Bipolar soft connected, bipolar soft disconnected and bipolar soft compact spaces

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1 Sogklaakari J Si Tehol 39 (3) May - Ju 07 Origial Artile Bipolar soft oeted bipolar soft disoeted ad bipolar soft ompat spaes Muhammad Shabir ad Ayreea Bakhtawar* Departmet of Mathematis Quaid-i-Azam Uiversity Islamabad 4530 Pakista Reeived: 5 February 06; Revised: 6 Jue 06; Aepted: 4 Jue 06 Abstrat Bipolar soft topologial spaes are mathematial expressios to estimate iterpretatio of data frameworks Bipolar soft theory osiders the ore features of data graules Bipolarity is importat to distiguish betwee positive iformatio whih is guarateed to be possible ad egative iformatio whih is forbidde or surely false Coetedess ad ompatess are the most importat fudametal topologial properties These properties highlight the mai features of topologial spaes ad distiguish oe topology from aother Takig this ito aout we explore the bipolar soft oetedess bipolar soft disoetedess ad bipolar soft ompatess properties for bipolar soft topologial spaes Moreover we itrodue the otio of bipolar soft disjoit sets bipolar soft separatio ad bipolar soft hereditary property ad study o bipolar soft oeted ad disoeted spaes By givig the detailed piture of bipolar soft oeted ad disoeted spaes we ivestigate bipolar soft ompat spaes ad derive some results related to this oept Keywords: bipolar soft topology bipolar soft disjoit set bipolar soft ope set bipolar soft losed set bipolar soft oeted spae bipolar soft disoeted spae bipolar soft ompat spae Itrodutio We reate models of reality that are improvemets of aspets of the geuie world Lametably these sietifi models are exessively ovoluted ad we aot loate the exat solutios The vulerability or istability of iformatio while modelig issues i soial siees medial siees artifiial itelligee egieerig atural siees et makes the utilizatio of ovetioal lassial method suessful Therefore traditioal set theories whih were based o the risp ad exat ase may ot be ompletely suitable for takig are of issues of ambiguity/vagueess To surpass these istabilities the sorts of theory have bee proposed (Gau et al 993; Pawlak 98; Zadeh 965) et all these speulatios have their atural troubles The reaso * Correspodig author address: ayreea_kha@yahooom behid these troubles is potetially the isuffiiey of the parameterizatio istrumet of the theory as stated by Molodtsov (999) Molodtsov (999) popularized the idea of the theory of soft sets as a ew effetive ad stroger mathematial tool for dealig with istabilities whih is free from the above halleges I his paper he exhibited the ruial after effets of the ew theory ad effetively oeted it to a few headigs; for example game theory Riema itegratio probability smoothess of futio ad so forth Soft frameworks give a exeptioally broad system with the otributio of parameters A lot of work o soft set theory ad its appliatio i differet fields have bee arried out by a umber of researhers (Aktas et al 007; Ali et al ; Ju 008; Ju et al 008; Maji et al 003; Shabir et al 009) If we review the history of soft topologial spaes the foudatio of whih was laid by Shabir et al (0) we fid may remarkable authors followig them (Ayguoglu et al 0; Cagma et al 0; Hussai et al 0 04; Khalil et al 05; Li 03; Mi

2 360 M Shabir & A Bakhtawar / Sogklaakari J Si Tehol 39 (3) ; Peygha 03 04; Varol et al 0; Zakari et al 06 Zorlutua et al 0) The oept of bipolar soft sets (a hybridizatio of the struture of soft set ad bipolarity) with its appliatio i deisio makig was itrodued ad disussed i detail by Shabir et al (03) ad studied exhaustively by Karaasla et al (04) A bipolar soft set is aquired by viewig ot oly a preisely hose set of parameters but also a assoiated set of oppositely meaig parameters alled the ot set of parameters Due to the quality of providig positive ad egative aspets of iformatio at a time the idea of the bipolar soft set is gaiig mometum amog researhers Hayat et al (05) applied the oept of bipolar