonmembershp (namely γ (x )) or each element x X to the set, respectvely, and 0 µ ( x) + 1 or each x X Obvously, every uzzy set on a nonempty set X s an IFS o the ollowng orm, = { x, 1 µ ( x) For the sake o smplcty, we shall use the symbol = { x, 314 = x, µ, γ or the ntutonstc uzzy set Denton 22 [1] Let X be a nonempty set and the IFSs and n the orm = { x,, = { x, Then 1 and only µ ( x) ( x) and or all x X ; µ 2 = { x,, µ ( x) x ; 3 = { x, µ ( x) ; 4 = { x, µ ( x) γ ( x) ; Denton 23 [1] The IFS s 0 and 1 are dened by, 0 = { x, 0,1 x and 1 = { x,1, 0 Denton 24 [3] n ntutonstc uzzy topology (IFT) n Coker s sense on a nonempty set X s a amly T o ntutonstc uzzy sets n X satsyng the ollowng axoms 1 0, 1 T ; 2 G 1 G 2 T, or any G 1, G 2 T ; 3 G T or arbtrary amly { G J} T In ths paper by ( X, T ) or smply by X we wll denote the Coker s ntutonstc uzzy topologcal space (IFTS) Each IFS whch belongs to T s called an ntutonstc uzzy open set (IFOS) n X The complement o an IFOS n X s called an ntutonstc uzzy closed set (IFCS) n X Denton 25 [3] Let ( X, T ) be an IFTS and = { x, µ, ν be an IFS n X Then the ntutonstc uzzy closure and ntutonstc uzzy nteror o are dened by 1 IFcl( ) = { C C sanifcsn Xand C } ; 2 IFnt( ) = { D D sanifosn Xand D } ; Proposton 21 [1] For any IFS n ( X, T ) we have 1 cl ( ) = nt( ) 2 nt ( ) = cl( )

3 Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons Corollary 21 [3] Let, ( J ) be IFSs n X,, j ( j K) IFSs n Y and X Y a uncton Then 1 1 ( ( )) (I s njectve, then = ( ( ))) 2 ( ( )) (I s surjectve, then ( ( )) = ) 3 ( ) = ( ) 4 j j ( j ) = ( j ) 5 (1 ) = 1 6 (0 ) = 0 7 ( ) = ( ) Denton 26 [4] Let X be a nonempty set and x X a xed element n X I r I 0, s I1 are xed real numbers such that r + s 1, then the IFS xr, s = x, xr,1 x 1 s s called and ntutonstc uzzy pont (IFP) n X, where r denotes the degree o membershp o x, s denotes the degree o non membershp o x and x X the support o x The IFP x s contaned n the IFS ( xr, s ) and only r < µ ( x), s > Denton 27 [6] n IFS U o an IFTS X s called 1 neghborhood o an IFP c ( a, b), there exsts an IFOSG n X such that c( a, b) G U 2 q -neghborhood o an IFP c ( a, b), there exsts an IFOSG n X such that c( a, b) qg U Denton 28 [3] Let X and Y be two nonempty sets and X Y be a uncton 1 I = { y, µ ( y), γ ( y) y Y} s an IFS n Y, then the premage o under (denoted by 1 ( ) ) s dened by ( ) = { x, ( µ )( x), ( γ )( x) x 2 I = { x, λ ( x), ν ( x) s an IFS n X, then the mage o under (denoted by () ) s dened by ( ) = { y, ( λ ( y)), (1 (1ν ))( y) Denton 29 [9] Let be IFS n an IFTS ( X, T ) s called an 1 ntutonstc uzzy regular open set (brely IFROS ) = ntcl ( ) 2 ntutonstc uzzy regular closed set (brely IFRCS ) = clnt( ) Denton 210 [2] Let ( X, T ) be a uzzy topologcal space and λ be a uzzy set n X λ s called uzzy F σ G set = λ = λ where each λ T The complement o uzzy G λ s 1 315

