Applied Matheatics 23 4 296-3 http://ddoiog/4236/a234975 Published Olie Septebe 23 (http://wwwscipog/joual/a) Modula Spaces Topology Ahed Hajji Laboatoy of Matheatics Coputig ad Applicatio Depatet of Matheatics aculty of Scieces Mohaed V-Agdal Uivesity Rabat Moocco Eail: hajid2@yahoof Received Apil 29 23; evised May 29 23; accepted Jue 7 23 Copyight 23 Ahed Hajji This is a ope access aticle distibuted ude the Ceative Coos Attibutio Licese which peits uesticted use distibutio ad epoductio i ay ediu povided the oigial wok is popely cited ABSTRACT I this pape we peset ad discuss the topology of odula spaces usig the filte base ad we the chaacteize closed subsets as well as its egulaity Keywods: Topology of Modula Spaces; 2 -Coditio; ilte Base Itoductio I the theoy of the odula spaces X the otio of 2 -coditio depeds o the covegece of the sequeces i odula space X Moe pecisely it eads: fo ay sequece i X if 2 li we have li 2 This coditio has bee used to study the topology of odula spaces see J Musielak [] ad to establish soe fied poit theoes i odula spaces see [2-7] Soe fied poit theoes without 2 -coditio ca be foud i [89] I this pape we peset a ew equivalet fo fo the 2 -coditio i the odula spaces X which is used to show that the coespodig topology is sepaate ad to establish soe associated topological popeties icludig the chaacteizatio of the -closed subsets as well as its egulaity The peset wok is a ipoved Eglish vesio of a pevious pepit i ech [] 2 Peliiaies We begi by ecallig soe defiitios Defiitio 2 Let X be a abitay vecto space ove K o ) A fuctioal : X is called odula if iplies a) fo ay X whe K ad b) e it fo ay eal t whe K c) y y fo ad 2) If we eplace c) by the followig y y fo ad the the odula is called cove 3) o give odula i X the X X as is called a odula space 4) a) If is a odula i X the if u u u is a -o b) Let be a cove odula the is called the Luebug o if u u 3 Topology τ i Modula Spaces I this sectio we itoduce the popety fo a odula which will be used to show that the coespodig topology oted by o odula space X is sepaate ad to chaacteize thei closed subsets We begi with the followig Popositio 3 Coside the faily B whee B X The ) The faily is a filte base 2) Ay eleet of is balaced ad absobig utheoe if is cove the ay eleet of is cove Poof ) is a filte base Ideed we have Copyight 23 SciRes
A HAJJI 297 a) because ay B b) Let B ad B 2 be i ad set if The fo ay z we have 2 ad theefoe B z z 2 2 z B B 2 That is B B B Hece is a filte base fo the eistece of B B B B 2 2) Let B a) B is balaced Ideed fo give e i with ad ad give B we have e i < This eas that B b) B is absobig Ideed fo give X we have li Whece fo all > thee eists > < < ad < Hece thee eists > B This shows that B is absobig Now assue that is i additio cove ad let B o give y B ad we have the y y < y B Thece B is cove Defiitio 3 We say that satisfies the popety if fo all > thee eist L > ad > such that y fo evey y satisfyig L ad y< Theoe 3 Assue that the odula satisfies the popety The X is a sepaate topological vecto space Poof I Popositio 3 we have see that the faily is a filte base ad futheoe ay eleet of is balaced ad absobig O the othe had fo ay B thee eists > B B B I fact let ; > > Sice satisfies the popety thee ae L > ad > fo < L ad y< we have y Thus if we set < if L we see that fo z y B B y with We obtai y z B This iplies z ad L Thece z This ifes that z B ad so B B B Hece the faily is a fudaetal syste of eighbohoods of zeo the the uique topology defied by i X is