Compactness in Multiset Topology

Transcription

1 opatess ultset Topolog Sougata ahata S K Saata Depatet of atheats Vsva-haat Satketa-7335 Ida Abstat The pupose of ths pape s to todue the oept of opatess ultset topologal spae e vestgate soe bas esults opat ultset topologal spae sla to the esults opat topologal spae Futheoe we edefe the oept of futos betwee two ultsets Kewods ultsets -topolog opatess otut I INTRODTION The oto of a ultset s well establshed both atheats ad opute see I atheats a ultset s osdeed to be the geealato of a set I lassal set theo a set s a well-defed olleto of dstt objets If epeated ouees of a objet s allowed a set the a atheatal stutue kow as ultset set fo shot s obtaed lassal set theo states that a gve eleet a appea ol oe a set t assues that all atheatal objets ou wthout epetto So the ol possble elato betwee two atheatal objets s ethe the ae equal o the ae dffeet The stuato see ad oda lfe s ot alwas lke ths I the phsal wold t s obseved that thee s eoous epetto Fo stae thee ae a hdoge atos a wate oleules a stads of DNA et A wde applato of sets a be foud vaous bahes of atheats Algeba stutues fo ultset spae have bee ostuted b Ibah et al[9] Applato of set theo deso akg a be see [7] I 0 Gsh ad Sul [5] todued ultset topologes dued b ultset elatos The sae authos futhe studed the otos of ope sets losed sets bass sub-bass losue teo otut ad elated popetes -topologaal spaes [6] I 05 S A El-Shekh R A-K Oa ad Raafat todue oto of sepaato aos o ultset topologal spae [8] I 05 El-Shekh et al [] todue soe tpes of geealed ope sets ad the popetes I [] J ahata ad D Das have woked o se opatess of -topologal spaes ut se opatess ples opatess Ths pape deals wth the oto of opatess -topolog A defto of opatess - topolog s todued alog wth seveal eaples Relatos aog opatess losedess ad Hausdoffess ae studed A oto of futo betwee two ultsets s todued ad otut of suh futos ude the -topologal otet s defed ehavou of opatess ude suh otuous appgs s also studed II PRELIINARIES DEFINITION [5]: A set daw fo the set s epeseted b a futo out o defed as : whee epesets the set of o-egatve teges Hee s the ube of ouees of the eleet the ultset Let be a set fo the set = { } wth appeag tes It s deoted b o / A set daw fo the set s deoted b { k / } whee k e = k Howeve those eleets whh ae ot luded the set have eo out DEFINITION [5]: Let be a set daw fo a set The suppot set of deoted b s a subset of ad { 0} e s a oda set ad t s also alled oot set DEFINITION 3 [5]: A doa s defed as a set of eleets fo whh sets ae ostuted The set spae s the set of all sets whose eleets ae suh that o eleet the set ous oe tha tes The set spae s the set of all sets ove doa suh that thee s o lt o the ube of ouees of a eleet a set If } the { k {{ / k} k / k } DEFINITION [5]: Let {0 } N The: = N f = N fo all N e s a subset of N f N fo all 3 P = N f P = a{ N } fo all P = N f P = { N } fo all 5 P = N f P = { + N } fo all ISSN: Page 75

2 6 P = N f P = a{ - N 0 } fo all hee ad epesets set addto ad set substato espetvel DEFINITION 5 [5]: Let The the opleet of s a eleet of suh that Ad f N the the opleet N of N s N DEFINITION 6 [5]: Let be a set daw fo the set ad f 0 fo all the s alled ept set ad deoted b e 0 fo all DEFINITION 7 [5]: A subset N of s a whole subset of f N fo all N DEFINITION 8 [5]: A subset N of s a patal whole subset of f N fo at least oe N DEFINITION 9 [5]: A subset N of s a full subset of f N REARK [5]: s a whole subset of eve set but t s ethe a full subset o a patal whole subset of a oept set DEFINITION 0 [5]: Let be a set The powe whole set of deoted b P s defed as the set of all the whole subsets of e fo ostutg powe whole subsets of eve eleet of wth ts full ultplt behaves lke a eleet a lassal set The adalt of P s whee s the adalt of DEFINITION [5]: Let be a set The powe full set of deoted b PF s defed as the set of all the full subsets of The adalt of PF s the podut of the outs of the eleets DEFINITION [5]: Let be a set The powe set P of s the olleto of all the subsets of e have N P ff N If N = the N P If N the N k P whee [ ] k = [ N] the podut s take ove dstt eleets of N ad [] = ff [N] = ff N ad [ ]! [ N]!! The Powe set of a set s the suppot set of the powe set ad s deoted b P The followg theoe shows the adalt of the powe set of a set DEFINITION 3 [5]: Let P be a powe set daw fo the set { / / } ad P be the powe set of a set The adp = DEFINITION [5]: The au set s defed as Z whe k Z a{ } DEFINITION 5 [5]: Let be a set spae ad { I} be a olleto of sets daw fo The the followg opeato opeatos ae possble ude a abta olleto of sets / I a I I I 3 I / I I I I I / I I I Z / Z Ths s alled set opleet DEFINITION 6 [5]: Let ad be two sets daw fo a set the the atesa podut of ad s defed as { / / / } Hee the pa s epeated tes DEFINITION 7 [5]: A sub set R of s sad to be a set elato o f eve ebe / / of R has a out the podut of ad / elated to / s deoted b / R / hee = ad = ISSN: Page 76

