Fuzzy Soft Semi Pre-Connected Properties in Fuzzy Soft Topological Spaces

Transcription

1 Fuy Sot Smi Pr-onnctd Proprt in Fuy Sot Topoloical Spacs Ponslvakumari R Slvi Dpartmnt o Matmatics nna Univrsity Tuticorin ams Tuticorin Tamilnadu India Dpartmnt o Matmatics Sri Parasakti oll or womn ourtallam Tamilnadu India-6780 bstract: In t prsnt papr w continu t study on uy sot topoloical spacs invstiat t proprt o uy sot smi pr connctd sts uy sot smi pr sparatd sts uy sot smi pr s-connctd sts av stabld svral intrstin proprt supportd by xampls Kywords: Fuy sot st Fuy sot toploicalspac Fuysot smi connctd Fuy sot pr connctd Introduction Molodtsov[4] initiatd t novl concpt o sot st as a nw matmatical tool or dalin wit uncrtaint in 999 Mai t al[] introducd t concpt o Fuy Sot St som proprt rardin uy sot union intrsction complmnt o a uy sot st Dmoran Law tc Ts rsults wr urtr rvd improvd by mad Karal[] In 0 o t al[5] t orward som mor propositions rardin uy sot st tory ty introducd t notions o union intrsction o two uy sot sts ovr two uy sot classstanay t al[9] introducd t dinition o uy sot topoloy in a subst o t initial univrs st wil Roy Samanta[7] av t dinition o uy sot topoloy ovr t initial univrs st Karal mad []av bn introducd t concpts o uy sot bictiv mappin uy sot idntity mappin uy sot continuous mappin anasrora[4] studid som propositions xampls rlatd to uy sot continuous mappin Som uy sot topoloical proprt basd on uy smi rsp β-)opn sot sts wr introducd in [890] bd l-lati t al[9] introducd t concpt o Fuy sot smi connctd Fuy sot pr connctd proprt in uy sot topoloical spacsin t prsnt papr w continu t study on uy sot topoloical spacs invstiat t proprt o uy sot smi pr connctd sts uy sot smi pr sparatd sts uy sot smi pr s-connctd sts av stabld svral intrstin proprt supportd by xampls Prliminar Dinition [0] uy st in U a st o ordrd pairs = { x x)) : xu} wr x ) : U [0] I a mappin x ) or x stats t rad o blonnss o x in T amily o all uy sts in U dnotd by U I olum 5 Issu 8 uust 06 Licnsd Undr rativ ommons ttribution Y Dinition [4] pair F ) calld a uy sot st ovr U wr F : P U) a mappin rom into PU ) Dinition [4] Lt Tn t mappin F : P U ) dind by F ) F a uy subst o U ) calld uy U wr F 0 i sot st ovr ) F 0 i U ) dnotd by FS U ) T st o all uy sot st ovr Dinition 4[] Lt X ) b a uy sot topoloical spac uy sot st ovr X said to b uy closd sot st in X / i its rlativ complmnt uy opn sot st W dnot t st o all uy closd sot sts by FS X ) or FS X ) Dinition 5[] Lt X ) b a uy sot topoloical spac FSS X ) T uy sot closur o dnotd by Fcl ) t intrsction o all uy closd sot supr sts o Tat impl Fcl ) { : uy D D D closd sot sts } T uy sot intrior o dnotd by Fint ) t uy sot union o all uy opn sot substs o Tat impl F int ) { : uy opn sot st D D D } Dinition 6[4] Lt U ) U* * * ) b uy sot topoloical spacs Lt : U U * : * b mappins = ) : U) U**) b a uy sot Papr ID: RT064 6