4 G Saravanakumar, S Tamlselvanand Vadvel Denton 211 [5] subset o a space ( X, T ) s called locally closed (brely lc ) = C D, where C s open and D s closed n ( X, T ) Denton 212 [9] Let ( X, T ) be an IFTS and = x, ν ( x) be a IFS n X Then the uzzy closure o are denoted and dened by cl ( ) = { K K s an IFRCS n X and K} and nt ( ) = { G G s an IFROS n X and G } Denton 213 [8] Let be an IFS n an IFTS ( X, T ) s called an ntutonstc uzzy e -open set (IFeOS, or short) n X clnt ( ) ntcl ( ) Denton 214 [3] Let ( X, T ) and ( Y, S) be two IFT s and let X Y be a uncton Then s sad to be ntutonstc uzzy contnuous the premage o each IFS n S s an IFS n T Denton 215 [8] Let ( X, T ) and ( Y, S) be two IFT s and let X Y be a uncton Then s sad to be ntutonstc uzzy e -contnuous the premage o each IFS n S s an IFeOS n T Denton 216 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space Let = { x, be an ntutonstc uzzy set on an ntutonstc uzzy topologcal space ( X, T ) Then s sad to be ntutonstc uzzy e - locally closed set (n short, IF lcs ) = C D, where C = { x, s an ntutonstc uzzy e -open set and D = { x, x s an ntutonstc uzzy e -closed set n ( X, T ) Denton 217 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space Let = { x, x be an ntutonstc uzzy set on an ntutonstc uzzy topologcal space X Then s sad to be an ntutonstc uzzy D D C C eg - set =, 1 where = { x, x s an ntutonstc uzzy e -open set n an ntutonstc uzzy topologcal space ( X, T ) Denton 218 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space Let = { x, x be an ntutonstc uzzy set on an ntutonstc uzzy topologcal space ( X, T ) Then s sad to be an ntutonstc uzzy eg -locally = 316

5 Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons closed set (n short,if- eg - lcs ) s an ntutonstc uzzy = C D, where C = { x, eg set and D = { x, x s an ntutonstc uzzy e -closed set n ( X, T ) The complement o an ntutonstc uzzy eg -locally closed set s sad to be an ntutonstc uzzy eg -locally open set (n short, IF eg -los) Denton 219 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space Let = { x, x be an ntutonstc uzzy set on an ntutonstc uzzy topologcal space ( X, T ) Then s sad to be an ntutonstc uzzy G locally closed set (n short,if G lcs ) = C, where = { x, x s an ntutonstc uzzy G set and C = { x, s an ntutonstc uzzy e -closed set n ( X, T ) The complement o an ntutonstc uzzy G locally closed set s sad to be an ntutonstc uzzy G locally open set (n short, IFG los) Denton 220 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space Let = { x, be an ntutonstc uzzy set on an ntutonstc uzzy topologcal space ( X, T ) The ntutonstc uzzy G locally closure o s denoted and dened by IFG -e -lcl( ) = { = x, µ ( x), γ ( x) X s an ntutonstc uzzy G locally closed set n X and } Denton 221 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space Let = { x, be an ntutonstc uzzy set on an ntutonstc uzzy topologcal space ( X, T ) The ntutonstc uzzy G locally nteror o s denoted and dened by IFG lnt( ) = { = { x, µ ( x), γ ( x) x s an ntutonstc uzzy G -e -locally open set n X and Proposton 22 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space For any two ntutonstc uzzy sets = { x, x and = { x, o an ntutonstc uzzy topologcal space ( X, T ) then the ollowng statements are true 1 IFG lcl (0 ) = 0 2 IFG lcl( ) IFG lcl () C C D D C C 317