soft sets to hemirigs Reetly Shabir ad Bakhtawar iitiated the study of bipolar soft topologial spaes They defied bipolar soft topology as a olletio of bipolar soft sets over the uiverse U Cosequetly they defied basi otios of bipolar soft topologial spaes bipolar soft ope ad bipolar soft losed sets bipolar soft subspae bipolar soft losure bipolar soft iterior bipolar soft eighborhood of a poit ad ivestigated their several properties Further Shabir ad Bakhtawar explored ad studied i detail bipolar soft separatio axioms I the preset study we iitiated some ew ideas i bipolar soft topologial spaes suh as bipolar soft oeted spaes bipolar soft disoeted spaes bipolar soft ompat spaes Setio presets Prelimiaries o basi oepts related to soft sets soft topologial spaes bipolar soft sets ad bipolar soft topologial spaes Setio 3 is devoted towards the idea of bipolar soft disjoit sets bipolar soft separatio of a set bipolar soft oeted spaes bipolar soft disoeted spaes ad bipolar soft hereditary property ad some examples are give for the better uderstadig of these ideas Setio 3 studies the oept of bipolar soft ompat spaes ad some results related to these oepts are exhibited These ewly defied ideas i bipolar soft topologial spaes will hopefully promote the future work ad studies to be held i the bipolar soft topology ad a be applied effetively to ope with uertaities Prelimiaries I this setio we reall some basi defiitios ad results related to bipolar soft sets soft topologial spaes ad bipolar soft topologial spaes Let U be a iitial uiverse E be the set of parameters E be the ot set of parameters Let P(U) deotes the power set of U ad A B C be o-empty subsets of E Defiitio (Maji et al 003) Let E { e e e e } 3 be the set of parameters The ot set of E deoted by E is defied by E { e e e e } 3 where for all i e ot e i i Defiitio (Shabir et al 03) A triplet F G A is alled a bipolar soft set over U where F ad G are mappigs F : A P U ad G : A P U suh that F e G e (Empty set) for all e A Defiitio 3 (Shabir et al 03) For two bipolar soft sets F G A ad F G B over a uiverse U we say that F G A is a bipolar soft subset of F G B if ) A B ad ) F e F e ad G e G e for all e A This relatio is deoted by F G A F G B Defiitio 4 (Shabir et al 03) Two bipolar soft sets F G A ad F G B over the uiverse U are said to be equal if F G A is a bipolar soft subset of F G B ad F G B is a bipolar soft subset of F G A Defiitio 5 (Shabir et al 03) The omplemet of a bipolar soft set F G A is deoted by F G A ad is defied by F G A F G A where F : A PU ad G : A P U are give by F e G e ad G e F e for all e A Defiitio 6 (Shabir et al 03) Exteded Uio of two bipolar soft sets F G A ad F G B over the ommo uiverse U is the bipolar soft set H I C over U where C A B ad for all e C F e if e A B H e F e if e B A F e F e if e A B G e if e A B I e G e if e B A G e G e if e A B We deote it by F G A F G B H I C

3 M Shabir & A Bakhtawar / Sogklaakari J Si Tehol 39 (3) Defiitio 7 (Shabir et al 03) Exteded Itersetio of two bipolar soft sets F G A ad F G B over the ommo uiverse U is the bipolar soft set H I C over U where C A B ad for all e C F e if e A B H e F e if e B A F e F e if e A B G e if e A B I e G e if e B A G e G e if e A B We deote it by F G A F G B H I C Defiitio 8 (Shabir et al 03) Restrited Uio of two bipolar soft sets F G A ad F G B over the ommo uiverse U is the bipolar soft set H I C over U where C A B is o-empty ad for all e C H e F e F e ad I e G e G e We deote it by F G A F G B H I C R Defiitio 9 (Shabir et al 03) Restrited Itersetio of two bipolar soft sets F G A ad F G B over the ommo uiverse U is the bipolar soft set H I C over U where C A B is o-empty ad for all e C H e F e F e ad I e G e G e We deote it by F G A F G B H I C R Remark If we take A B E i Defiitios the exteded uio oiides with restrited uio ad exteded itersetio oiides with restrited itersetio Propositio (Shabir et al 03) Let F G A ad F G B be two bipolar soft sets over the ommo uiverse U with the uiverse set of parameters E The