&nbsp; بواسطة&nbsp; مفهوم&nbsp; المجموعات&nbsp; المفتوحة من النمط&nbsp; nano&nbsp; topological&nbsp; spaces)&nbsp; &nbsp;) في الفضاءات التبولوجية&nbsp; من&nbsp; النمط -open&nbsp; sets)&nbsp; &nbsp;( nano&nbsp; &nbsp;في هذا العمل قدمنا موضوع جديد في نظرية&nbsp; البعد&nbsp; (حسب علمنا )هو)( &nbsp;و بالإضافة&nbsp;&nbsp; لذلك&nbsp; سوف&nbsp;&nbsp; سندرس&nbsp; سلوك&nbsp; هذه&nbsp; الفضاءات&nbsp; تحت&nbsp;&nbsp; تأثير&nbsp; نوع&nbsp;&nbsp; معين&nbsp; من&nbsp;&nbsp; الدوال&nbsp; المستمرة&nbsp; و التي&nbsp; تدعى -continuous mapping ) &nbsp;&nbsp;&nbsp;( nanoومن خلال بحثنا في الخواص المقدمة في البحث&nbsp; رأينا ان هذه الخواص موازية الى بعض خواص نظرية البعد في التبولوجيا العامة.In this paper the concept of nano topology is studied in a different way . The notation of nano&nbsp; -open is established. Also the concept -covering dimension is defined. The aim of this paper is to&nbsp; introduce and define a new type of covering dimension by using nano -open sets namely, a nano -covering dimension in a nano topological space and find some relations to other concepts. Some properties and characterization of this covering dimension are obtained

Jhonny, Saied A.

Mohammed, Sanaa N.