3 The doa ad age of the set elato R o s defed as follows: s DoR { s t / R s / } DoR sup{ } s RaR { s t / R s / } RaR s sup{ } DEFINITION 8 [5]: A set elato f s alled a set futo f fo eve eleet / Do f thee s eatl oe / Ra f st / / s f wth the pa oug as the podut of ad ad P The s alled a ultset topolog o f satsfes the followg popetes: The set ad ae The set uo of the eleets of a subolleto of s 3 The set teseto of eleets of a fte subolleto of s Ad the odeed pa s alled a - topologal spae Eah eleet s alled ope set DEFINITION 0 [5]: Let be a - topologal spae ad N be a subset of The olleto N { N } s a -topolog o N alled the subspae -topolog DEFINITION [5]: If s a set the the - bass fo a - topolog o [] s a olleto of subsets of alled -bass eleets suh that: fo eah fo soe > 0 thee s at least oe -bass eleet suh that f whee The 3 DEFINITION 9 [5]: Let suh that 3 REARK [5]: If a olleto satsfes the odtos of -bass the the -topolog geeated b a be defed as follows A subset of s sad to be ope set f foe eah k thee s a -bass eleet suh that k Note that eah -bass eleet s tself a eleet of DEFINITION [5]: A sub olleto S of s alled a sub -bass fo whe the olleto of all fte tesetos of ebes of S s a -bass fo DEFINITION 3 [5]: The -topolog geeated b the sub -bass S s defed to be the olleto of all uos of fte tesetos of eleets of S THEORE [5] Let ad be a -bass fo a -topolog o The equals the olleto of all set uos of eleets of the -bass THEORE [5] If s a -bass fo the fo the -topolog o the the olleto N = { N } s a -bass fo the subspae -topolog o a subset N of DEFINITION [5]: A subset N of a - topologal spae s sad to be losed f the set N s ope e N s ope THEORE 3 [5]: Let be a - topologal spae The the followgs hold: The set ad the ept set ae losed sets Abta set teseto of losed sets s a losed set 3 Fte set uo of losed sets s a losed set THEORE [5]: Let N be a subspae of a - topologal spae The a set A s losed N fff t equals the teseto of a losed set of wth N DEFINITION 5 [8]: A set s alled - sgleto ad deoted b {k/} f : N s suh that = k ad = 0 fo all -{} Note that f k the {k/} s alled whole - sgleto subset of ad {/} s alled - sgleto whee 0 < < k DEFINITION 6 [8]: Let be a - topologal spae If fo eve two -sgletos {k / } {k / } suh that thee s G H suh that {k / } G {k / } H ad G H = the s alled -T -spae DEFINITION 7 [8]: Let ad N bed two -topologal spaes The set futo f : N s sad to be otuous f fo eah ope subset V of N the set f V s a ope subset whee f V s the set of all pots / fo whh f / V fo soe A ove III OPAT -TOPOLOGY ad be a -topologal spae A olleto of sub-sets of s sad to be a ove of f N DEFINITION 3: Let N EAPLE 3: Let ={//3/} ad ={ {/}{//}} Take ={{/3/}{//}} leal s a ove of Sub-ove ad be a -topologal spae Ad let be a ove of If DEFINITION 3: Let ISSN: Page 77

