Interval Valued Fuzzy Neutrosophic Soft Structure Spaces

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1 Neutrosop Sets ad Systems Vol Iterval Valued Fuzzy Neutrosop Sot Struture Spaes Iroara & IRSumat Nrmala College or Wome Combatore- 608 Tamladu Ida E-mal: sumat_rama005@yaooo bstrat I ts paper we trodue te topologal struture o terval valued uzzy eutrosop sot sets ad obta some o ts propertes We also vestgate some operators o terval valued uzzy eutrosop sot topologal spae Keywords: Fuzzy Neutrosop sot set Iterval valued uzzy eutrosop sot set Iterval valued uzzy eutrosop sot topologal spae Itroduto I 999[9] Molodsov tated te ovel oept o sot set teory w s a ompletely ew approa or modelg vagueess ad uertaty I [6] Maj et al tated te oept o uzzy sot sets wt some propertes regardg uzzy sot uo terseto omplemet o uzzy sot set Moreover [78] Maj et al exteded sot sets to tutost uzzy sot sets ad Neutrosop sot sets Neutrosop Log as bee proposed by Florete Smaradae[5] w s based o ostadard aalyss tat was gve by braam Robso 960s Neutrosop Log was developed to represet matematal model o uertaty vagueess ambguty mpreso udeed ompleteess osstey reduday otradto Te eutrosop log s a ormal rame to measure trut determay ad alseood I Neutrosop set determay s quated expltly wereas te trut membersp determay membersp ad alsty membersp are depedet Ts assumpto s very mportat a lot o stuatos su as ormato uso we we try to ombe te data rom deret sesors Yag et al[6] preseted te oept o terval valued uzzy sot sets by ombg te terval valued uzzy set ad sot set models JagY et al[5] trodued terval valued tutost uzzy sot set I ts paper we dee terval valued uzzy eutrosop sot topologal spae ad we dsuss some o ts propertes Prelmares Deto []: uzzy eutrosop set o te uverse o dsourse X s deed as = were [0 ] ad 0T ( x) I ( x) F ( x) Deto []: terval valued uzzy eutrosop set (IVFNS sort) o a uverse X s a objet o te orm x T ( x) I ( x) F ( x) were T (x) = X It ([0]) I (x) = X It ([0]) ad F (x) = X It ([0]) {It([0]) stads or te set o all losed subterval o [0] satses te odto xx supt (x) + supi (x) + supf (x) Deto []: Let U be a tal uverse ad E be a set o parameters IVFNS(U) deotes te set o all terval valued uzzy eutrosop sets o U Let E par (F) s a terval valued uzzy eutrosop sot set over U were F s a mappg gve by F: IVFNS(U) Note : Iterval valued uzzy eutrosop sot set s deoted by IVFNS set Deto []: Te omplemet o a INFNSS (F) s deoted by (F) ad s deed as (F) = (F ) were F : IVFNSS(U) s a mappg gve by F ( = <x F F( (x) (I F( (x)) F F( (x) > or all xu ad e (I F( (x)) = - I F( (x) = [ - I F( ( x) - I F( ( x) ] Deto 5[]: Te uo o two IVFNSS (F) ad (GB) over a uverse U s a IVFNSS (HC) were C = BeC ~ H C ( were Deto 6[]: T ~ ( ) I ~ ( ) F~ ( ) T ~ ( ) I ~ ( ) F~ ( ) T ~ ( ) I ( ) F ~ ( ) H ( H ~ ( H ( m{ } e B e B e B I roara & IRSumat Iterval Valued Fuzzy Neutrosop Sot Struture Spaes

2 7 Neutrosop Sets ad Systems Vol5 0 Te terseto o two IVFNSS (F) ad (GB) over a uverse U s a IVFNSS (HC) were C = BeC ~ H C ( were T ~ ( ) I ~ ( ) F~ ( ) T ~ ( ) I ~ ( ) F~ ( ) T ~ ( ) I ( ) F ~ ( ) H ( H ~ ( H ( max e B e B e B INTERVL VLUED FUZZY NEUTROSOPHIC SOFT TOPOLOGY Deto : Let (F be a elemet o IVFNS set over (UP(F be te olleto o all INFNS subsets o (F sub-amly o P(F s alled a terval valued uzzy eutrosop sot topology ( sort IVFNStopology) o (F te ollowg axoms are satsed: () ( (F () {( / K} mples ( E