6 G Saravanakumar, S Tamlselvanand Vadvel 3 IFG lcl( IFG lcl ( )) = IFG -e - lcl () 4 IFG lcl ( ) = ( IFG lcl ( )) ( IFG lcl ()) Remark 21 [7] 1 IFG lcl ( ) = and only s an ntutonstc uzzy G locally closed set 2 IFG lnt( ) IFG lcl () 3 IFG lnt (1 ) = 1 4 IFG lnt (0 ) = 0 5 IFG lcl (1 ) = 1 3 Intutonstc uzzy G locally contnuous unctons Denton 31 Let ( X, T ) be an ntutonstc uzzy topologcal space Let = x, µ, γ be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( X, T ) Then s sad to be an ntutonstc uzzy G -e -locally neghbourhood o an ntutonstc uzzy pont x there exsts an ntutonstc uzzy G locally open set n an ntutonstc uzzy topologcal space ( X, T ) such that x, It s denoted by IFG lnbd Denton 32 Let ( X, T ) be an ntutonstc uzzy topologcal space Let = x, µ, γ be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( X, T ) Then s sad to be an ntutonstc uzzy G locally quas neghbourhood o an ntutonstc uzzy pont x there exsts an ntutonstc uzzy G locally open set n an ntutonstc uzzy topologcal space ( X, T ) such that x s q, It s denoted by IFG lqnbd r, Remark 31 1 The amly o all ntutonstc uzzy G locally neghbourhood o an ntutonstc uzzy pont x, s denoted by N x ) r s IFG el ( r,s 2 The amly o all ntutonstc uzzy G locally quas neghbourhood o an ntutonstc uzzy pont x, s denoted by N x ) r s IFG elq ( r,s Denton 33 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy mappng Then s sad to be an ntutonstc uzzy G locally contnuous uncton, or each ntutonstc uzzy pont ( ) x, n X and N ), there exsts N x ) such that r s ( x r,s IFG elq ( r,s 318

7 Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons Theorem 31 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy mappng Then the ollowng are equvalent 1 s an ntutonstc uzzy G locally contnuous uncton 2 ( ) s an ntutonstc uzzy G locally open set n an ntutonstc uzzy topologcal space ( X, T ), or each ntutonstc uzzy open set n an ntutonstc uzzy topologcal space ( Y, S) 3 ( ) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( X, T ), or each ntutonstc uzzy closed set n an ntutonstc uzzy topologcal space ( Y, S) 4 IFG lcl( ( )) ( IFcl( )), or each ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( Y, S) 5 ( IFnt( )) IFG lnt( 1 ( )), or each ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( Y, S) Proo () () Let be an ntutonstc uzzy open set n an ntutonstc uzzy topologcal space ( Y, S) Let x be an ntutonstc uzzy pont n an ntutonstc uzzy topologcal space ( X, T ) such that x 1 s q ( ) Snce s an ntutonstc uzzy r, IFG elq ( r,s G locally contnuous uncton, there exsts N x ) such that ( ) Then xr, s (1) 1 ( ( )) (2) Thus, ( ( )) ( ) Ths mples 1 ( ) whch x IFG elq x shows that ( ) N ( xr, s ) Hence ( ) s an ntutonstc uzzy G -e - locally open set n an ntutonstc uzzy topologcal space ( X, T ) () () Ths can be proved by takng complement o () () (v) Let be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( Y, S) Snce IFcl( ), ( ) ( IFcl( )) y (), ( IFcl( )) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( X, T ) Thus, IFG lcl( ( )) ( IFcl( )) (v) (v) Usng (v), IFG -e - lcl( ( )) ( IFcl( )) Then IFG e lcl( ( )) ( IFcl( )), IFG lnt( ( )) ( IFnt( )), IFG e lnt( ( )) ( IFnt( )) mples that ( IFnt( )) IFG e lnt( 1 ( )), puttng =, we have ( IFnt( )) IFG e lnt( ( )) 319