4 thee s soe oveg the we sa s a sub-ove of oveg EAPLE 3: Let ={//3/} ad ={ {/}{//}} Take ={{/3/}{/}{//}{//}} leal s a ove of Ad ={{/}{//}{/3/}} s a sub-ove of oveg Ope ove ad be a -topologal spae The s alled a ope ove of f oves EAPLE 33: Let ={//3/} ad ={ {/}{/}{//}{3/}{/3/}{/ 3/}{//3/}{//}} The s a -topologal spae osde ={{/}{//}{3/}{/3/}{//}} the s a ope ove of Aga ={{/} {//}{3/}} s a sub-ove of oveg EAPLE 3: [R] ad =R whee R s the set of eal ubes Ad let be a gve topolog o R Let us osde a whole sub-ultset of suh that = Now we show that the olleto PF DEFINITION 33: Let s a -bass Let The fo soe ad hee β Let β ad leal PF ad PF fo soe V It s lea that = V = V Hee 3 = PF V β Hee β s a -bass ad wll geeate a -topolog dued fo the PF gve topolog o R deoted b Ad f ={ α α Δ} s a ope ove of R the ={ β α Δ} s a ope ove of EAPLE 35: Slal we a show that f s a topologal spae Ad f [] wth = The PF whee s a whole sub-ultset of s a -bass ad hee wll geeate a -topolog dued fo the gve topolog o deoted b PF Ad f ={ α α Δ} s a ope ove of the ={ β α Δ} s a ope ove of D opat -topolog DEFINITION 3: Let [] ad let be a - topolog o The s sad to opat f fo a ope ove of thee s a fte sub-ove of oveg It s lea fo the defto that a - topologal spae whose suppot set s fte spae s opat A sub-ultset N of -topologal spae s sad to be opat f t s opat the subspae topolog dued fo EAPLE 36: Let us osde [R] ad =[0] whee R s the set of eal ubes Ad osde the usual topolog [0] [0] [0] o [0] The ={ ab ab = ab [0]} fos a bass of Now osde the whole sub-ultset of a b suh that = ab The ={ ab [0] a b } s a -bass se a b ab fo soe ab [0 ] a b let whee = a ad b = d The t obvous that u=a{a} < v={bd} ad u< othewse = Ad t s lea that = d u v Take 3 = u v The 3 = a b Hee wll geeate a topolog deoted b Now f [0 ] ={ } be a ope ove of [0 ] the leal ={ [0] } s a ope ove of [0] ad hee has a fte sub-ove ={ = 3 } oveg [0] se [0] [0] s opat The ={ = 3 } s also a fte sub-ove of oveg Hee s opat PROPOSITION 3: Let be a topologal spae suh that s a fte set Ad let st ad thee ae ftel a st > The PF s ot a opat -topologal spae PROOF: e kow Fo let PF us defe as a full subset of suh that Se so PF fo all s a ope ove of e wll ow show that t has o fte sub-ove Suppose t has a fte sub-ove ad let s the fte Theefoe ISSN: Page 78

5 sub-ove of oveg Let us hoose a st } ad > { The Ad theefoe a otadto se > Hee s ot a opat PF THEORE 3: A subset N of a -topologal spae s opat f ad ol f eve oveg of N b ope sets otas a fte oveg of N PROOF: If N s opat ad { } s a olleto of ope sets of oveg N the N s a olleto of elatvel ope sets N oveg N A fte subolleto N oves N b defto ad hee the olleto oves N ovesel f { V } s a olleto of elatvel ope sets oveg N thefo eah thee s a ope set suh that N V The olleto oves N so has a fte sub-ove oveg N The V N oves N THEORE 3: Let ad be a -topologal spae Ad let be a fal of losed sets possessg fte teseto popet The s opat ff V V fo soe The poof of ths theoe s ve eas ad sla to THEORE [9] so I skp ths poof THEORE 33: Let ad be a - topolog o The s opat ff eve bas ope ove has a fte sub-ove oveg PROOF: If s opat the the ase s tval ovesel let be a ope ove of ad be a base fo Eah s the uo of eta s s leal a bas ope ove of hpothess ths lass of s has a fte sub-ove Fo eah set ths fte subove we a selet a whh otas t The lass of s whh ase ths wa s evdetl a fte sub-ove of the ogal ope ove of ad hee s opat Now we wll pove that eve opat sub-set of set eas opat ude otuous tasfoato ut befoe dog ths we edefe the futo betwee sets DEFINITION 35: Let ad N Y ' A futo f : N s a elato st fo eah / a uque / wth / N st f / / e fo eah / a uque N suh that f / / whee N e wte a futo as f / / / f / / Defe age of f R hee Let R f N f { f / } f / f : N be a futowhee ad Y ' N Ad let A Let us deote A to be the whole subset of wth A A e defe the estto of f upo A b the estto of f o A e f A: A N s defed b f A k / f / whee k / A ad / A e k Hee we defe f A f A Let Y ' N N N ad f : N be a futo Defe f N f N { / { f / } N } Hee s ethe a whole subset of o a eot set Let f : N be a futo fo ' to N Y the t a be easl show that f A be two subsets of st A the f A f ad f A be two subsets of N st A the f A f THEORE 3: Let f : N be a futo fo a set to a set N Y ' The fo subsets A of ad D of N f A f A f f D f f D PROOF: It a be easl show that A A ISSN: Page 79