the followig are true ) F G A F G B F G A F G B ) F G A F G B F G A F G B 3) F G A R F G B F G A R F G B 4) F G A F G B F G A F G B R R Defiitio 0 (Shabir et al 03) The Uio of two bipolar soft sets F G A ad F G A over the ommo uiverse U is the bipolar soft set ( H I A ) over U where for all e E H e F e F e ad I e G e G e We deote it by F G A F G A H I A Defiitio (Shabir et al 03) The Itersetio of two bipolar soft sets F G A ad F G A over the ommo uiverse U is the bipolar soft set H I A over U where for all e A H e F e F e ad I e G e G e We deote it by F G A F G A H I A Defiitio (Shabir et al 03) A bipolar soft set over U is said to be relative ull bipolar soft set (with respet to the parameter set A deoted by ( u A) if for all e A e ad u ( e) U for all e A The relative ull bipolar soft set with respet to the uiverse set of parameters E is alled the ull bipolar soft set over U ad is deoted by ( u E) A bipolar soft set F G E over U is said to be a o ull bipolar soft set if F ( e) for some e E Defiitio 3 (Shabir et al 03) A bipolar soft set over U is said to be relative absolute bipolar soft set (with respet to the parameter set A) deoted by A if for all e A e U ad e for all e A The relative absolute bipolar soft set with respet to the uiverse set of parameters E is alled the absolute bipolar soft set over U ad is deoted by E

4 36 M Shabir & A Bakhtawar / Sogklaakari J Si Tehol 39 (3) Obviously a bipolar soft set F G E over U is said to be a o absolute bipolar soft set over U if F e U for some e E Defiitio 4 (Shabir ad Bakhtawar) Let F G E be a bipolar soft set over U ad u U The u F G E if u F e for all e E If u F G E the automatially u G( e) for all e E Note that for ay u U u F G E if u F e for some e E Defiitio 5 (Shabir ad Bakhtawar) Let be a o-empty subset of U The ( E) deotes the bipolar soft set over U defied by ( e) for all e E e for all e E ad u U The Defiitio 6 (Shabir ad Bakhtawar) Let F G E deotes the bipolar soft set over U defied by u u F e u ad G e U \ u for eah e E u u Defiitio 7 (Shabir ad Bakhtawar) Let be the olletio of bipolar soft sets over U with E as the set of parameters The is said to be a bipolar soft topology o U if () ( u E) ( E) belogs to () the uio of ay umber of bipolar soft sets i belogs to (3) the itersetio of ay two bipolar soft sets i belogs to The U E E is alled a bipolar soft topologial spae over U ad the members of are said to be bipolar soft ope sets i U Propositio (Shabir ad Bakhtawar) Let U E E be a bipolar soft topologial spae over U The the olletio F e F G E for eah e E defies a topology o U e Defiitio 8 (Shabir ad Bakhtawar) Let U E E be a bipolar soft topologial spae over U ad F G E be a bipolar soft set over U The F G E is said to be bipolar soft losed if ad oly if F G E belogs to A bipolar soft set F G E over U is said to be bipolar soft lope if it is both a bipolar soft losed ad a bipolar soft ope set over X Defiitio 9 (Shabir ad Bakhtawar) Let F G E be a bipolar soft set over U ad be a o-empty subset of U The the bipolar sub soft set of F G E over deoted by F G E is defied as follows F e F e ad G e G e for eah e E Propositio 3 (Shabir ad Bakhtawar) Let U E E be a bipolar soft topologial spae over U ad be a oempty subset of X The { F G E F G E } is a bipolar soft topology o Theorem (Shabir ad Bakhtawar) Let U E be a soft topologial spae over U (Shabir et al 0) The s the olletio osistig of bipolar soft sets F G E suh that F E ad G e Fe U \ F e for all e E defies a bipolar soft topology over U 3 Bipolar Soft Coeted ad Bipolar Soft Disoeted Spaes I this setio we disussed ad explored oe of the most importat property of bipolar soft topologial spaes alled the bipolar soft oetedess ad bipolar soft disoetedess Defiitio 0 Two bipolar soft sets F G E F G E are said to be bipolar soft disjoit if F e F e for all e E Defiitio Let U E E be a bipolar soft topologial spae over U A bipolar soft separatio of E is a pair F G E F G E of o-ull disjoit bipolar soft ope sets over U suh that F e F e U for all e E Defiitio A bipolar soft topologial spae U E E is said to be a bipolar soft disoeted spae if there exists a bipolar soft separatio of E Further U E E is said to be a bipolar soft oeted spae if ad oly if it is ot a bipolar soft disoeted spae Example Let U m m m m be the uiverse set represetig markets Let E { e e } 3 4 dresses formal dresses} ad E { e e } mahie embroidery ausal dresses Let F G E F G E {had embroidery represets the preferees of markets for seletio of lothes by two wome The the bipolar soft topology over U geerated