2 mappin Tn = ) said to b uy sot continuous i t invrs ima undr = ) o anyg *) a uy sot st F tat wnvr G *) G ) Torm 7[] Lt FSS X ) FSS Y ) K b two amil o uy sot sts For t uy sot unction t ollowin statmnts : FSS X ) FSS Y) K old a) b) c) ) ) )) ) FSS Y) K FSS Y I ))) ) ) ) K surctiv tn t quality olds ) ))) ) FSS X ) I inctiv tn t quality olds d) 0 ) 0 K ) ) K t quality olds K) I 0 K) 0 surctiv tn ) I ) ) tn ) ) ) I ) ) tn ) ) ) ) ) FSS Y) K ) ) ) J J ) ) ) ) FSS Y) J J K i) J ) ) J ) FPcl ) consquntly ) ) ) ) FSS X) FPcl ) { D : D FPOS X ) D} I J J inctiv tn t quality olds Dinition 8[] Lt X ) b a uy sot topoloical spac uy sot sparation o a pair o non null propr uy opn sot sts suc tat 0 Dinition 9[] uy sot topoloical spac X ) said to b uy sot connctd i only i tr no uy sot sparations o X Otrw X ) said to b uy sot dconnctd spac Dinition 0[] Lt X ) b a sot topoloical spac sot smi sparation on X a pair o non null propr smi opn sot sts F G suc tat FG X F G Dinition [] sot topoloical spac X ) said to b sot smi connctd i only i tr no sot smi sparations o X Otrw X ) said to b sot smi dconnctd spac Dinition [] Lt X ) b a uy sot topoloical spac FSS X ) I F int Fcl )) tn calld uy pr opn sot st W dnot t st o all uy pr opn sot sts by FPOS X ) or FPOS X ) t st o all uy pr closd sot sts by FPS X ) or FPS X ) Dinition [] Lt X ) b a uy sot topoloical spac FSS X ) FSS X ) Tn ) calld uy pr intrior sot point o i FPOS X ) suc tat T st o all uy pr intrior sot points o uy pr sot intrior o calld t dnotd by FPint ) consquntly FPint ) : FPOS X )} ) calld uy pr closur sot points o t uy pr sot closur calld dnotd by Dinition 4[] Lt X ) b a uy sot topoloical spac uy sot pr sparation on a pair o non null propr uy pr opn sot sts suc tat 0 Dinition 5[] uy sot topoloical spac X ) said to b uy sot pr connctd i only i tr no uy sot pr sparations o Otrw X ) said to b uy sot pr dconnctd spac olum 5 Issu 8 uust 06 Licnsd Undr rativ ommons ttribution Y Papr ID: RT064 64

3 Fuy Sot Smi Pr- onnctdnss Dinition Lt U ) b a uy sot topoloical spac uy sot smi pr sparation on a pair o non null propr uy smi pr opn sot sts suc tat 0 Dinition uy sot topoloical spac U ) said to b uy sot smi pr connctd i only i tr no uy sot smi pr sparations o Otrw U ) said to b uy sot smi pr dconnctd spac XMPL ) T dcrt uy sot topoloical spac o mor tan on mmbr always uy dconnctd ) T indcrt uy sot topoloical spac always uy sot connctd uy sot subspac ) o uy sot topoloical spac U ) said to b uy smi pr opn sot subspac i FSPOS X) uy sot subspac o U ) wr Torm 5 Lt ) b a uy sot smi pr connctd subspac o uy sot topoloical spac U ) suc tat FSPOS X) FSPOS X ) I as a uy sot smi pr sparations Tn itr or ProoLt b uy sot smi pr sparation on