8 G Saravanakumar, S Tamlselvanand Vadvel (v) () Let be an ntutonstc uzzy open set n an ntutonstc uzzy topologcal space ( Y, S) Then IFnt = Usng (v), ( IFnt( )) IFG -e - ( 1 lnt ( )) mples that ( ) IFG lnt( 1 ( )) ut, IFG lnt( ( )) ( ) mples that ( ) = IFG lnt( 1 ( )) that s, ( ) s an ntutonstc uzzy G locally open set n an ntutonstc uzzy topologcal space ( X, T ) Let x be any ntutonstc uzzy pont n 1 ( ) Then x 1 ( ) We have 1 x s q ( ) mples that ( x ) q ( ( )) ut ( ( )) Thus, r, r, s or any ntutonstc uzzy pont x and N ( x r,s ), there exsts IFG elq = r, s ( ) N ( x ) such that ( ( )) Thereore, ( ) Thus, s an ntutonsc uzzy G locally contnuous uncton Theorem 32 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy bjectve uncton Then s an ntutonstc uzzy G locally contnuous uncton and only IFnt ( ( )) ( IFG lnt ()), or each ntutonstc uzzy set o an ntutonstc uzzy topologcal space ( X, T ) Proo ssume that s an ntutonstc uzzy G locally contnuous uncton and be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( X, T ) Hence, ( IFnt( ( ))) s an ntutonstc uzzy G locally open set n an ntutonstc uzzy topologcal space ( X, T ) From Theorem (v) o (01) ( IFnt ( )) IFG lnt( ( ( ))) Snce s an ntutonstc uzzy surjectve uncton, ( IFnt ( )) IFG, lnt () ( ( IFnt ( ))) ( IFG - e - lnt ()) That s, IFnt ( ( )) ( IFG lnt ()) Conversely, assume that IFnt ( ( )) ( IFG lnt ()), or each ntutonstc uzzy set o an ntutonstc uzzy topologcal space ( X, T ) Let be an ntutonstc uzzy open set n an ntutonstc uzzy topologcal space ( Y, S) Then = IFnt( ) Snce s an ntutonstc uzzy surjectve uncton, = IFnt( ) = IFnt( ( ( ))) ( IFG lnt( 1 ( ))) Snce s an ntutonstc uzzy njectve uncton, ( ) ( ( IFG - lnt( 1 ( )))) From the act that, s an ntutonstc uzzy njectve uncton, we have ( ) IFG e lnt( ( )) (3) but - e 320

9 Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons IFG e lnt( ( )) ( ) (4) From (3) and (4) mples that ( ) = IFG lnt( 1 ( )) That s, ( ) s an ntutonstc uzzy G locally open set n an ntutonstc uzzy topologcal space ( X, T ) Thus, s an ntutonstc uzzy G locally contnuous uncton Theorem 33 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy bjectve ucton Then s an ntutonstc uzzy G locally contnuous uncton and only ( IFG lcl( )) IFcl( ( )), or each ntutonstc uzzy set o an ntutonstc uzzy topologcal space ( X, T ) Proo ssume that s an ntutonstc uzzy G locally contnuous uncton and be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( X, T ) Hence, ( IFcl( ( ))) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( X, T ) From Theorem (v) o (01) IFG lcl( ( ( ))) ( IFcl ( )) Snce s an ntutonstc uzzy njectve uncton, IFG lcl( )) ( IFcl ( )) Takng on both sdes, ( IFG -e - lcl( )) ( ( IFcl ( ))) Snce s an ntutonstc uzzy surjectve uncton, ( IFG lcl( )) IFcl( ( )) Conversely, assume that ( IFG lcl( )) IFcl( ( )), or each ntutonstc uzzy set o an ntutonstc uzzy topologcal space ( X, T ) Let be an ntutonstc uzzy closed set n an ntutonstc uzzy topologcal space ( Y, S) Then = IFcl( ) Snce s an ntutonstc uzzy surjectve uncton, and by assumpton = IFcl( ) = IFcl( ( ( ))) ( IFG lcl ( ( ))), ( ) ( ( IFG lcl( 1 ( )))) Snce s an ntutonstc uzzy njectve uncton, ut ( ) ( ( )) (5) IFG e lcl ( ( )) IFG e cl( ( )) (6) From (5) and (6) mples that s an ntutonstc uzzy ( ) = IFG lcl( 1 ( )) That s, ( ) G locally closed set n an ntutonstc uzzy topologcal space ( X, T ) Thus, s an ntutonstc uzzy G locally contnuous uncton Theorem 34 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy bjectve ucton I s an 321