5 M Shabir & A Bakhtawar / Sogklaakari J Si Tehol 39 (3) is a bipolar soft diso- by F G E ad F G E is give by {( u E) ( E) F G E F G E F G E } where F G E F G E F G E are bipolar soft sets over U defied as follows: F e m F e m m ad G e m G e m 4 F e m m m F e m m ad G e G e m F e F e ad G e m G e m m The U E E eted spae beause F G E ad F G E form a bipolar soft separatio of E Example Let U w w w E { e e e } be the uiverse set represetig weddig marques Let 3 3 {Expesive best food servie ideal deoratio faility} be the set of parameters ad E { e e e } {heap average food 3 servie poor deoratio faility} be the ot set of parameters Let F G E F G E represets the hoies made by two differet families for the seletio of weddig marques The {( u E) ( E) F G E F G E } F G E where F G E F G E is the bipolar soft topology over U geerated by F G E 4 4 F e w w F e w w F e w w ad G e w G e w w G e w 3 4 F e w w F e w w w F e w w ad G e w w G e w G e 4 3 F e w F e w w F e w ad G e w w G e w w G e w F e w w w F e w w w F e w w w ad G e w G e w G e We ote that the bipolar soft topologial spae geerated by F G E F G spae beause there does ot exist a bipolar soft separatio of E Propositio 4 Let F G E be a bipolar soft set The ) F G E F G E H I E where H e F e F e U I e G e G e for eah e E ) F G E F G E H I E where H e F e F e I e G e G e U for eah e E Further F G E F G E will always satisfy F e F e G e G e 3) F G E ( E) F G E ad F G E ( E) ( E) 4 for eah e E ad for eah e E ad E is a bipolar soft oeted for all e E Proof Straightforward Theorem A bipolar soft topologial spae U E E is bipolar soft disoeted spae if ad oly if there exist two bipolar soft losed sets F G E F G E with G e G e for some e E suh that G e G e U for all e E ad G e G e for all e E Proof First suppose U E E is a bipolar soft disoeted spae The there exist a bipolar soft separatio of E Let F G E ad H I E forms a bipolar soft separatio of E The F e H e U for all e E () F ( e) for some e E () H ( e) for some e E (3) F( e) H ( e) for all e E (4) From equa- for some e E As F e G e ad H e I e therefore from equatio () we have G e I e U tio () () ad (3) we have G e I e for all e E where G e I e Further F G E ad H I E are bipolar soft losed sets sie F G E ad H I E

6 364 M Shabir & A Bakhtawar / Sogklaakari J Si Tehol 39 (3) Coversely suppose that there exist two bipolar soft losed sets F G E F G E with G e G e for some e E suh that G e G e U for all e E ad G e G e for all e E The F G E ad F G E are bipolar soft ope sets with F e G e F e G e for some e E F e F e G e G e U for all e E F e F e for all e E Therefore suh that ad F G E ad F G E form a bipolar soft separatio of E Thus U E E is a bipolar soft disoeted spae Remark The uio of two bipolar soft oeted spaes over the same uiverse eed ot to be a bipolar soft oeted spae Example 3 Let U h h E e e E e e {( u E) ( E) F G E } ad {( u E) ( E) F G E } F e U F e G e G e U F e F e U where ad ad G e U G e The U E E U E E oeted spae beause F G E F G E form a bipolar soft separatio of ( u E) i are bipolar soft oeted spaes over U But we ote that is ot a bipolar soft Propositio 5 The itersetio of two bipolar soft oeted spaes over a same uiverse is a bipolar soft oeted spae Proof Let U E E ad U E E be two bipolar soft oeted spaes Suppose to the otrary that U E E is ot a bipolar soft oeted spae The there exist bipolar soft sets to whih forms a bipolar soft separatio of ( u E ) i U E E Sie the F G E F G E ad F G E F G E This implies that soft