6 A A Now / f A / A st f / / whee / A o / st f / / se A A / A st f / / o / st f / / { / } f A o { / } f { / } f A f f A f A f Aga / f A f / f A o / f / A st f / / o / st f / / / A st f / / / } f A f A { A A se f A f f A Hee f A f A f Theefoe A ae two subsets of a set the f A f A f A f f A f Slal t a be show that f A f A f A f f A f Poof Is sla THEORE 35: Let f : N be a otuous futo fo the set to the set N Y ' whee ad N be two -topologal spaes Ad let be opat subset The f s opat N PROOF: Let be a ope ove of f The f Now we show that f f Let / f / } f { / f f f f Theefoe f f f f f Eah f s a ope whole subset of Now Theefoe { f } s a ope ove of ad hee has a fte sub-ove { f } oveg Theefoe f f f f Hee has a fte sub-ove f As oveg s abta so f s opat THEORE 36: Let ad be a - topolog o The eve losed whole subset of s opat The poof s tval so I skp t REARK: If we do t osde losed whole subset the the above theoe a be false Let us osde whee R set of eals st [0] ad w> ad > Now suppose the usual topolog [0] o [0] s gve Let be a sub-ultset of suh that ad ad ad osde [0 ] Now we wll show that [0] Fo a -bass Let The t s lea that suh that [0 ] Let whee ISSN: Page 80

7 ISSN: Page 8 asei: If ] [0 The 3 ] [0 aseii: If The st ] [0 The Now take 3 3 whee 3 ] [0 The leal 3 aseiii: If ] [0 ad The leal 3 Hee s a bass ad wll geeate a -topolog Let } { be a ope ove of otag bas ope sets If } { [0] oves the ad hee has a fte sub-ove oveg beause ] [0 s opat othewse st / Theefoe 0 st 0 0 whee 0 ] [0 Now se so o Hee ] [0 st ad ad Now osde } { The s a ope ove of st [0] Aga we kow that ] [0 s opat so has a fte sub-ove oveg Let us take # } \{ the leal # s a fte sub-ove of oveg Now osde the sets [0] 0 whee 0 0 ad ad The 0 N s ope ad hee N s losed Ad leal N ad N N Now osde ] [0 G 8 the G s a ope ove of N havg o fte sub-ove oveg N othewse 8 G wll have a fte sub-ove oveg the teval a otadto Hee N s ot opat

8 be a -T -spae { k / } ad F be a opat subset of st { k / } F The ope sets V st { k / } F V ad V THEORE 37: Let ad PROOF: Se s T so fo F ope sets V suh that { k / } ad { / } ad The the V V olleto { V F} s a ope ove of F Se F s opat so suh that { } ad F V V V Take The s a ope set suh that { k / } ad leal V Hee ope sets V suh that { k / } F V ad V IVONLSION I ths pape a oept of opatess s todued -topologal spae ad soe of ts popetes ae studed Thee s a huge sope of futue wok studg othe topologal oepts ths settg AKNOLEDGEENT The authos ae thakful to the Edtoal oad ebes fo aeptg ou pape The fst autho s thakful to the Depatet of atheats Vsva- haat The seod autho s thakful to G SAP New Delh Ida [Gad No F 50/3/DRS-II/05 SAP-I] REFERENES [] lad ad D ae ultset theo Note Dae Joual of Foal Log a [] lad ad D ae Real-valued ultsets ad fu sets Fu Sets ad Sstes b [3] lad ad D ae The developet of ultset theo ode Log [] S A El-Shekh R A-K Oa ad Raafat γ-opeato -topologal spae Ge ath Notes [5] K P Gsh ad Sul Jaob Joh ultset topologes dued b ultset elatos Ifoato Sees [6] K P Gsh ad Sul Jaob Joh O ultset Topologes Theo ad Applatos of atheats & opute See [7] K P Gsh ad Sul Jaob Joh Relatos ad futos ultset otet Ifoato Sees [8] K P Gsh ad Sul Jaob Joh Rough ultsets ad foato ultsstes Advaes Deso Sees 0-7 [9] A Ibah D Sgh J N Sgh A outle of ultset spae Algeba Iteatoal Joual of Algeba [0] S P Jea S K Ghosh ad K Tpath O the theo of bags ad lsts Ifoato Sees [] D Sgh A ote o the developet of ultset theo ode Log [] J ahata D Das Se opatess ultset Toplog av pept av:03560-avog ISSN: Page 8