K ) () I ( (g te ( (g Te te par ((F ) s alled terval valued uzzy eutrosop sot topologal spae (IVFNSTS) Te members o are alled -ope IVFNS sets or ope sets were :IVFNS(U) s deed as ( = {<x [00][00][]>:xU e} ad F :IVFNS(U) s deed as F ( = {<x [][][00]>:xU e} Example : Let U={ } E = {e e e e } = {e e e } (F = {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} ( ={e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} ( = {e ={< [0506] [005] [00]> < [005] [0506] [000]> < [] [] [00]>} {e ={< [005] [0506] [00]> < [005] [0708] [00]> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} ( = {e ={< [00] [0506] [00]> < [0607] [0506] [00]> < [] [] [00]>} {e ={< [00] [005] [000]> < [] [] [00]> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} ( = {e ={< [00] [005] [00]> < [005] [0506] [00]> < [] [] [00]> {e ={< [00] [005] [00]> < [005] [0708] [00]> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} ( = {e ={< [0506] [0506] [00]> < [0607] [0506] [000]> < [] [] [00]>} {e ={< [005] [0506] [000]> < [] [] [00]> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} Here ={( (F ( ( ( ( } o P(F s a IVFNS topology o (F ad ((F ) s a terval valued uzzy eutrosop sot topologal spae Note: Te subamly ={( (F ( ( ( } o P(F s ot a terval valued uzzy eutrosop sot topology o (F se te uo ( ( = ( does ot belog to Deto : s every IVFNS topology o (F must ota te sets ( ad (F so te amly ={( (F } orms a IVFNS topology o (F Ts topology s alled dsrete IVFNS- topology ad te par ((F ) s alled a dsrete terval valued uzzy eutrosop sot topologal spae Teorem : Let { ; I} be ay olleto o IVFNS-topology o (F Te ter terseto s also a topology I o (F I roara & IRSumat Iterval Valued Fuzzy Neutrosop Sot Struture Spaes

3 Neutrosop Sets ad Systems Vol5 0 8 Proo: () Se ( (F or ea I ee ( (F I () Let {( / K} be a arbtrary amly o terval valued uzzy eutrosop sot sets were ( or ea K Te or ea I I ( or K ad se or ea I s a topology tereore ( or ea I Hee K ( K I () Let ( (g te ( ad I (g or ea I ad se or ea I s a topology tereore ( (g or ea I Hee ( (g Tus I satses all I te axoms o topology Hee orms a topology I But te uo o topologes eed ot be a topology w s sow te ollowg example Remar 5: Te uo o two IVFNS topology may ot be a IVFNS- topology I we osder te example te te subamles ={( (F ( } ad ={( (F ( } are te topologes (F But ter uo ={( (F ( topology o (F ( } s ot a Deto 6: Let ((F ) be a IVFNS-topologal spae over (F IVFNS subset ( o (F s alled terval valued uzzy eutrosop sot losed (IVFNS losed) ts omplemet ( s a member o Example 7: Let us osder example te te IVFNS losed sets ((F ) are ( = {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} (F = {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} ( = {e ={< [00] [0506] [0506]> < [000] [005] [005]> < [00] [00] []>} {e ={< [00] [005] [005]> < [00] [00] [005]> < [] [] [00]>} {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} ( ={e ={< [00] [005] [00]> < [00] [005] [0607]> < [00] [00] []>} {e ={< [000] [0506] [00]> < [00] [00] []> < [] [] [00]>} {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} ( = {e ={< [00] [0506] [00]> < [00] [005] [005]> < [00] [00] []>} {e ={< [00] [0506] [00]> < [00] [00] [005]> < [] [] [00]>} {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} ( = {e ={< [00] [005] [0506]> < [000] [005] [0607]> < [00] [00] []>} {e ={< [000] [005] [005]> < [00] [00] []> < [] [] [00]>} {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} are