y ypots FSPOS X ) FSPOS X ) [ ] [ ] Sinc uy sot smi pr connctdtn itr 0 or Dinition 4 0 Tror itr or xampl 6 Lt U ) b uy sot topoloical spacs Lt U { x y } sts Lt { { x / 0 y / 0 / 0}) { x / 0 y / 0 / 0}) { x / 0 y / 0 / 0})} U ) classs o uy sot { { x / y / /}) { x / y / /}) { x / y / /})} b uy sot smi pr sparation on uy sot subspac { { x / 0 y / 0 / 05}) { x / 0 y / 04 / 0}) { x / 04 y / 0 / 0})} y ypots { { x / 0 y / 0 / 0}) { x / 0 y / 0 / 0}) { x / 0 y / 0 / 0})} { { x / 0 y / 0 / 05}) { x / 0 y / 04 / 0}) { x / 04 y / 0 / 0})} FSPOS X ) { { x / y / /}) { x / y / /}) { x / y / /})} { { x / 0 y / 0 / 05}) { x / 0 y / 04 / 0}) { x / 04 y / 0 / 0})} FSPOS X ) [ ] [ ] Sinc uy sot smi pr connctd Tn itr 0 or 0 Tror itr or Torm 7 I U ) a uy sot smi pr connctd spac uy sot coarsr tan tn U ) also a uy sot smi pr connctd Proo Lt b uy sot smi pr sparation on U ) Tn Sinc suc tat uy sot smi pr sparation on U ) wic a contradiction wit t uy sot smi pr connctdnss o U ) Hnc U ) uy sot smi pr connctd Rmark 8 olum 5 Issu 8 uust 06 Licnsd Undr rativ ommons ttribution Y Papr ID: RT064 65

4 T convrs o torm 7 not tru in nral as sown in t ollowin xampl xampl 9 Lt U { x y } U ) classs o uy sot sts Lt b t indcrt uy sot topoloy tn uy sot smi pr connctd on t otr lt wr k s v uy sot sts ar ovr U dind as ollows { 0 k s v} { { x / 0 y / 0 / 0}) { x / 0 y / 0 / 0}) { x / 0 y / 0 / 0})} k { { x / y / /}) { x / y / /}) { x / y / /})} { { x / 0 y / 05 / 0}) { x / 0 y / 07 / 04}) dconnctd { x / 0 y / 08 / 09})} FSPOS X) FSPOS X) { { x / 05 y / 0 / 0}) { x / y / 08 / 0}) { x / 05 y / 0 / 0})} s { { x / 0 y / 05 / 08}) { x / 0 y / 06 / 07}) { x / y / /})} v { { x / y / /}) { x / 05 y / 0 / 0}) { x / y / 08 / 0})} Tn dins a uy sot topoloy on U suc tat ow k ar uy smi pr opn sot sts in wic orm a uy sot smi pr sparation o U ) wr k 0 k Hnc U ) uy sot smi pr dconnctd Torm 0 uy sot subspac ) o uy sot smi pr dconnctdnss spac U ) uy sot smi pr Proo Lt ) b uy sot smi pr connctd spac Sinc U ) uy sot smi pr dconnctd Tn tr xt uy sot smi pr sparation on U ) y ypots FSPOS X) FSPOS X ) [ ] [ ] wic a contradiction wit t uy sot smi pr connctdnss o ) Tror ) uy sot smi pr dconnctd Rmark uy sot smi pr dconnctdnss proprty not rditary proprty in nral as in t ollowin xampl xampl In xampl 9 lt { x y} U W considr t uy sot st ovr ) dind as ollows: i Tn w ind { { x / y / / 0}){ { x / y / / 0)} { x / y / / 0})} as ollows { : } wr 0 0 { { x / 0 y / 0 / 0}) { x / 0 y / 0 / 0}) { x / 0 y / 0 / 0})} wr W W { { x / 0 y / 05 / 0}) { x / 0 y / 07 / 0}) { x / 0 y / 08 / 0})} k wr R { { x / y / / 0}) { x / y / / 0}) R { x / y / / 0})} wr r wr olum 5 Issu 8 uust 06 Licnsd Undr rativ ommons ttribution Y Papr ID: RT064 66

