CHARACTERIZATION OF SOFT COMPACT SPACES BASED ON SOFT FILTER

Transcription

1 CHRCTERIZTION O SOT COMPCT SPCES BSED ON SOT ILTER 1,2 PEI WNG, 1 JILI HE 1 Departmet of Mathematcs ad Iformato Scece, Yul Normal versty, Yul, Guagx, , PRCha 2 School of Mathematcs ad Iformato Scece; Guagx verstes Key Lab of Complex System Opt mzato ad Bg Data Processg, Yul Normal versty, Yul , Cha E-mal: 1 wagpe131@sacomc BSTRCT I ths paper, we defe soft flter ad maxmal soft flter ad gve some characterzatos about them Zorlutua, kdag, M, tmaca[2]troduced ad studed the oto of soft compact space I partcular, we vestgate the soft compact space based o soft flter ad maxmal soft flter ad get some ew resultsit provdes theoretcal supports for the developmet of the soft topologcal spaces Keywords: Soft Compact space, Soft lter, Maxmal Soft lter, Soft Cluster Pot, Soft Set 1 INTRODCTION I our lfe, may complcated questos egeerg, evromet, medce, ecoomcs ad socology, ca ot be well dealt wth by the classcal methods because there are varous ucertates for these questos I order to solve these ucertates, LZadeh[1] troduced fuzzy set theory, ZPawlak[3] troduced rough set theory ad MBGorzalzay[4] troduced terval mathematcs theory lthough they have bee successfully appled patter recogto, data mg, mache learg, ad so o ll the above theores have ther ow dffcultes, they are stll restrctve for may applcatos because the ca oly deal wth complete formato systems DMolodtsov[5] troduced the cocept of a soft set as a mathematcal tool for dealg wth ucertates whch s relatvely easy from the above dffcultes Because t eedt to fd ay relatoshp o t Soft set theory has rch potetal for practcal applcatos several domas, a few of whch are dcated by DMolodstov [6] I recetly years, soft set theory has bee researched may felds ZKog et al[7]troduced reducto of the oto of ormal parameter of soft sets Zou ad Xao[8]dscussed the soft data aalyss method Pe ad Mao[9]proof that the soft sets ca form a specal formato systems There are may people who worked o soft group, soft semgroup, soft semrg [10,11,12], soft deals[13] etc der the specal structure of the ature of the soft set attracts a lot of people to research The topologcal structures of set theores dealg wth ucertates were frst studed by Chag[14] ELash et al[15] geeralzed rough set theory the framework of topologcal spaces Shabr ad Naz[19]are the frst perso who troduce the cocept of soft topologcal spaces whch are defed over a tal uverse wth a fxed parameters Zorlutua, kdag, M, tmaca[2] troduced some ew cocepts soft topologcal spaces ad gve some ew propertes about soft topologcal spaces Compact spaces are oe of the most mportat classes geeral topologcal spaces[16] They have may well kow propertes whch ca be used may dscples Zorlutua, kdag, M ad tmaca[2] troduced compact soft spaces aroud a soft topology Sabr ad Bashr[17] vestgated the propertes of soft ope, soft eghborhood ad soft closure, they also dscussed the propertes of soft teror ad so o Wo Keu M[18]troduced the oto of the soft regular Besde these, we wll dscuss compact soft spaces aroud a soft topology I ths paper, we cosder oly soft sets (, ) over a uverse whch all the parameter set are same We deote the famly of these soft sets by SS( ) or terms whch are ot defed here, please refer to[20] ad relate refereces 431