10 G Saravanakumar, S Tamlselvanand Vadvel ntutonstc uzzy G locally contnuous uncton Then uzzy closed set, then ( ) = IFG lcl( 1 ( )) 322 Y I s an ntutonstc Proo Let be an ntutonstc uzzy closed set n an ntutonstc uzzy topologcal space ( Y, S) y Theorem(v)o (01) Snce = IFcl( ) ut From (7) and (8) mples that IFG e lcl IFcl ( ( )) ( ( )) = ( ) ( ) ( ( )) IFG e lcl ( ) = IFG lcl( 1 ( )) Proposton 31 Let ( X, T ), ( Y, S) and ( Z, R) be any three ntutonstc uzzy topologcal spaces Let ( X T ) ( Y, S) be an ntutonstc uzzy G locally contnuouus uncton I ( X ) Z Y then g ( X, T ) ( Z, R) where R = S/ Z restrctng the range o s an ntutonstc uzzy G locally contnuous uncton Proo Let be an ntutonstc uzzy closed set n an ntutonstc uzzy topologcal space ( Z, R) Then = S/ Z, or some ntutonstc uzzy closed set o an ntutonstc uzzy topologcal spaces ( Y, S) I ( X ) Z Y, ( ) = g ( ) Snce ( ) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( X, T ) Hence, g ( ) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( X, T ) Thereore, g s an ntutonstc uzzy G locally contnuous uncton Proposton 32 Let ( X, T), ( X1, T1 ) and ( X 2, T2 ) be any three ntutonstc uzzy topologcal spaces and P X X X be an ntutonstc uzzy projecton o 1 2 X 1 X 2 onto X I X X1 X 2 s an ntutonstc uzzy G locally contnuous uncton Then P X X s also an ntutonstc uzzy G locally cotnuous uncton Proo Let be an ntutonstc uzzy closed set n an ntutonstc uzzy topologcal spaces ( X, T ) ( = 1, 2), ( P ) ( ) = ( P ( )) Snce P s an ntutonstc uzzy mappng P ( ) s an ntutonstc uzzy closed set n an ntutonstc uzzy topologcal spaces X 1 X 2 Hence, ( P ( )) s an ntutonstc uzzy G -e - locally closed set n an ntutonstc uzzy topologcal space ( X, T ) Hence, P s an ntutonstc uzzy G -e -locally contnuous uncton Proposton 33 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces I ntutonstc uzzy graph uncton g X X Y s an ntutonstc uzzy (7) (8)

11 Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons G locally contnuous uncton Then ( X, T ) ( Y, S) s an ntutonstc uzzy G locally contnuous uncton Proo Let g be an ntutonstc uzzy G locally contnuous uncton and x be any ntutonstc uzzy pont n an ntutonstc uzzy topologcal space ( X, T ) I IFG elq N x ) n an ntutonstc uzzy topologcal space ( r,s ( Y, S), X N g( xr,s ) n an ntutonstc uzzy topologcal space X Y Snce g s an ntutonstc uzzy G - e -locally contnuous uncton, there exsts N ) such that g ( ) X y 323 IFG elq ( x r,s Denton 024, we have ( ) Thereore, s an ntutonstc uzzy G locally contnuous uncton Denton 34 Let ( X, T ) and ( Y, S) be two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy mappng Then s sad to be an 1 Intutonstc uzzy G -e -locally rresolute uncton, or each ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal 1 ( space ( Y, S), ) s an ntutonstc uzzy ntutonstc uzzy topologcal space ( X, T ) G locally closed set n an 2 Intutonstc uzzy weakly G locally contnuous uncton, or each ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( Y, S), 1 ( ) s an ntutonstc uzzy closed set n an ntutonstc uzzy topologcal space ( X, T ) Example 31 Let X = { a, b, c} = Y, and = x,(, ),(, ), = x,(, ),(, ), = x,(, ),(, ), = x,(, ),(, ), C = x,(, ),(, ) Now, the amly T = {0,1,,,, } o IFS s n X s an IFT on X and the amly S = {0,1, C} o IFS s n Y s an IFT on Y I we dene the uncton X Y be the dentty uncton Now, s an ntutonstc uzzy G locally rresolute uncton, because C s an IFG locally closed set n Y, ( C ) s also an IFG locally closed set n X