5 { { x / 05 y / 0 / 0}) { x / 0 y / 06 / 0}) r { x / y / / 0})} s wr P { { x / 0 y / 05 / 0} { x / 0 y / 06 / 0}) P { x / y / / 0})} Tus t collction { : } a uy sot topoloy on Y ) in wic tr no uy sot smi pr sparation on ) Tror ) uy sot smi pr connctd at t tim tat U ) uy sot smi pr dconnctd as sown in xampl 9 Dinition Lt U ) K) b uy sot topoloical spacs : FSS U) FSS ) b a sot unction Tn calld; K ) Fuy smi pr continuous sot i ) FSPOS U) ) Fuy smi pr opn sot i ) FSPOS ) ) Fuy smi pr closd sot i ) FSPS ) 4) Fuy smi pr irrsolut sot i ) FSPOS U ) FSPOS ) 5) Fuy smi pr irrsolut opn sot i ) FSPOS ) FSPOS U) 6) Fuy smi pr irrsolut closd sot i ) FSPS ) FSPS ) Torm 4 Lt U ) U K) b uy sot topoloical : U ) U K) b a uy spacs smi pr irrsolut subctiv sot unction I U ) uy sot smi pr connctd tn U K) also a uy sot smi pr connctd Proo Lt U K) b a uy sot smi pr dconnctd spac Tn tr xt pair o non null propr uy smi pr opn sot substs o K 0K K smi pr irrsolut sot unction tn Sinc suc tat uy ) ) ar pair o non null propr uy smi pr opn sot substs o suc tat ) ) ) 0 K) 0 ) ) ) K) rom Torm 7 T mans tat ) ) orms a uy sot smi pr sparation o wic a contradiction wit t uy sot smi pr connctdnss o U ) Tror U K) uy sot smi pr connctd 4 Fuy Sot Smi Pr s-onnctd Spacs Dinition 4 non null uy sot substs o uy sot topoloical spac U ) ar said to b uy sot smi pr sparatd sts i FSPcl ) FSPcl ) 0 Torm 4 Lt kd k ar uy sot smi pr sparatd substs o uy sot topoloical spac U ) Tn ar uy sot smi pr sparatd sts Proo: Lt tn FSPcl ) FSPcl ) It ollows tat FSPcl ) FSPcl ) k FSPcl ) k 0 D D lso sinc kd Tn FSPcl ) FSPcl kd) Hnc FSPcl ) FSPcl k ) 0 D Tus ar uy sot smi pr sparatd sts Torm 4 Twouy smi pr closd sot substs o uy sot topoloical spac U ) ar uy sot smi pr sparatd sts i only i ty ar doint Proo Lt ar uy sot smi pr sparatd sts Tn FSPcl ) FSPcl 0 Sinc ar uy smi pr closd sot sts Tn D 0 onvrsly lt ar doint uy smi pr closd sot sts Tn FSPcl ) 0 olum 5 Issu 8 uust 06 Licnsd Undr rativ ommons ttribution Y Papr ID: RT064 67

6 FSPcl ) 0 It ollows tat ar uy sot smi pr sparatd sts Dinition 44 uy sot topoloical spac U ) said to b uy sot smi pr s-connctd i only i can not xprssd as t uy sot union o two uy sot smi pr sparatd sts in U ) Torm 45 Lt ) b a uy sot subspac o uy sot topoloical spac U ) Tn only i ar uy sot smi pr sparatd on i ar Fuy sot smi pr sparatd on wr t uy sot subspac or Proo Suppos tat on ar uy sot smi pr sparatd FSPcl ) 0 itr 0or 0 It ollows tat or T impl tat FSPcl ) 0 [ ) ] ) 0 S FSPcl FSPcl or [ FSPcl ) ] FSPcl ) 0 ar uy sot smi pr sparatd on Torm 46 Lt b a uy sot subst o uy sot topoloical spac U ) Tn uy sot pr s-connctd wrt U ) i only i uy sot smi pr s- connctd wrt ) Proo Suppos tat not uy sot smi pr s- connctd wrt ) Tn wr ar uy sot smi pr sparatd on Lt ) b a uy sot subspac o uy sot topoloical spac U ) Tn only i ar uy sot smi pr sparatd on i ar Fuy sot smi pr sparatd on olum 5 