give by G G X if G the V V G so that X is a topological vecto space To show that X is sepaate let y i X y ad assue that fo ay V eighbohood of ad V y eighbohood of y we have V Vy So that oe ca coside fo cetai zb yb * The z y z Sice satisfies the popety the thee eist fo ay two eals L ad y fo evey y satisfyig < L < 2 ad y < Now set Y y ad X z ad ote that we have X z Y X y z It follows that fo ay if L we have 2 Y y z 2 2 2 This ifes that y Thus y fo abitay ad the = y a cotadictio sice by hypothesis y Theefoe thee eist eighbohoods V of ad eighbohood V y of y Copyight 23 SciRes
298 A HAJJI V V y τ Covegece ad Chaacteizatio of τ-closed Subsets of X ρ We begi by ecallig soe eeded defiitios of the -covegece ad the -closed subsets of the the odula space X (see fo eaples [2-8]) Defiitio 32 Let X be a odula space ) A sequece i X is said to be -co- veget to deoted by 2) A subset B of ay sequece if as X is said to be -closed if fo B we have B We deote by B the closue of B i the sese of 3) A odula is said to be satisfyig the atou popety if y ad y y liif y as I this sectio we defie the -covegece the - closed subsets of X ad we show that the topology defied by -closed i the defiitio befoe oted by ad the topology ae the sae topology The atuel covegece i the sese of the topology ad -closed subsets of X ae give by the followig defiitios Defiitio 33 A sequece i X is said to be coveget to i the sese of the topology (o siply -coveget) if fo ay > thee eists N B wheeve > N Note that the popety is a ecessay coditio to show the uiqueess of the liit whe eists Thus the -covegece eed the popety ad it is easy to see that -covegece ad -covegece ae equivalet Defiitio 34 Let be a odula satisfyig the popety A subset B of X is said to be -closed if ad oly if the coplietay of B i X oted by B is a eleet of The followig lea shows that the popety akes sese i the theoy of odula spaces Lea 3 Let be a odula ad X be a odula space The satisfies the 2 -coditio if ad oly if satisfies the popety Poof To pove if let be a sequece i X as This iplies that fo all > thee eists fo ay > we have 2 if L Now take X ad Y 2 fo ay It follows 2 X Y X if L whe- This yields Y 2 2 Whece the sequece 2 eve teds to zeo as goes to ad theefoe satisfies the 2 -coditio o oly if let be a odula satisfyig the 2 -coditio ad suppose that thee eists > such that fo ay L > ad fo ay > thee eist y X satisfyig < L y< ad y I paticula fo L thee eist y X ad y y which iplies ad y as Howeve we have y y 2 y2 Now sice satisfies the 2 -coditio the as It follows that y y as which cotadicts the fact that y > fo ay ially fo all > thee ae L > ad > if < ad y < we have y < This copletes the poof of Lea 3 I the followig theoe we show that the -topology ad the -topology ae the sae Theoe 32 Let be a odula satisfyig the 2 - coditio ad X the is -closed if ad oly if is -closed The followig esult is eeded to show Theoe 32 Popositio 32 Let be a odula satisfyig the 2 -coditio ad a -closed subset of X The Poof o B X we have C C is a ope set of the -topology X X B B B B > Copyight 23 SciRes