te terval valued uzzy eutrosop sot losed sets ((F ) Teorem 8: Let ((F ) be a terval valued uzzy eutrosop sot topologal spae over (F Te () ( (F are terval valued uzzy eutrosop sot losed sets () Te arbtrary terseto o terval valued uzzy eutrosop sot losed sets s terval valued uzzy eutrosop sot losed set () Te uo o two terval valued uzzy eutrosop sot losed sets s a terval valued uzzy eutrosop losed set Proo: () Se ( (F mples ( ad (F are losed I roara & IRSumat Iterval Valued Fuzzy Neutrosop Sot Struture Spaes

4 9 Neutrosop Sets ad Systems Vol5 0 () Let {( / K} be a arbtrary amly o IVFNS losed sets ((F ) ad let ( = ( e ( Now ( = E ( ) K E ) K K K ( ad ( or ea K so ( Hee ( Tus ( s IVFNS losed set () Let { / } be a te amly o IVFNS losed sets ((F ) ad let ( g Now ( ( g ( So ( ( E Tus (g s a IVFNS losed set ) ( ad Hee (g Remar 9: Te terseto o a arbtrary amly o IVFNS ope set may ot be a IVFNS- ope ad te uo o a arbtrary amly o IVFNS losed set may ot be a IVFNS losed Let us osder U= { }; E = {e e e e } = {e e e } ad let (F = {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} ( = {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} For ea N we dee e e We observe tat = {(F ( ( } s a IVFNS topology o (F But ( ={e ={< [00] [00] [0505]> < [][][00]> < [00] [00] []>} {e ={< [00] [00] [00]> < [00] [00] []> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>}} s ot a IVFNS-ope set IVFNS topologal spae ((F ) se ( Te IVFNS losed sets te IVFNS topologal spae ((F ) are (F ( ad ( or (= ) But ( ={e ={< [0505] [] [00]> < [00][00][]> < [] [] [00]>} {e ={< [00] [] [00] > < [] [] [00]> < [] [] [00]>} {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} s ot a IVFNS-losed set IVFNS topologal spae ((F ) se ( Deto 0: Let ((F ) ad ((F ) be two IVFNS topologal spaes I ea ( mples ( te s alled terval valued uzzy eutrosop sot er topology ta ad s alled terval valued uzzy eutrosop sot oarser topology ta I roara & IRSumat Iterval Valued Fuzzy Neutrosop Sot Struture Spaes

5 Neutrosop Sets ad Systems Vol5 0 0 Example : I we osder te topologes ={( (F ( ( ( ( } as example ad ={( (F ( ( } o (F Te s terval valued uzzy eutrosop sot er ta ad s terval valued uzzy eutrosop sot oarser topology ta Deto : Let ((F ) be a IVFNS topologal spae o (F ad B be a subamly o I every elemet o a be expressed as te arbtrary terval valued uzzy eutrosop sot uo o some elemet o B te B s alled a terval valued uzzy eutrosop sot bass or te terval valued uzzy eutrosop sot topology Example : I example or te topology ={( (F ( ( ( ( } te subamly B ={( (F ( ( P(F s a bass or te topology ( } o Deto : Let be te IVFNS topology o (F IVFNS(U ad ( be a IVFNS set P(F s a egborood o a IVFNS set (g ad oly tere exst a -ope IVFNS set ( e ( su tat (g ( ( Example 5: Let U={ } E = {e e e e } = {e }I a IVFNS topology ={( (F ( } were (F = {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} ( = {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} ( = {e ={< [005] [0506] [005]> < [00] [005] [0506]> < [005] [00] [00]>}} Te IVFNS set ( = {e ={< [0506] [0607] [00]> < [00] [005] [0506]> < [005] [005] [00]>} s a egbourood o te IVFNS set (g = {e ={< [00] [005] [005]> < [00] [00] [0607]> < [005] [00] [00]>}} beause tere exst a -ope IVFNS set ( su tat (g ( ( Teorem 6: IVFNS set ( P(F s a ope IVFNS set ad oly ( s a egbourood o ea IVFNS set (g otaed ( Proo: Let ( be a ope IVFNS set ad (g