Bitopological spaces via Double topological spaces

Transcription

1 topologcal spaces va Double topologcal spaces KNDL O TNTWY SEl-Shekh M WFE Mathematcs Department Faculty o scence Helwan Unversty POox 795 aro Egypt Mathematcs Department Faculty o scence Zagazg Unversty Egypt Mathematcs Department Faculty o Educaton n Shams Unversty Egypt Modern cademy For Engneerng &Technology n Maad Egypt dral_kandl@yahoocom bstract: n ths paper we shall study some btopologcal propertes va double topologcal spaces We characterze the notons o parwse contnuous resp parwse open parwse closed P contnuous P - open P - closed or short by a double contnuous resp double open double closed mappngs between double topologcal spaces lso we characterze the notons o P - contnuous resp P -open P - closed by a supra double contnuous resp open closed mappngs between supra double topologcal spaces Fnally we nvestgate the relatonshps between these types o mappngs and gve some counter examples [ KNDL O TNTWY SEl-Shekh M WFE topologcal spaces va Double topologcal spaces Journal o mercan Scence 0;7:79-798] SSN: Keywords: btopologcal spaces parwse contnuous mappngs supra- topologcal spaces parwse open mappngs parwse closed mappngs ntroducton The noton o a btopologcal space X that s a set X equpped wth two topologes and was ormulated by J Kelly n [] There are several hundred works ndcated to the nvestgaton o btopologcal spaces The book [] s a versatle ntroducton to the theory o btopologcal spaces and ts applcatons Flou set stems rom some lngustc consderatons o Yves Gentlhomme about the vocabulary o a natural language [6] E E Kerre [3] ntroduced the mathematcal denton o lou sets and bnary operatons on t n ths paper we ollow the suggeston o J G Garca and S E Rodabaugh [5] that "double uzzy set" s a more approprate name 'than "ntutonstc uzzy set" and thereore adopt the term "double-set" or the lou set and "double-topology" or the lou topology There are several hundred works ndcated to the nvestgaton o double topology eg [9 0 5] n ths paper makng use the relaton between btopologcal spaces TS`s or short and double topologcal spaces DTS or short we characterze the notons o P - contnuous resp P -open P - closed mappngs by a double contnuous resp open closed mappngs lso we ntroduce the noton o supra double topologcal space SDTS or short and characterze the noton o P - contnuous resp P -open P - closed mappngs by supra double contnuous resp open closed mappngs Fnally we nvestgate the relatonshp between these types o mappngs and gve same counter examples Note that or the concepts and results that are used but not stated here we reer to [[] [8] [4]] Prelmnares: n ths secton we shall present the undamental dentons and concepts whch wll be needed n the sequel Denton [9] double set D- set or short s an ordered par P X P X such that The amly o all double sets on X wll be denoted by DX e DX = { P X P X: } The double set X =X X s called the unversal double set and ϕ = ϕ ϕ s called the empty double set Denton [9] Let = = D X Then: = = = = 3 Υ = Υ Υ 79 edtor@amercanscenceorg

2 4 = 5 = where s the complement o 6 Let η η P X The double product o η and η s denoted byη η and s η } Denton 3[9] Let : X Y be a mappng and let DX and DY The mage o dened dened by η η = { η : by = and the premage s dened by = Denton 4 [9] Let X be a non- empty set Then a amly D X s called a double topology on X t satses the ollowng axoms: = = X X η η then η 3 { s : s S} η then Υ s S s η η satses the axoms 3 then t s called a supra double topology The par X η s called a double topologcal space Each member o η s called an open double set n X The complement o an open double set s called a closed double set For any D X the double closure o s denoted by and s dened by = { η and } Denton 5 [9] mappng : X η Y s called: doubl contnuous D contnuous or short η whenever doubl open D open or short whenever η doubl closed D closed or short whenever η Remark: [] Every DTS X η dene a TS whch s X π π where π ={ V X : V X st V V η } and π ={ V X : V X ST V V η } onversely every TS X dene a DT = { D X: } on X assocated wth Theorem 6[9] : X η Y contnuous uncton then : X = are contnuous unctons Theorem 7[9] Let X η be any DTS Then η s a D- π Y π be a DTS and Y : X Y η s a D- contnuous uncton : X Y π = are contnuous unctons Denton 8[3] mappng : X Y s called P-contnuous resp P -open P- 79 edtor@amercanscenceorg