separatio of ( u E) i U E E ad U E E whih is a otraditio to give hypothesis Thus U E E F G E F G E belogig F G E F G E F G E F G E form a bipolar F G E F G E form a bipolar soft separatio of ( u E) i is a bipolar soft oeted spae Remark 3 The itersetio of two bipolar soft disoeted spaes over the same uiverse eed ot to be a bipolar soft disoeted spae U h h h E e e E e e = ( u E) E F G E Example 4 Let 3 F G E F G E } ad {( u E) ( E) H I E H I E H3 I3 E } where F e h F e h h ad G e h G e h 3 F e h h F e h ad G e G e h 3 3 F e F e ad G e h G e h h H e h h H e h h ad I e h I e h 3 3 H e h H e h ad I e h I e h H e H e ad I e h h I e h h The U E E U E E are bipolar soft disoeted spaes over U Now {( u E) ( E)} We ote that U E E beause there do o exist ay two o ull disjoit bipolar soft ope sets F G E ad H I E belogig to that F e H e U for all e E is ot a bipolar soft disoeted spae suh Propositio 6 The uio of two bipolar soft disoeted spaes over the same uiverse is a bipolar soft disoeted spae Proof Straightforward Theorem 3 Let U E E be a bipolar soft topologial spae over U ad let the bipolar soft sets F G E ad F G E form a bipolar soft separatio of E If E E U E E the F e for all e E or F e for all e E Proof Sie F G E ad F G E form a bipolar soft separatio of E U ( F ( e) F ( e)) U for eah e E () is a bipolar soft oeted subspae of we have

7 M Shabir & A Bakhtawar / Sogklaakari J Si Tehol 39 (3) F ( for some e E () F ( for some e E (3) F ( e) F ( e)? for all e E (4) As U E E From equatio () This implies we have F G E ad ( F e F e ) for eah e E F G E are ope i ( F e ) ( F e ) for all e E (5) Also from equatio (4) ( F e) ( F e) for all e E is oeted so either F e for all e E or F e for all e If F e for all e E the from equatio (5) F e for all e E ad this implies F e As E E all e E If all e E F e for all e E the from equatio (5) Remark 4 The overse of Theorem 3 does ot hold i geeral F e for all e E for ad this implies F e for Example 5 Let U h h h h E e e E e e ad u E E F 3 4 G E F G E F G E F G E F G E F G E } where F e h F e h ad G e h h h G e h h h F e h F e h ad G e h h h G e h h h F e h h F e h h ad G e h h G e h h F e h h F e h h ad G e h h G e h h F e h h h F e h h h ad G e h G e h F e h h h F e h h h ad G e h G e h The U E E is a bipolar soft topologial spae over U Also ote that F G E F G E soft separatio of {( ) ( ) 4 4 E form a bipolar Now let h h the { h h } the { E E F G E F G E F G E F G E 4 4 F G E 5 5 F G E 6 6 } is a bipolar soft topology over where F e h ad G e h G e h F e h ad G e h G e h F e ad G e h h G e h h F e h h ad G e G e F e h ad G e h G e h F e h ad G e h G e h F e for all e E But E E F e 3 h Oe a easily ote that ( ) is ot a bipolar soft oeted spae beause 4 F G E form a bipolar soft separatio of ( E) Remark 5 If there exist a o ull o-absolute bipolar soft lope set over U the U E E eed ot be a F G E ad bipolar soft disoeted spae Propositio 7 Let U E E be a bipolar soft topologial spae over U If there exist a o-ull o-absolute bipolar soft lope set F G E over U suh that F e F ( e) U for eah e E the U E E is a bipolar soft

8 366 M Shabir & A Bakhtawar / Sogklaakari J Si Tehol 39 (3) disoeted spae Proof Sie F G E is a bipolar soft lope set over U the F G E is a bipolar soft lope set over U Now by give hypothesis ad by propositio 4 we have ( ) e E ad ( ) F e F e U for all e E F e F e for eah e E ad G e G ( e) U for eah e E Therefore F G E ad F G E form a bipolar soft separatio of E Thus U E E is a bipolar soft disoeted spae ad ( ) G e G e for eah Example 6 Let U h h h E e e E e e ad {( u E) ( E) F 3 G E F G E F G E F G E } where 4 4 F e h F e h h ad G e h G e h 3 F e h F e h ad G e h G e h h 3 F e h h F e U ad G e G e F e F e ad G e h h G e U The U E E is a bipolar soft topologial spae over U Note that F G E is a o-ull o-absolute bipolar soft lope set over U but U E E is ot a bipolar soft disoeted spae sie there do ot exist a bipolar soft separatio of E Theorem 4 Let U E be a soft topologial spae over U ad s U E E be a bipolar soft