A HAJJI 299 ially > B Poof of Theoe 32 Let be -closed ad be a sequece i The fo ay > thee eists fo evey > we have B This iplies that > B Whece akig use of Popositio 3 we get that Covesely assue that is ot -closed the is ot a ope set fo the -topology Thee eists the satisfyig B C X ad so B fo ay > Theefoe fo k thee eists k B k Thece the obtaied sequece satisfies This iplies which is i cotadictio with the fact that I coclusio is -closed Reak 3 Obseve that satisfies the -coditio satisfies the popety As cosequece we see that ude the assuptio that satisfies the popety we have topology topology The defiitios of -covegece ad -closed subsets of X eed the hypothesis that satisfies the 2 -coditio The followig esult shows that the odula space X is a egula space Theoe 33 Let be a odula satisfyig the 2 - coditio A be a -closed subset of X ad A The thee eists a ope eighbohood V of V A I ode to show the theoe above we eed the followig esult Popositio 33 Let be a odula satisfyig the 2 -coditio ad A The X A if y ya if ad oly if whee A is the closue of A fo the -topology Poof We have The fo ay A A if y ya 2 thee eists y A y < this iplies that thee eists a sequece y A Ivesely let ists a sequece A y y Whece A the by Theoe 32 thee e- A foe fo ay thee eists Hece y A y ; A thee- Poof of the Theoe 33 By Popositio 33 A if ad oly if A > Net sice satisfies the 2 -coditio the by Lea 3 fo > 3 thee eist L ad if L ad y we have y Moe- * ove thee eists if L wheeve Now let a 3 ad we coside the ope eighbohood of V B Suppose et that V A ad let y V A Sice A is closed we ake use of Popositio 3 to ehibit a sequece y A y y So that oe cosides X y y ad Y y Sice y A ad A the Y O the othe had ote that X y y if L wheeve ad X Y y if L Theefoe 2 Y y y 3 3 wheeve a cotadictio Thus V A Reak 32 If satisfies atou popety the B B X is a closed ball of the topology We ote by B all closed ball ceteed at with the adius (see [7]) Coollay 3 Ude the sae hypotheses of Theoe f Copyight 23 SciRes
3 A HAJJI 33 ad if the odula satisfies atou popety the V A Poof Makig appeal of Theoe 33 thee eists V B V The we A have f V B Ideed let y V ad ote that fo Popositio 3 thee eists a sequece y Bf which iplies that that y y y y Ideed it is easy to see Y y y ad sice satisfies the 2 -coditio we have also X 2 y y Thece fo thee ae L ad such that 2 X if L By Popositio 3 thee eists y y y Moeove the sequece B y V satisfyig y y Hece y V ially we take the sae aguets as i the poof of Theoe 33 we have V A REERENCES [] J Musielak Olicz Spaces ad Modula Spaces Lectue Notes i Matheatics Vol 34 983 [2] A Ait Taleb ad E Haebaly A ied Poit Theoe ad Its Applicatio to Itegal Equatios i Modula uctio Spaces Poceedigs of the Aeica Matheatical Society Vol 28 2 pp 49-426 doi:9/s2-9939-99-5546-x [3] A Razai ad R Moadi Coo ied Poit Theoes of Itegal Type i Modula Spaces Bulleti of the Iaia Matheatical Society Vol 35 No 2 29 pp -24 ad [4] A Razai E Nabizadeh M B Mohaadi ad S H Pou ied Poit of Noliea ad Asyptotic Cotac- Y X Yif L tios i the Modula Space Abstact ad Applied Aaly- 2 sis Vol 27 27 Aticle ID: 4575 wheeve the [5] A P aajzadeh M B Mohaadi ad M A Noo ied Poit Theoes i Modula Spaces Matheati- Y y y cal Couicatios Vol 6 2 pp 3-2 wheeve It follows ad hece if L 2 2 2 2 Theefoe yy B Bf Ivesely let [6] M A Khasi Noliea Seigoups i Modula uctio Spaces Thèse d'état Dépateet de Mathéatiques Rabat 994 [7] M A Khasi W Kozlowski ad S M-Reich ied Poit Theoy i Modula uctio Spaces Noliea Aalysis Theoy Methods ad Applicatios Vol 4 No 99 pp 935-953 [8] Lael ad K Nououzi O the ied Poits of Coespodeces i Modula Spaces ISRN Geoety Vol 2 2 Aticle ID: 53254 doi:542/2/53254 y y Bf [9] M A Khasi Quasicotactio Mappigs i Modula Spaces without 2 -Coditio ied Poit Theoy ad Applicatios Vol 28 28 Aticle ID: 9687 V Bf y Bf [] A Hajji oe Equivalete à la Coditio 2 et Cetais Résultats de Sépaatios das les Espaces Modulaies 25 http://axivog/abs/atha/59482 Copyight 23 SciRes