3 closed s - contnuousresp open closed = Denton 9 [4] mappng : PX PX s called supra- nteror operator t satses the ollowng axoms : X = X 3 4 = Proposton 0 [4] Let X be a TS Then = = { U Υ U : U } s a supra topology on X and X s the STS-assocated to X The operator : PX PX dened by = 0 Υ 0 0 s the - nteror o = s a supra operator n whch = { X: = } Proposton [4] Let X be a TS and X ts assocated STS Then : P X P X dened by = s a supra- closure operator whch nduces the supra topology Remark n [] the author used the noton o parwse open n a TS whch means that: s p- open = U Υ U U = n [4 7] we used the same noton under the name o P - open or supra open n X where s a supra topology generated by and We say that X s P - open supra open Denton 3[4] mappng : X Y s called -contnuous P U P -open V U whenever wheneverv P -closed V whenever V 3 Operaton on DTS and SDTS: Denton 3 mappng : DX DX s a called double closure operator t satses the ollowng axoms : ϕ = ϕ 3 Υ = Υ 4 = satses the axoms 4 and the ollowng axom t s called a supra double closure operator: 3 Υ Υ Theorem 3 Let X be a TS The operator : DX DX dened by 0 0 =[ ] 0 D X s a double nteror operator whch generates the double topology on X Proo: s a sample we prove the dualty o the property 3 denton 3 above e we prove that = The proo o the other parts are smlar `` s a well dened map snce `` 0 [ ] [ ]= = [[ ] [ ] 0 ] 0 ] 0 = 0 0 [ 0 0 =[ ] [ 0 ] 0 = [ = [ ] ] 0 Then s a double nteror operator and hence t generates a double topology η on X where η ={ = }= { [ 0 ] 0 ={ 0 0 = } 0 0 { 0 = 0 ={ }= = }= 0 = } orollary 33 Let X be any TS Then the operator : D X D X dened by: = Υ D X s a double closure 793 edtor@amercanscenceorg

4 operator generates the double topology on X Theorem 34 Let X be a TS and let X ts assocated supra topologcal space Then the : D X DX dened by operator = where 0 = Υ 0 = s a supra double nteror operator such that = Proo: The proo that s a supra-nteror operator ollows rom the denton o and the act that s a supra nteror operator prop 9 For the proo o = Then let = = and = onversely Let = SO = Then = = Hence = and thereore = So consequently = and orollary 35 Let X be a TS Then the operator : D X D X such that = where = = s a supra double closure operator such that = Theorem 36 Every double closure operator : D X D X generates a TS X where = { X: = D X} = Proo: Straghtorward 4 The relatons between P contnuous resp P open P closed mappngs and Duble contnuous resp duble open double closed mappngs: n ths secton we characterze the noton o P - contnuous resp P -open P -closed by a D- contnuous resp D-open D-closed mappngs Y Theorem 4 mappng : X s parwse contnuous Y s double : X contnuous Proo: Let : X contnuous and = and = Y t So Hence : X Y s double contnuous onversely: Let : X be P - Then Y be D-contnuous and let G Then G Y So G Y = G X Hence G and s contnuous uncton lso : X Y let G Then ϕ ϕ G = ϕ G G So Hence G and : X Y s contnuous uncton Thereore : X Y s P - contnuous Theorem 4 Let : X Y be a mappng Then the ollowng condtons are equvalent: : X contnuous Y s double D X edtor@amercanscenceorg