2 2 BSIC CONCEPTS Defto 11[2] par (, ) s called a soft set over, where s a mappg gve by : P ( ) Defto 12[21] or two soft sets(, ) ad ( G, B ) over a commo uverse, (, ) s a soft subset of ( G, B ), deoted by (, ) ( G, B), f B ad ( e) G( e) for ay e Defto 13[21] or two soft sets (, ) ad ( G, B ) over a commo uverse are sad to be soft equal f(, ) s a soft subset of ( G, B ) ad ( G, B) s a soft subset of (, ) Defto 14[22] The complemet of a soft sets(, ), deoted by(, ) c, s defed by (, ) c c c =(, ) Where : P ( ) s a c mappg gve by ( α) \ ( α) =, for c s called the soft complemet c c ayα fucto of Clearly, ( ) s the same as c c ad [(, ) ] =(, ) Defto 15[12] soft set (, ) over s sad to be a ull soft set, deoted byφ, f ( e) = φ for aye ; a soft set (, ) over s sad to be a absolute soft set, deoted by, f ( e) =, for ay e It s obvous,( ) c = φ ad ( φ ) c = Defto 16[6] The uo of two soft sets (, ) ad ( G, B ) over a commo uverse s a soft set ( H, C ), where C= B ad for e C ( e) e \ B H( e) = G( e) e B\ ( e) G( e) e IB Ths relatoshp s wrtte as = (, ) ( G, B) ( H, C) Defto 17[6] The tersecto of two soft sets (, ) ad ( G, B ) over a commo uverse s a soft set ( H, C ),, H( e) = ( e) I G( e), where C= I B ad for all e C (, ) I ( G, B) = ( H, C) It ca be deoted as Defto 18[2] The soft set (, ) SS ( ) s called a soft pot, deoted by e, f ( e) φ ad elemet e ad all e e ( e) = φ for the ( \{ }) Defto 19[2] The soft pot e s sad to be the soft set( G, ), deoted by e % ( G, ) the elemet e ad ( e) G( e) 3 MIN RESLTS, f for Zorlutua, kdag, M ad tmaca[2] troduced the soft compact spaces They get a characterzato theorem that a soft topologcal space s compact f ad oly f each famly of soft closed sets wth the fte tersecto property has a oull tersecto Besde ths, we gve some characterzatos based o soft flter Defto 30[19] Let τ be a collecto of soft sets over a uverse wth a fxed set of parameters, the τ SS( ) s called a soft topology o wth set f (T1) φ, belog to τ ; (T2) the uo of ay umber of soft sets τ belog to τ ; (T3) the tersecto of ay two soft sets τ belog to τ Defto 31[19] soft set ( G, ) a soft topologcal space (,, ) τ s called a soft eghborhood of the soft pot e %, f there exsts a soft ope set ( H, ) such that e % ( H, ) ( G, ) 432

3 Defto 32[6] soft set ( G, ) a soft topologcal space (, τ, ) s called a soft eghborhood of the soft set(, ),f there exsts a soft ope set ( H, ) such that (, ) ( H, ) ( G, ) Defto 33[6] sequece of soft sets, say{(, ): N}, s evetually cotaed a a soft set (, ) f ad oly f there a a teger m such that (, ) (, ) whe m The sequece s frequetly cotaed (, ) f ad oly f for each tegerm, there s a teger such that (, ) (, ) whe m The sequece s a soft topologcal space(, τ, ), the we say that the sequece coverges to a soft set(, ), f t s evetually cotaed each eghborhood of (, ) Defto 34[6] famly Ψ of soft sets s a cover of a soft set (, ) f (, ) {(, ) Ψ: I} If each member of Ψ s a soft ope set, the Ψ s called a soft ope cover Theorem1 famly of soft sets has the fte tersecto property The there exsts a maxmal soft flter such that Proof Let = { : ad of soft sets has the fte tersecto property}, we defe a relato o by 1 2 whe 1 ad 2 belog to, the s a poset It s obvous that every cha has a upper boud rom Zors lemma, has a maxmal elemet We shall proof that s a soft flter Obvously satsfes (1) Let (, )% ad ( G, )%, sce has the fte tersecto property, the φ (, ) I ( G, ) % ad = {(, ) I ( G, )} has the fte tersecto property, the %, therefore (, ) I ( G, ) % as s a maxmal elemet The satsfes (3) (, )% ad (, ) ( G, ) the ( G, )} %, sce s a maxmal elemet, the ( G, ) % satsfes (2) Theorem2 The followg are equvalet over Defto 35[6] soft topologcal space (, τ, ) s called soft compact space f each soft ope cover of has a fte subcover Defto 36 Let be a oempty famly of soft sets over satsfes: (1) φ % ; (2) (, )% ad ( G, ) % ; (, ) ( G, ), the (3) If (, )% ad ( G, )%, the (, ) I ( G, )% s called soft flter urther, s called maxmal soft flter f for ay soft flter H such that H, the H= =SS ( ) (1) soft flter s a maxmal soft flter (2) for ay soft set ( G, ) ad (, ) I ( G, ) φ for ay(, ) %, the ( G, ) % Proof (1 2) Let s a maxmal soft flter, ( G, ) s a soft set ad (, ) I ( G, ) φ for ay(, ) % Let = {( H, ):(, ) I ( G, ) ( H, ),(, ) % } It s easy to proof that s a soft flter ad Sce s a maxmal soft flter, the G % = =, therefore (, ) (2 1) Let s a soft flter ad satsfes (1) ad for ay soft flter prove We shall Let ( G, ) %, from (3) ad 433