Issu 8 uust 06 Licnsd Undr rativ ommons ttribution Y wr t uy sot subspac or Tror not uy sot smi pr s-connctd wrt U ) Torm 47 Lt ) b a uy sot smi pr s-connctd subspac o uy sot topoloical spac U ) b uy sot smi pr sparatd o wit Tn itr or Proo Lt or som uy sot smi pr sparatd substs o Sinc ) ) Tn ) FSPcl ) FSPcl ) 0 lso FSPcl ) ) FSPcl ) 0 Sinc ) uy sot smi pr s-connctd Tus Torm 48 Lt ) Y M) b uy sot smi pr s- Y connctd subspacs o uy sot topoloical spac U ) suc tat non o tm uy sot smi pr sparatd Tn YM uy sot smi pr s- connctd Proo Lt ) Y M) b uy sot smi pr s-connctd subspacs o suc tat YM not uy sot smi pr s-connctd Tn tr xt two non null uy sot smi pr sparatd sts k o suc tat YM kd Sinc Y M ar uy sot smi pr s-connctd Y kd y torm 47 itr kdor also itr YM kdorym I kdor Tn k 0 or kd 0 Tror [ Y ] k [ k ] [ Y k ] [ Y k ] 0 Y k Y Y M D D M D M D M D M M kd or YM w t [ YM ] Y M D D sinc YM kd Similarly i ow [ Y ) ] FSPcl[ Y ) k ] [ Y ) ] [ FSPcl[ Y ) FSPcl k )] M M D M M D Papr ID: RT064 68

7 [ YM ] [ FSPcl kd)] Proo 0 Suppos tat uy sot smi pr s-dconnctd Tn FSPcl[ YM ) ] [ YM ) kd] [ FSPcl[ YM ) FSPcl )] [ YM ) kd] [ YM ] [ FSPcl ) kd] wr ar Fuy sot smi pr 0 It ollows tat [ Y ] k sparatd sts It ollows 0 So M D [ Y ] Y ar uy sot smi pr sparatd M M wic a contradiction Hnc YM uy sot smi pr s-connctd Torm 49 Lt ) b a uy sot smi pr s-connctd subspac o uy sot topoloical spac U ) S SS X ) I S FSPcl ) Tn M M S S M) uy sot smi pr s-connctd subspac o U ) Proo Suppos tat S S M) not uy sot smi pr s- connctd subspac o U ) Tn tr xt uy sot smi pr sparatd sts on suc tat SM So w av uy sot smi pr s- connctd subst o uy sot pr s-dconnctd spac y torm 47 itr or I Tn FSPcl ) FSPcl ) It ollows tat FSPcl ) FSPcl ) 0 Hnc FSPcl ) 0 wic a contradiction I y a similar way w can t 0 wic a contradiction Hnc S S M) uy sot smi pr s-connctd subspac o U ) orollary 40 I ) uy sot smi pr s-connctd subspac o uy sot topoloical spac U ) Tn FSPcl ) uy sot smi pr s-connctd Proo It obvious rom Torm 49 Torm 4 I or all pair o dtinct uy sot point tr xt a uy sot smi pr s-connctd st wit tn uy sot smi pr s-connctd olum 5 Issu 8 uust 06 Licnsd Undr rativ ommons ttribution Y Sinc 0 Tn ar dtinct uy sot point in y ypots tr xts a uy sot smi pr s-connctd st suc tat Morovr w av uy sot smi pr s-connctd subst o a uy sot smi pr s-dconnctd spac It ollows by Torm 47 itr or bot cass a contradiction wit t ypots Tror uy sot smi pr s- connctd Torm 4 Lt { ) : J} b a non null amily o uy sot smi s-connctd subspacs o uy sot topoloical spac U ) J ) 0 tn J ) also a uy sot smi pr s-connctd J uy subspac o U ) Proo Suppos tat ) ) J J uy sot smi pr s-dconnctd Tn or som uy sot smi pr sparatd substs o Sinc ) 0 Tn ) It J J ollows tat So itr or Suppos tat Sinc ) J ) So w av ) uy sot smi pr s-connctd subst o uy sot smi pr s-dconnctd st or y Torm 47 itr ) ) J I ) J Tn T impl tat 0 wic a contradiction Tror ) = ) J J dconnctd uy sot smi pr s- Torm 4 Lt U ) U K) b uy sot topoloical : U ) U K) b a uy spacs smi pr irrsolut surctiv sot unctioni U ) Papr ID: RT064 69