5 4 D Y 5 D Y Proo: : Let = Then ] [ ] [ 3: Let D X Snce Then [ ] [ ] [ ] by 3 4: Let D Y Take = usng 3 we have [ ] 4 : Let G Then lso ϕ G = usng 4 we have: ϕ ϕ G G = ϕ G Y ϕ G G ϕ G Hence So G X ϕ G Thereore G and the mappng : X Y s contnuous Smlarly we can show that : X Y s contnuous So : X Y s P- contnuous ccordng to theorem 4 s double contnuous 5: Let be double contnuous So : X Y s P contnuous Let D Y Then 0 0 [ = 0 ] 0 = : Let Y Y Y Y Then Y and thereore 0 = 0 = Y Y = Hence : X Y then ϕ Y Y and the mappng s contnuous Smlarly and apply the condton we have ϕ Hence and the s contnuous mappng : X Y So : X Y s P contnuous So accordng to theorem 4 s double contnuous y a smlar way as n theorems 4 4 we have the ollowng theorems 795 edtor@amercanscenceorg

6 Theorem 43 mappng : X Y s P- open : X Y s D open mappng : X Y s D- open [ H ] [ H ] H D X Theorem 44 surjecton mappng : X Y s D- open and D-contnuous [ H ] = [ H ] H D X Theorem 45 mappng : X Y s P- closed : X Y s D-closed mappng : X Y s D- closed [ H ] [ H ] H DX Theorem 46 mappng : X Y s a D- closed and D-contnuous [ H ] H ] H DX = [ orollary 47 Let X and Y be double topologcal spaces Then : X Y : X homeomorphsm s a double homeomorphsm Y s P- 5 The relatons between P -contnuous resp P - open P -closedmappngs and supra double contnuous rep supra double open supra double closed mappngs: n ths secton we shall study the relaton between P -contnuous resp P -open P - closed mappngs and supra- double topologcal spaces The proos o the ollowng results are smlar to the proo o the results n secton 4 So we prove theorem 5 as an example and we shall omtte the proo o the others Theorem 5 Let X and Y be btopologcal spaces lso let X and Y be ther assocated supra-topologcal spaces The ollowng equvalent: : X Y s P - contnuous s SD- : X contnuous 3 Y 4 D X 5 [ D Y 6 ] D Y Theorem 5 Let X and Y be btopologcal spaces Then the ollowng are equvalent: : X Y s P open s a SD- : X Y open 3 [ H ] [ H ] H D X Proo: Let = = So = Then Hence : X Y s a SD-open uncton 3 Snce H H H D X Then [ H ] H [ H ] = [ H ] H Hence [ H ] [ H ] H D X 3 Straghtorward 796 edtor@amercanscenceorg

7 Theorem 53 Let X and Y be btopologcal spaces mappng : X Y s SD- open and SD-contnuous [ H ] = [ H ] H D X Theorem 54 Let X and Y be btopologcal spaces Then the ollowng are equvalent: s a SD- : X closed : X Y Y s P closed H [ H ] H DX Theorem 55 Let X and Y be btopologcal spaces mappng : X Y s SD- closed and SD- contnuous H = [ H ] H DX orollary 56 Let X and Y be btopologcal spaces Then : X Y s a SDhomeomorphsm : X Y s P - homeomorphsm uncton 6 Relaton between D contnuous open closed mappngs and SD contnuous open closed mappngs: Theorem 6 Let X and Y be btopologcal spaces and let : X Y be P- contnuous resp P- open Then : X Y s SDcontnuous resp SD-open Proo: t ollows rom the denton o P- contnuous resp P- open and SD-contnuous resp SD-open Theorem 6 Let X and Y be btopologcal spaces and let : X Y be P- closed and njecton Then : X Y Proo: Let : X Y P- closed and njecton Let H s SD-closed be a Then H = K K G G such that K G = Then H = K K G G = K K G G K K G G K G H Y Hence : s SD-closed X Note that the mappng may be SD-contnuous SD-open and SD-closed but not double contnuous double open and double closed mappng as shown n the ollowng example: Example 63 Let X={a b c} ={ ϕ X {a} {b c}} ={ ϕ X {c} {a b}} lso let Y= {p q r} ={ ϕ Y {r} {p q}} ={ ϕ Y {p} {q r}} Then = {ϕ X ϕ {c} ϕ {a b} ϕ X {a} {a b} {a} X {b c} X} = { ϕ Y ϕ {p} ϕ {q r} ϕ Y {r} {q r} {r} Y {p q} Y} let : X Y such that a = p b= q c=r Then s not D-open snce ϕ {c} ϕ {r} let ϕ {c} = = { ϕ X {a} {b c} {c} {a b}{a c}} Then s not a topology snce {b c} {a b}= {b} lso = {ϕ Y {r} {p q} {p} {q r} {r p}} s not a topology snce {p q} {q r}= {q} Then ={ϕ ϕ {a} ϕ {b c}ϕ {c} ϕ {a b} ϕ {a c} ϕ X {a} {a} {a} {a b} 797 edtor@amercanscenceorg