be ay IVFNS set otaed ( Se we ave (g ( ( t ollows tat ( s a egbourood o (g Coversely let ( be a egbourood or every IVFNS set otaed t Se ( ( tere exst a ope IVFNS set ( su tat ( ( ( Hee ( = ( ad ( s ope Deto 7: Let ((F ) be a terval valued uzzy eutrosop sot topologal spae o (F ad ( be a IVFNS set P(F Te amly o all egbouroods o ( s alled te egbourood system o ( up to topology ad s deoted by N ( Teorem 8: Let ((F ) be a terval valued uzzy eutrosop sot topologal spae I ( s te egbourood system o a IVFNS set ( Te () Fte tersetos o members o N ( belog to N ( () Ea terval valued uzzy eutrosop sot set w otas a member o N( belogs to N( Proo: () Let (g ad ( are two egbouroods o ( so tere exst two ope sets ( g ( su tat ( ( g (g ad ( ( ( Hee ( ( g ( (g ( ad ( g ( s ope Tus (g ( s a egbourood o ( () Let (g s a egbourood o ( ad (g ( so tere exst a ope set ( su tat g ( ( (g By ypotess (g ( g ( g so ( (g ( w mples tat ( ( g ( ad ee ( s a egbourood o ( Deto 9: N I roara & IRSumat Iterval Valued Fuzzy Neutrosop Sot Struture Spaes

6 Neutrosop Sets ad Systems Vol5 0 Let ((F ) be a terval valued uzzy eutrosop sot topologal spae o (F ad ( (g be IVFNS sets P(F su tat (g ( Te (g s alled a teror IVFNS set o ( ad oly ( s a egbourood o (g Deto 0: Let ((F ) be a terval valued uzzy eutrosop sot topologal spae o (F ad ( be a IVFNS set P(F Te te uo o all teror IVFNS set o ( s alled te teror o ( ad s deoted by t( ad deed by t( = {(g / ( s a egbourood o (g } Or equvaletly t( ={ (g / (g s a IVFNS ope set otaed ( } Example : Let us osder te IVFNS topology ={( (F ( ( ( ( } as example ad let ( = {e ={< [005] [0607] [00]> < [0708] [0607] [00]> < [] [] [00]>} {e ={< [00] [0506] [00]> < [] [] [00]> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} be a IVFNS set Te t( = { (g / (g s a IVFNS ope set otaed ( } = ( ( =( Se ( ( ad ( ( Teorem : Let ((F ) be a terval valued uzzy eutrosop sot topologal spae o (F ad ( be a IVFNS set P(F Te () t( s a ope ad t( s te largest ope IVFNS set otaed ( () Te IVFNS set ( s ope ad oly ( = t( Proo: Proo ollows orm te deto Proposto : For ay two IVFNS sets ( ad (g s a terval valued uzzy eutrosop sot topologal spae ((F ) o P(F te () (g ( mples t(g t( () t( = ( ad t(f = (F () t(t( ) = t( (v) t((g ( ) = t(g t( (v) t((g ( ) t(g t( Proo: () Se (g ( mples all te IVFNS ope set otaed (g also otaed ( Tereore { ( g / ( g s a IVFNS ope set otaed (g } { ( / ( s a IVFNS ope set otaed ( } So t(g t( () Proo s obvous () t(t ( ) = { (g / (g s a IVFNS ope set otaed t( } ad se t( s te largest ope IVFNS sset otaed t( Tereore t(t( ) = t( (v) Se t (g (g ad t ( ( we ave t (g t ( (g ( ----() ga se (g ( (g ad (g ( ( we ave t ((g ( )t (g ad t ((g ( )t ( Tereore t((g ( ) t (g t ( () From () ad () t((g ( )=t (g t ( (v) Se (g (g ( ad ( (g ( so t(g t((g ( ) ad t ( t((g ( ) Hee t(g t ( t((g ( Deto : Let ((F ) be a IVFNS topologal spae o (F ad let ( (g be two IVFNS set P (F Te (g s alled a exteror IVFNS set o ( ad oly (g s a teror IVFNS set o te omplemet ( Deto 5: Let ((F ) be a terval valued uzzy eutrosop sot topologal spae o (F ad ( be a IVFNS set P (F Te te uo o all exteror IVFNS set o ( s alled te exteror o ( ad s deoted by ext ( ad s deed by ext ( ={ (g / ( s a egbourood o (g } Tat s rom deto ext ( = t(( ) Proposto 6: For ay two IVFNS sets ( ad (g a terval valued uzzy eutrosop sot topologal spae ((F ) o P(F te () ext ( s ope ad s te largest ope set otaed ( () ( s ope ad oly ( = ext ( () (g ( mples ext ( ext (g (v) ext((g ( )ext (g ext ( (v) ext((g ( )=ext (g ext ( Proo: Proos are stragt orward I roara & IRSumat Iterval Valued Fuzzy Neutrosop Sot Struture Spaes