topologial spae B over U ostruted from U E as i Theorem db If U E is a soft disoeted spae (Hussai 04) over U s s the U E E is a bipolar soft disoeted spae over U B Proof Sie U E is a soft disoeted spae therefore there exist o-ull soft ope sets (say) F E H E over U suh that s ad U E ( F E) ( H E) F E H E E Further F G E H I E are o-ull bipolar bipolar soft ope sets (beause F( e) H ( e) ) where for all e E G e U \ F e ad I e U \ H e sie F E H E belogs to s Now F e H ( e) U for all e E ad G e I ( e) ( U \ F e) ( U \ H e) F e H ( e) for eah e E ad G e I( e) ( U \ F e) ( U \ H e) U for eah e E F G E H I E belogig to forms a bipolar soft separatio of E Thus U E E disoeted spae Propositio 8 Let E E The E E B for eah e E Also B This implies that is a bipolar soft ad Z E E be two bipolar soft subspaes of U E E is a bipolar soft subspae of Z E E Proof As Z so Z Moreover eah bipolar soft ope set F G E of E E ad G e G e for all e E where F G E is a bipolar soft ope set of F e F e Now for eah e E F e Z F e ad G e Z G e F e ( Z F e) ad G e Z G e Z Z F e F e ad G e G e Z E E ( ) Z Z where F ad let Z is of the form U E E G E is a bipolar soft ope set i

9 M Shabir & A Bakhtawar / Sogklaakari J Si Tehol 39 (3) Theorem 5 Let E E be the olletio of bipolar soft oeted subspaes of a bipolar soft topologial spae U E E If E E is a bipolar soft oeted subspae of U E E J Proof Let E E the J J J be a olletio of bipolar soft oeted subspaes of U J E E ad E E be a disoeted subspae of U E E J F G E be a bipolar soft separatio of E The F e F e ( F e F e) for all e E () F e for some e E () F e for some e E (3) F e F e F e F e for all e E (4) Suppose that J Cosider a fixed The from equatio () F e F e for all e E From equatio (4) or F e F e for all Sie E E e E is a bipolar soft oeted subspae of U E E so either F e e E F e for all Now there are three ases: (i) (ii) Case: (i) F e for all e E ad for all J F e for all e E ad for all J (iii) for some J If F e ad for other some J F e F e for all e E ad for all J This otradits equatio () Case: (ii) If F e for all e E ad for all J This otradits equatio (3) Case: (iii) this implies As so there exist some x J for all J the that is F e J the that is F e J suh that Let F G E for all e E F e for all e E F e for all e E for all e E Now by equatio () x F e F e x F e or x F e the F e ad if x F e the F e E E is a bipolar soft oeted subspae of U E E U E E E E of U E E also possesses the property P If x F e So the ase (iii) is ot possible Hee our suppoitio is wrog ad Defiitio 3 A property P of a bipolar soft topologial spae is said to be a bipolar soft hereditary iff every bipolar soft subspae property Remark 6 The bipolar soft oetedess (respet bipolar soft disoeted) is ot a bipolar soft hereditary Example 7 Let U h h h E e e E e e ad {( u E) ( E) F 3 G E F G E F G E } where

10 368 M Shabir & A Bakhtawar / Sogklaakari J Si Tehol 39 (3) F e h F e h ad G e h h G e h h 3 3 F e h F e h ad G e h h G e h h 3 3 F e h h F e h h ad G e h G e h The U E E is a bipolar soft oeted spae Now take h h The { E E F G E F G E F G E } F e h F e h ad G e h G e h F e h F e h ad G e h G e h F e F e ad G e G e The E E 3 3 subspae of bipolar soft oeted spae where 3 3 is a bipolar soft disoeted F G E } where 3 3 Example 8 Let U h h h E e e E e e ad {( u E) ( E) F 3 G E F G E F e h F e h ad G e h G e h h 3 F e h h F e h h ad G e G e h 3 3 F e F e ad G e h G e U The U E E is a bipolar soft disoeted spae Now take h 3 The { E E F G E F G E F G E } where F e F e ad G e G e F e F e ad G e G e F e F e ad G e G e The E E is a bipolar soft oeted subspae of bipolar soft disoeted spae Bipolar Soft Compat Spaes I this setio we study aother importat property of bipolar soft topologial spaes alled the bipolar soft ompatess Bipolar soft ompat spaes are ivestigated ad some results related to this oept are derived Defiitio 4 A family F G E F G E F G E of bipolar soft sets is alled the bipolar soft over of a bipolar soft set J F G E if ( ) J ( ) Further it is alled the bipolar soft ope