8 {a} {a c} {a} X {c} {c} {c} {b c} {c} {a c} {c} X {b c} {b c} {b c} X {a b} {a b} {a b} X {a c} {a c} {a c} X X } and ={ϕ ϕ {r} ϕ {p} ϕ {p q} ϕ {q r} ϕ {r p} ϕ Y {r} {r} {r} {r q} {r} {r p} {r} Y {p} {p} {p} {p q} {p} {r p} {p} Y {p q} {p q} {p q} Y {q r} {q r} {q r} Y {r p} {r p} {r p} Y Y } Let : X Y s SD-contnuous but : X contnuous snce Y ϕ {p} but ϕ {a} Let : X s SD-open but : X snce ϕ {c} Y Then s not D- ϕ {p} = Then Y s not D- open ϕ {r} Fnally Let : X s SD-closed but : X snce {a b} X but ϕ {c} = Y Then Y s not D-closed but {a b} X = {p 4 El-Shekh S new approach to uzzy btopologcal space normaton Scence Garca JG and Rodabaugh SE Ordertheoretc topologcal ategorcal redundances o nterval-valued sets grey sets vague sets nterval-valued `` ntutonstc`` sets `` ntutonstc`` uzzy sets and topologes` Fuzzy sets and Systems Gentlhomme Y 968 Les ensembles lous en lngustque ahers Lngustque Theoretque pplqee 5 pp Kandl HFawaz and S El-Shekh On btopologcal spaces Proc O ssut Frst ntern-conpart V Kandl Nouch and SEl-Shekh On uzzy btopologcal spaces Fuzzy sets and systems Kandl and Tantawy O and Waae M On lou NTUTOST topologcal spaces JFuzzy mathematcs Vol5 No Kandl Tantawy O and Waae M On Flou NTUTOST ompact Space J o Fuzzy Mathematcs Vol7 No Kandl Tantawy O and Waae M Flou separaton axoms J The Mathematcal nd Physcal Socety O Egypt submtted Kelly J : btopologcal space Pro London MathSoc Kerre EE Fuzzy sets and approxmate reasonng Lectures notes Unversty o Gent elgum 98833p 4 Qangyuan Yuab Dayou Luab and Janzhong henc uzzy spatal regon model based on lou set dvances n Spato- Temporal nalyss Volume 5008 pp Sadk ayhan and Doğan Çoker Parwse separaton axoms n ntutonstc topologcal spaces Hacettepe Journal o mathematcs and Statstcs vol 34S //0 q} Y Reernces DvalshvlP topologcal Spaces: Theory Relatons wth Generalzed lgebrac Structures and pplcaton Mathematcs studes 005 Datta M : Projectve btopologcal space J ustralan Math Soc Datta M: Projectve btopologcal space J ustralan Math Soc edtor@amercanscenceorg