4 (1), the ( H, ) I ( G, ) (, ) I ( G, ) for ay φ % ( H, ), φ for ay(, ) % rom (2), ( G, ) %, therefore Defto 37 soft flter coverges to a soft pot % a soft topologcal space e (,, ) τ, f every soft eghborhood of the soft pot e belog to the soft flter It ca be deoted by e Remark 1 rom (2) of the defto 26, e f ad oly f every soft eghborhood e of the soft pote, there exsts a soft set (, ) % such that(, ) e Defto 38 s a soft flter a soft topologcal spaces(, τ, ), a soft pot e s called soft cluster pot of, f e % ( G, ) ay ( G, ) % for Theorem3 soft flter coverges to a soft pot e, the e s the soft cluster pot of; f s a maxmal soft flter ad e s a soft cluster pot of, the the soft flter coverges to the soft pot e Proof Let e ad( G, ) %, from the Defto27, every soft eghborhood e of the soft pot e belog to soft flter, from (3) ad (1), I ( G, ) φ e check e % ( G, ), t s easy to ; coversely, let soft pot e be a soft cluster pot of, from the Defto 28, I ( G, ) φ e for every soft eghborhood e of e ad ( G, ) %, from the Theorem 2, % e, from the Defto 27, e Ispred by Zorlutua, kdag, M ad tmaca[2]proposed some deftos ad propertes about soft compact spaces, We wll vestgate some others propertes ad ts applcatos τ s a soft compact space Theorem 4 (,, ) ad (, ) s a soft closed set over, the (, ) s a soft compact space Proof It s clear from the Deftos Lemma 1[2] soft topologcal space s compact f ad oly f each famly of soft closed sets wth the fte tersecto property has a oull tersecto Theorem 5 The followg are equvalet for a soft topologcal space(, τ, ) (1) soft topologcal space (, τ, ) s compact; (2) Every soft flter over has a soft cluster pot; (3) Every maxmal soft flter over coverges to a soft pot Proof (1 2) Let be a soft flter of the soft compact space (, τ, ), the = {( G, ):( G, ) % } s a closed famly ad has the fte tersecto property rom the Lemma 1, I {( G, ):( G, ) % } φ, there exsts a soft pot e % ( G, ) for every ( G, )% rom the Defto 28, e s a soft cluster pot of (2 3) Every maxmal soft flter over has a soft cluster pote, from the Theorem 3, e the (3 1) Let Ψ s a famly of soft sets a soft topologcal space (, τ, ), Ψ s a ope cover of a soft set(, ), f there s o fte subcover, the = { \( G, ):( G, ) Ψ % } has the fte tersecto property rom the Theorem 1, there exsts a maxmal soft flter such that rom (2), coverges to a soft pot, therefore has a cluster pot e I other words, e % (, ) for ay (, )%, t s true for ay 434

5 elemet, so e % \( G, ) = \( G, ) for ay ( G, ) Ψ % It s a cotradcto Defto 39 soft topologcal space(, τ, ) s called soft coutable compact space f each coutable ope cover of has a fte subcover rom the Defto, t s obvous that a soft compact space s a soft coutable compact space The followg theorem s a specal case as Lemma 1 Theorem 6 soft topologcal space s coutable compact f ad oly f each famly of soft closed sets wth the fte tersecto property has a oull tersecto Defto 310 soft pot e s called ω - cluster pot of soft set ( G, ), f every soft eghborhood of e cotas fte elemets of ( G, ) Defto 311[6] soft pot e a soft topologcal space (, τ, ) s a soft cluster pot of a sequece f the sequece s frequetly cotaed every eghborhood of e Theorem 7 The followg are equvalet for a soft topologcal space (, τ, ) (1) soft topologcal space(,, ) coutable compact τ s a soft (2) Every soft sequece over has a soft cluster pot (3) Every soft set wth fte soft pot has a ω-cluster pot Proof (1 2) Let {( G, ): N} be a sequece of soft sets a soft coutable compact space(, τ, ), let = {( G, ): = 0,1,2,} ( = 0,1,2,), + the = { } s a closed famly of soft sets wth the fte tersecto property rom Theorem 6, I { : % } φ, there exsts a soft pot e % for every % rom the Defto 28, e s the soft cluster pot of (2 3) Let ( G, ) be a soft set wth fte soft pot, we select dfferet soft pot ( G, ) formg a sequece{ e }, the the soft sequece { e }s cluster pot s also the ω-cluster pot (3 1) Let { } s a closed soft sets wth famly wth the fte tersecto property, let I = { : = 1,2,} e = 1,2,, % I { } e % { e } = 1,2,, or ay,( = 1,2,) If the sequece e s a fte set, the there exsts a soft pot such that e s the soft cluster pot of e ; f the sequece { e } s the sequece { } fte, the the sequece { e } has a ω-cluster pot rom the above we kow that the ω-cluster pot s also a soft cluster pot, the the sequece { e } has the soft cluster pot e, therefore e % ( = 1,2,), from the Theorem 6, the (, τ, ) s a soft coutable compact space CKNOWLEDGEMENT Ths work s supported by (No2014GXNSB118015), (No 2012YJZD20) ad (NO201204LX335) RERENCES: [1]LZadeh, uzzy sets, formato ad cotrol, Vol 8,1965, pp [2]IZorlutua, Mkdag, WKM ad Stmaca, Remarks o soft topologcal spaces, als of uzzy mathematcs ad romatcs Vol3, 2012, pp \\ [3]ZPawlak, Rough sets, Iteratoal Joural of Computer Scece, Vol 11, 1982, pp

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