over of a bipolar soft set F G E if eah member of is a bipolar soft ope set A bipolar soft subover of is a subfamily of whih is also a bipolar soft over Defiitio 5 A bipolar soft topologial spae U E E is alled a bipolar soft ompat spae if eah bipolar soft ope over of E has a fiite bipolar soft subover Example 9 Let U h h h E e e E e e ad {( u E) ( E) F 3 G E F G E F G E F G E F G E F G E F G E } where F G E F G E F G E F G E 4 4 F G E F G E F G E are bipolar soft sets over U defied as follows: F e h F e h ad G e h G e h 3 3 F e h F e h ad G e h G e h 3 3 F e h h F e h h ad G e h G e h F e h h F e h h ad G e G e F e h h F e h h ad G e G e

11 M Shabir & A Bakhtawar / Sogklaakari J Si Tehol 39 (3) F e ad G e h G e h F e h ad G e G e The U E E U E E F e spae over U Further we a easily observe that E has a fiite subover Example 0 Let U N be the uiverse set of atural umbers let E e e is a bipolar soft topologial is a bipolar soft ompat spae beause every ope over of ad E e e be the set of parameters ad the ot set of parameters respetively Let be the bipolar soft topology over N osistig of all bipolar soft sets defied o the parameter set E geerated by the bipolar soft sets F G E F G E F G E where F e F e ad G e U \ G e U \ F e 3 F e 3 ad G e U \ 3 G e U \ 3 F e 3 4 F e 3 4 ad G e U \ 3 4 G e U \ F e F e ad G e U G e U \ \ The the bipolar soft topologial spae U E E over U geerated by the bipolar soft sets is ot a bipolar soft ompat spae sie { F G E : N} soft subover Defiitio 6 Let U E E { F G E : N} is a bipolar soft ope over of N with o fiite bipolar ad U E E be two bipolar soft topologial spaes over the uiverse U If the is said to be fier tha If or the is omparable with Propositio 9 Let U E E be a bipolar soft ompat spae ad The U E E is a bipolar soft ompat Proof Let F G E J F G E J is a bipolar soft ompat spae Therefore be the bipolar soft ope over of E i U E E Sie the E by bipolar soft ope sets of U E E But U E E is the bipolar soft ope over of ( U E ) ( F G E ) ( F G E ) ( F G E ) for some J HeeU E E is a bipolar soft ompat spae Theorem 6 Let E E be a bipolar soft subspae of U E E The E E is a bipolar soft ompat spae if ad oly if every over of ( E) by bipolar soft ope sets i U otais a fiite subover Proof Let E E F G E be a over of ( E ) by bipolar be a bipolar soft ompat spae ad F e for eah e E J F G E is a bipolar soft ope over of E J J soft ope sets i U Now? G e for eah e E Therefore Sie E E is a bipolar soft ompat spae therefore we have ( E) ( F G E) ( F G E) ( F G E) for some J This implies that F G E is a bipolar subover of E by bipolar soft ope sets i U i i i Coversely suppose F G E F G E J J is a bipolar soft ope over of E ad we have ( E ) ( F G E ) ( F G E ) ( F G E ) for some is a bipolar soft ope over of E It is easy to see that by bipolar soft ope sets i U Therefore by give hypothesis J Thus F G E i i i

12 370 M Shabir & A Bakhtawar / Sogklaakari J Si Tehol 39 (3) is a bipolar subover of E Hee E E is a bipolar soft ompat spae Defiitio 7 Let U E E be a bipolar soft topologial spae over U ad let If every elemet of a be writte as the uio of the elemets of the is alled a bipolar soft basis for bipolar soft topology Eah elemet of is alled a bipolar soft basis elemet U h h h h E e e E e e {( u E) ( E) F G E ad 3 4 F G E F G E F G E F G E F G E F G E F G E } where F G E F G E F G E F G E F G E F G E F G E are bipolar soft sets over U defied as follows: Example Let F e h h ad G e h G e h h F e h h ad G e h G e h h F e ad G e h h G e U F e h h ad G e h h G e h h F e ad G e h h h G e U F e ad G e h h h G e U F e h h ad G e G e h h F e h h ad G e h h G e h h h h 3 h h 4 h h F e U 7 h The U E E is a 8 bipolar soft topologial spae over U Now if we take {( u E) F G E F G E F G E F G E F G E } the is a bipolar soft basis for 8 8 Next if we take where { u E F G E F G E F G E F G E} the is ot a bipolar soft basis beause E aot be writte as the uio of the elemets of Theorem 7 A bipolar soft topologial spae U E E is a bipolar soft ompat spae if ad oly