12 Example 32 Let G Saravanakumar, S Tamlselvanand Vadvel X = { a, b, c} = Y, and = x,(, ),(, ), = x,(, ),(, ), = x,(, ),(, ), = x,(, ),(, ), C = x,(, ),(, ) D = x,(, ),(, ) Now, the amly T = {0,1,,,,, C} o IFS s n X s an IFT on X and the amly S = {0,1, D} o IFS s n Y s an IFT on Y I we dene the uncton X Y be the dentty uncton, Now, s an ntutonstc uzzy weakly G -e - locally rresolute uncton, because D s an IFG locally closed set n Y, ( D ) s ntutonstc uzzy closed set n X Theorem 35 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy mappng Then the ollowng statements are equvalent 1 s an ntutonstc uzzy G locally rresolute uncton 2 or every ntutonstc uzzy set o an ntutonstc uzzy topologcal space ( X, T ), ( IFG lcl( )) IFG -e - lcl ( ( )) 3 or every ntutonstc uzzy set o an ntutonstc uzzy topologcal space ( Y, S), IFG lcl ( ( )) ( IFG lcl ()) Proo () () Let be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( X, T ) Suppose s an ntutonstc uzzy G locally rresolute uncton Now, IFG lcl ( ( )) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( Y, S) y hypothess, (IFG lcl ( ( ))) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( X, T ) and hence, ( ( )) ( IFG ( lcl ( ( )))) Now, IFG lcl ( ) ( IFG lcl ( ( ))) That s, ( IFG lcl( )) IFG -e - lcl ( ( )) () () Let be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( Y, S), then ( ) s an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( X, T ) y (), ( IFG lcl( ( ))) IFG lcl( ( 1 ( ))) Snce s an ntutonstc uzzy bjectve uncton, IFG lcl ( ( )) ( IFG lcl ()) 324

13 Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons () () Suppose s ntutonstc uzzy IFG locally closed set n an ntutonstc uzzy topologcal space ( Y, S) Then IFG lcl ( ) = y hypothess, IFG lcl ( ( )) ( IFG lcl ( )), IFG - e lcl( ( )) ( ) 4 Concluson Intutonstc uzzy topology s an mportant and a major area o mathematcs In ths paper, we ntroduce ntutonstc uzzy G locally neghborhood, ntutonstc uzzy G locally quas neghborhood, ntutonstc uzzy G locally contnuous and ntutonstc uzzy G locally rresolute unctons n ntutonstc uzzy topologcal spaces are studed cknowledgement The authors express ther earnest grattude to the revewers, edtor-nche and managng edtors or ther constructve suggestons and comments whch helped to mprove the present paper REFERENCES 1 KTtanassov, Intutonstc uzzy sets, Fuzzy Sets and Systems, 20 (1986) Galasubramanan, Maxmal uzzy topologes, Kybernetka, 31 (1995) DCoker, n ntroducton to ntutonstc uzzy topologcal spaces, Fuzzy Set and Systems, 88 (1997) DCoker and MDemrc, On ntutonstc uzzy ponts, Notes IFS, 2(1) (1995) MGanster and ILRelly, Locally closed sets and and LC-contnuous unctons, IntrJ Math and Math Sc, 12(3) (1989) IMHanay, Completely contnuous uncton n ntutonstc uzzy topologcal space, Czechoslovak Math J, 53 (2003), GSaravanakumar, STamlselvan and Vadvel, Intutonstc uzzy G locally closed sets, submtted 8 DSobana, VChandrasekar and Vadvel, On uzzy e -open sets, uzzy e - contnuty and uzzy e -compactness n ntutonstc uzzy topologcal spaces, Communcatons n Mathematcal nalyss (n Press) 9 SSThakur and SSngh, On uzzy sem-pre open sets and uzzy sem-pre contnuty, Fuzzy Sets and Systems, (1998) LZadeh, Fuzzy sets, Inormaton and Control, 8 (1965)