7 Neutrosop Sets ad Systems Vol5 0 Deto 7: Let ((F ) be a terval valued uzzy eutrosop sot topologal spae o (F ad ( be a IVFNS set P(F Te te terseto o all losed IVFNS set otag ( s alled te losure o ( ad s deoted by l ( ad deed by l ( ={ (g / (g s a IVFNS losed set otag ( } Tus l ( s te smallest IVFNS losed set otag ( Example 8: Let us osder a terval valued uzzy eutrosop sot topology ={( (F ( ( ( ( } as example ad let ( = {e ={< [00] [00] [0506]> < [00] [005] [0708]> < [00] [00] []>} {e ={< [00] [005] [0607]> < [00] [00] []> < [] [] [00]>} {e ={< [] [] [00]> < [] [] [00]> < [] [] [00]>} be a IVFNS set Te l( ={ (g / (g s a IVFNS losed set otag ( } = ( ( = ( Se ( ( ad ( ( Proposto 9: For ay two IVFNS sets ( ad (g s a terval valued uzzy eutrosop sot topologal spae ((F ) o P(F te () l ( s te smallest IVFNS losed set otag ( () ( s IVFNS losed ad oly ( = l ( () (g ( mples l(g l( (v) l(l( ) = l( (v) l( = ( ad l(f = (F (v) l((g ( ) = l(g l( (v) l((g ( ) l(g l( Proo: () ad () ollows rom te deto () Se (g ( mples all te losed set otag ( also ota (g Tereore { ( g / ( g s a IVFNS losed set otag (g }{ ( / ( s a IVFNS losed set otag ( } So l (g l ( (v) l(l ( ) = { (g / (g s a IVFNS losed set otag l ( } ad se l ( s te smallest losed IVFNS set otag l ( Tereore l(l ( ) = l ( (v) Proo s obvous (v) Se l(g (g ad l( ( we ave l(g l ( (g ( Ts mples l(g l( l((g ( ) -----() d se (g ( (g ad (g ( ( so l((g ( )l(g ad l((g ( )l( Tereore l((g ( )l(g l( -----() Form () ad () l(g l( = l(g l( (v) Se (g (g ( ad ( (g ( so l (g l((g ( ) ad l( l((g ( ) Hee l ((g l( l((g ( ) Teorem 0: Let ((F ) be a terval valued uzzy eutrosop sot topologal spae o (F ad ( be a IVFNS set P(F Te te olleto ( {( ( g /( g } s a terval valued uzzy eutrosop sot topology o te terval valued uzzy eutrosop sot set ( Proo: () Se ( (F ( = ( (F ad ( = ( ( Tereore ( ( () Let {( / } be a te amly o IVFNS ope sets ( te or ea = tere exst ( g su tat ( = ( ( g Now ( = [( ( g ] = ( ) [ E ] g E ad se [ g E ] so ( ( () Let {( / K} be a arbtrary amly o terval valued uzzy eutrosop sot ope sets ( te or ea K tere exst ( g su tat ( =( ( g Now ( = ( ( g = K K K ( ( g ad se ( g K I roara & IRSumat Iterval Valued Fuzzy Neutrosop Sot Struture Spaes

8 Neutrosop Sets ad Systems Vol5 0 So ( ( K Deto : Let ((F ) be a IVFNS topologal spae o (F ad ( be a IVFNS set P(F Te te IVFNS topology ( {( ( g /( g } s alled terval valued uzzy eutrosop sot subspae topology (IVFNS subspae topology) ad (( ( ) s alled terval valued uzzy eutrosop sot subspae o ((F ) Example : Let us osder te terval valued uzzy eutrosop sot topology ={( (F ( ( ( ( } as te example ad a IVFNS-set ( ={e ={< [00][00][00]> < [0506][005] [00]> < [00] [0506] [0607]>} {e ={< [00] [0506] [00]> < [005] [0607] [00]> < [005] [005] [00]>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} be a IVFNS set ( (F = ( ( ( = ( ( ( = ( g = {e ={< [00][00][00]> < [005][005] [00]> < [00] [0506] [0607]>} {e ={< [00] [0506] [00]> < [005] [0607] [00]> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} ( ( = ( g = {e ={< [00][00][00]> < [0506][005] [00]> < [00] [0506] [0607]>} {e ={< [00] [005] [00]> < [005] [0607] [00]> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} ( ( = ( g = {e ={< [00][00][00]> < [005][005] [00]> < [00] [0506] [0607]>} {e ={< [00] [005] [00]> < [005] [0607] [00]> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} ( ( = ( g = {e ={< [00][00][00]> < [0506][005] [00]> < [00] [0506] [0607]>} {e ={< [00] [0506] [00]> < [005] [0607] [00]> < [00] [00] []>} {e ={< [00] [00] []> < [00] [00] []> < [00] [00] []>} Tus ( = {( (F ( g ( g ( g ( g } s a terval valued uzzy eutrosop sot subspae topology or ad (( ( ) s alled terval valued uzzy eutrosop sot subspae o ((F ) Teorem : Let (( ) be a IVFNS topologal subspae o (( ) ad let (( ) be a IVFNS topologal subspae o ((F ) Te (( ) s also a IVFNS topologal subspae o ((F ) Proo: Se ( ( (F (( ) s a terval valued uzzy eutrosop sot topologal subspae o ((F ) ad oly ( = Let ( ow se (( ) s a IVFNS topologal subspae o (( ) e ( = so tere exst ( =( ( su tat But (( ) s a IVFNS topologal subspae o ((F )Tereore tere exst ( su tat ( =( ( Tus ( =( ( =( ( So ( = ( ( ( ( mples ( ---() Now assume (g ( e tere exst ( su tat (g = ( ( But ( ( ( = So ( (( ( ) = ( ( = (g We ave I roara & IRSumat Iterval Valued Fuzzy Neutrosop Sot Struture Spaes

9 Neutrosop Sets ad Systems Vol5 0 (g mples ( () From () ad () = ( Hee te proo Teorem : Let ((F )be a IVFNS topologal spae o (F B be a bass or ad ( be a IVFNS set P(F Te te amly B ( {( ( g /( g B} s a IVFNS bass or subspae topology ( Proo: Let ( ( te tere exst a IVFNS set (g su tat ( = ( (g Se B s a base or tere exst sub-olleto {{( / I} o B su tat (g = ( I Tereore ( = ( (g = ( ( ( ) = I (( ( ) I Se ( ( B ( mples B ( s a IVFNS bass or te IVFNS subspae topology ( Coluso I ts paper te oto o topologal spae terval valued uzzy eutrosop sot sets s trodued Furter some o ts operators ad propertes o topology IVFNS set are establsed Reerees []ja Muerjee joy Kat Das bjt Saa Iterval valued tutost uzzy sot topologal spaes als o Fuzzy Matemats ad Iormats Volume 6 No (November 0) pp [] Iroara IRSumatJMarta Jey Fuzzy Neutrosop Sot Topologal Spaes IJM-(0) [] Iroara & IRSumat Some results o terval valued uzzy eutrosop sot sets IJIRS Vol Issue [] KT taassov Iterval valued tutost uzzy sets Fuzzy Sets Systems () (989) -9 [5] YJag Y Tag QCe HLuJTag Itervalvalued tutost uzzy sot sets ad ter propertes Computers d Matemats Wt pplatos 60(00) [6] PKMaj R Bswas as RRoy Fuzzy sot sets Joural o Fuzzy Matemats Vol 9 o 00 pp [7] PKMaj R Bswas as RRoy Itutost Fuzzy sot sets Te joural o uzzy Matemats Vol 9 ()(00) [8] Pabtra Kumar Maj Neutrosop sot set als o Fuzzy Matemats ad Iormats Volume 5 No(0)57-68 [9] MolodtsovD Sot set teory Frst results Computers d Matemats Wt pplatos7 (-5) (999) 9- [0]S Roy ad T K Samata ote o uzzy sot topologal spae Fuzzy Mat Iorm() (0) 05- []M Sabr ad M Naz O sot topologal spaes Comput Mat ppl 6 (0) []T Smseler ad S Yusel Fuzzy sot topologal spaes Fuzzy Mat Iorm 5()(0) []B Taay ad M B Kademr Topologal struture o uzzy sot sets Comput Mat ppl6 (0) [] FSmaradae Neutrosopy ad Neutrosop Log Frst Iteratoal Coeree o Neutrosopy Neutrosop Log Set Probablty ad Statsts Uversty o New Mexo Gallup NM 870 US (00) [5] FSmaradae Neutrosop set a geeralzato o te tutuosts uzzy sets Iter J Pure pplmat (005) [6] XBYag TNL JYYag YL D Yu Combato o terval valued uzzy sets ad sot set Computer ad Matemats wt applatos 58 ()(009) 5-57 [7]L Zade Fuzzy sets Iormato ad otrol 8 (965) 8-5 Reeved: Jue 0 0 epted: ugust 0 0 I roara & IRSumat Iterval Valued Fuzzy Neutrosop Sot Struture Spaes