if there is a bipolar soft basis for suh that every bipolar soft over of E by the elemets of has a fiite bipolar soft subover Proof Let U E E be a bipolar soft ompat spae Obviously is a bipolar soft basis for Therefore every bipolar soft over of E by elemets of has a fiite bipolar soft subover Coversely to show U E E is a bipolar soft ompat let { L M E } be a bipolar soft ope over of J E L M E as a uio of basis elemet for eah J These elemets form a bipolar soft ope We a write over of E suh that F G E I U F e F e F e F e for eah e E ad 3 G e G e G e for eah e E F G E for some L M E i i i Now by give hypothesis for some I we have That is E F G E F G E Now F G E L M E for eah i I is a fiite bipolar subover of E Hee U E E This implies that is a bipolar soft ompat spae

13 M Shabir & A Bakhtawar / Sogklaakari J Si Tehol 39 (3) Colusios Durig the study we have goe ito detail about defiig ad fidig out the properties of bipolar soft oeted spaes bipolar soft disoeted spaes ad bipolar soft ompat spaes The redit of stregtheig the foudatios i the tool box of bipolar soft topology will be give to these ewly defied oepts The fidigs ad results whih we have draw a be applied to solve existig problems i various fields whih otai uertaity Referees Aktas H & Cagma N (007) Soft sets ad soft groups Iformatio Siees Ali M I & Shabir M (00) Commets o De Morga s law i fuzzy soft sets The Joural of Fuzzy Mathematis Ali M I Shabir M & Naz M (0) Algebrai strutures of soft sets assoiated with ew operatios Computers ad Mathematis with Appliatios Ali M I Feg F Liu X Mi W K & Shabir M (009) O some ew operatios i soft set theory Computers ad Mathematis with Appliatios Ayguoglu A & Aygu H (0) Some otes o soft topologial spaes Neural Computig ad Appliatios 3-9 Cagma N Karatas S & Egioglu S (0) Soft topology Computers ad Mathematis with Appliatios Gau W L & Buehrer D J (993) Vague sets IEEE Trasatios o Systems Ma ad Cyberetis 3() Hayat K & Mahmood T (05) Some appliatios of bipolar soft set: Charaterizatios of two isomorphi Hemi-Rigs via BSI-h-Ideals British Joural of Mathematis ad Computer Siee 3 - Hussai S (04) A ote o soft oetedess Joural of the Egyptia Mathematial Soiety 3 6- Hussai S & Ahmad B (0) Some properties of soft topologial spaes Computers ad Mathematis with Appliatios Ju B (008) Soft BCK/BCI-algebras Computers ad Mathematis with Appliatios Ju B & Park C H (008) Appliatios of soft sets i ideal theory of BCK/BCI-Algebras Iformatio Siees Karaasla F & Karatas S (05) A ew approah to bipolar soft sets ad its appliatios Disrete Mathematis Algorithms ad Appliatios 7(4) doi:04/ S Khalil O H & Ghareeb A (05) Spatial objet modelig i soft topology Sogklaakari Joural of Siee ad Tehology 37(4) Li F (03) Soft oeted spaes ad soft paraompat spaes Iteratioal Joural of Mathematial Siee ad Egieerig 7() -7 Maji P K Biswas R & Roy R (003) Soft set theory Computers ad Mathematis with Appliatios Mi W K (0) A ote o soft topologial spaes Computers ad Mathematis with Appliatios Molodtsov D (999) Soft set theory first results Computers ad Mathematis with Appliatios Pawlak Z (98) Rough sets Iteratioal Joural of Iformatio ad Computer Siee Peygha E Samadi B & Tayebi A (04) Some results related to soft topologial spaes Fata Uiversitatis Series: Mathematis ad Iformatis 9(4) Retrieved from php/fumathif/artile/view/4/pdf Peygha E Samadi B & Tayebi A (03) About soft topologial spaes Joural of New Results i Siee Shabir M & Ali M I (009) Soft ideals ad geeralized fuzzy ideals i semigroups New Mathematis ad Natural Computatio Shabir M & Naz M (0) O soft topologial spaes Computers ad Mathematis with Appliatios Shabir M & Naz M (03) O bipolar soft sets Retrieved from Varol B P Shostak A & Aygu H (0) A ew approah to soft topology Haettepe Joural of Mathematis ad Statistis Zadeh LA (965) Fuzzy sets Iformatio ad Cotrol Zakari A H Ghareeb A & Omra S (06) O soft weak strutures Soft Computig doi:0007/s Zorlutua I Akdag M Mi WK & Atmaa S (0) Remarks o soft topologial spaes Aals of Fuzzy Mathematis ad Iformatis