8 uy sot smi pr s- connctd tn U K) also a uy sot smi pr s- connctd Proo Lt U K) b a uy sot smi pr dconnctd spac Tn tr xt pair o non null propr uy smi pr opn sot substs o K K suc tat FSPcl ) FSPcl ) 0 Sinc uy smi pr irrsolut sot unction tn ) ) ar pair o non null propr uy smi pr opn sot substs o suc FSPcl FSPcl = tat )) ) )) ) ) 0 K) 0 ) FSPcl )) ) FSPcl )) ) 0 k) 0 ) ) ) K) rom Torm 7 [[0] Torm 4] T mans tat ) ) ar pair o non null propr uy smi pr opn sot substs o Wic a contradiction o t uy sot smi pr s- connctdnss o U ) Tror U K) uy sot smi pr s- connctd Rrncs [] bd l-lati M Fuy sot sparation axioms basd on uy β-opn sot stsnnfuy Mat Inormvolx)05 [] bd l-lati M Rodyna Hosny Fuy sot pr-connctd proprt in uy sot topoloical spacs Sout sian Journal o Matmatics 05ol5 5)0- [] mad Karal Mappins on uy sot classs dvfuy Syst009rtID pp[] [4] anasr ora On Fuy Sot ontinuous Mappin Intrnational Journal or asic Scincs Social Scincs IJSS) ISS:9-968olIssu uust-0 pp50-64 [5] akirtanay urclkmir M Topoloical structur o uy sot stsomt Matppl6)0) [6] in nsom Local Proprt o Sot Smi-Opn StsHindawi Publin orporation0-6) [7] an L Fuy topoloical spacs JMatnal4968)8-90 olum 5 Issu 8 uust 06 Licnsd Undr rativ ommons ttribution Y [8] Kil Tantawy O l-sik bd l-lati M lsaydsfuy sot connctdnss basd on uy β-opn sot stsjournal o Matmatics omtr pplications Rsarc5)05) 7-48 [9] Kil Tantawy O l-sik bd l-lati M lsaydsfuy sot smi connctd proprt in uy sot topoloical spacs MatSci Ltt4 ) 05)7-79 [0] Kil Tantawy O l-sik bd l-lati M lsaydssom uy sot topoloical proprt basd on usy smi opn sot sts Sout sian JMat44)04)54-69 [] Mai PK was R Roy R Fuy Sot Sts Journal o Fuy Matmatics ol 9nopp [] Mai PK Roy R Sot St tory omtrs Matmatics wit pplications 45 00pp [] MaantaJ Das PK rsults on Fuy Sot topoloical Spacs MatGM Marc 0pp-- [4] Molodstov D Sot St tory First Rsult omtrs Matmatics wit pplications ol 7pp [5] otsut DK Tory o uy sot sts rom a w rom a nw prspctiv Intrnational Journal o latst Trnds in comtin ol o Sptmbr 0pp [6] Paararol yunhfuy sot topoloy Hacttp Journal o Matmatics Stattics 4 ) 0) [7] Roy S SamantaTk ot on Fuy Sot Topoloical Spacs nnals o Fuy Matmatics Inormatics 0 [8] Roy S SamantaTk n introduction to opn closd sts on uy topoloical spacs nn Fuy MatInorm 6 ) ) [9] Tanay Kmir M urc Topoloical Structur o Fuy sot sts omtr Matmatics wit pplications 6 0) pp [0] TubaanSimsklrSaiyYuksl Fuy Sot Topoloical Spacs nnals o Fuy Matmatics Inormatics ol 5oJanuary 0)pp87-96 [] Yon an Kim Jun MiKo Fuy G-closur oprators communkoranmat Soc 8)008)5-40 Papr ID: RT064 60