Journal of University of Babylon for Pure and Applied Sciences

English

Journals of University of Babylon

Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٦٩On vague Order and Decomposition Mapping by -Open Sets Using Nano      Saied A. Jhonny    Al-Qadisiyah Education – Iraq          Sanaa N. Mohammed       Education of Babylon – Iraq  al.sharfawi @gmail.com  Abstract          In this paper the concept of nano topology is studied in a different way . The notation of nano  -open is established. Also the concept -covering dimension is defined. The aim of this paper is to  introduce and define a new type of covering dimension by using nano -open sets namely, a nano -covering dimension in a nano topological space and find some relations to other concepts. Some properties and characterization of this covering dimension are obtained.  Keywords: Nano -open , Nano -closure , Nano -interior , Nano -normal , Nano -continuous  .  2000 Mathematics Subject classifications 54 A 20 , 54 C08 , 54 C10 .   ةصلاخلا      طمنѧلا نѧم ةѧحوتفملا  تاѧعومجملا  موѧھفم  ةطѧساوبnano  topological  spaces)   (   نѧم  ةѧیجولوبتلا تاءاضѧفلا يѧفطمنلا -open  sets)   ( nano    نمدѧق لѧمعلا اذѧھ يف  دѧعبلا  ةѧیرظن يѧف دѧیدج عوѧضوم ا) اѧنملع بسѧح( وѧھ)(   و وѧس  كلذѧل   ةفاѧضلإاب ىعدѧت  يѧتلا و  ةرمتسѧملا  لاودѧلا   نѧم  نیѧعم   عوѧن  ریثأѧت   تѧحت  تاءاضѧفلا  هذѧھ  كولѧس  سردنѧس   ف-continuous mapping )    ( nano ضعب ىلا ةیزاوم صاوخلا هذھ نا انیأر  ثحبلا يف ةمدقملا صاوخلا يف انثحب للاخ نموةماعلا ایجولوبتلا يف دعبلا ةیرظن صاوخ.  لا تاѧملكلا ةѧیحاتفم:-   وناѧن-open ,  ، وناѧن و-closure  ، وناѧن و-interior  ، وناѧن و-normal  ، وناѧن و-    continuous.   Introduction The[ Lellis and Carmel, 2013] introduced of  nano topological space in relation to a subset  of a universal   around a lower and upper approximations of   . The boundary region of  with respect to an equivalence relation  is the set of all objects[ Alias and HajiM, 2011].  The  elements of a nano  topological space are known nano open sets , and also close  sets, nano interior  and nano closure of a set[ Cadas et al.,2003]. was also introduced by the same other the weak forms of nano open set namely, -open set. Throughout this paper  and  are non-empty sets, a finite universes  ) and ; )and  denote the families of equivalence classes by equivalence relations (  and (  respectively on  and ( .   and  are the nano topological spaces on  and  with respect to  and  resp. [Lellis and Carmel,2013]. It is clear that each open set is -open but the contrary is not true. [Najasted,2013]  has  shown  that the family of  all  -open sets is a topology on . The dimension function was investigated by Pears  A. [Pears, 1975].  The aim of this paper is  a new notation for nano -open sets are discussed. :Basic definitions and notations.  2            We introduce some elementary concept which we need in our work. Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٧٠2.1. Definition: [ Lellis and Carmel,2013]        Let (  be a non-empty finite set of objects called the universe and  be an equivalence relation on ( ) named as the indiscernibility relation. Elements belonging to the same equivalence class are said to be indiscernible with one another , i.e. (   denotes the family of equivalence classes of  by  ).  The pair  is said to be the approximation space. Let ).  (i) The lower approximation of  with respect to is the set of all objects, which can be for certain  classified as (  with respect to  and it is denoted by[  ].  [ ,where denotes the equivalence class determined by  .    (ii)  The upper approximation of  with respect to is the set of all objects which can be for possibly classified as  with respect to  and it is denoted by  .  That is  (iii) The boundary region of   with  respect  to  is  the  set of all objects , which can be  classified neither as  nor as not - with respect to  and denoted by , . ] 2.2. Theorem :-  [Lellis and Carmel, 2013]        If  is an approximation space and(  , then: (i)  ; (ii)  ; ; (iii)  ; (iv) ] ; (v)  ; (vi)[  ; (vii) If [ , then   and  .  (viii)[   and   .  (ix)[  ;[ ; i.e. The following diagram shows that the relations among the difference types of sets above .  Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٧١   2.3. Theorem :- [ Lellis and Carmel,2013 ]        Let(  be the universe,  be an equivalence relation on  and  ,  . Then satisfies the following axioms :    (i) }  ;  (ii)  The union of the elements of any sub collection of  is in . (iii) The intersection of the elements of any finite sub collection of  is in .  Thus[ is a topology called nano topology on  with respect to  . We call is a nano topological space . The elements of[  are called nano open ,   is called the dual nano topology of[ . 2.4. Example:         Let   and  where   with the nano topology[   ,then the nano closed sets are(  and  .   2.5. Definition:-  [ Lellis M.T.and Carmel R. , 2013 ]         Let  be a nano topological  space  with respect  to(   and if ( , then the nano  interior of(  is defined as the union of all nano open subsets of  and it is denoted by  .  Also, the nano closure of   is defined as  the intersection of all nano closed sets containing (  and it is denoted by . Now, we say that  is nano -open set , if  ( . The collection of all nano -open subsets of will be denoted by  and its need not to be the  forms a nano  -topology on . The complement of the nano -open  sets is nano -closed subset  i.e.(  is nano -closed set , if{ . 2.6. Remark:         The intersection of two a nano -open subsets need not to be a nano -open, as the following example shows: Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٧٢Let (  , with  and  .  Then nano topology on(  is , then the sets and are nano -) open sets in( , but their intersection is not nano -open . Also, the intersection of two a nano -closed sets need not to be a nano  -closed as the following example shows: Let(   , with   and  .    , the sets and are nano -closed sets but their intersection is not nano -closed. i.e. [ is supra nano topology on( .  2.7. Definition:         A nano topological space  is called nano -multiplicative , if arbitrary intersection of nano -open ( nano -closed ) sets of is also nano  -open ( nano -closed ) set and it is denoted by ( -space). If  be an - space, then the[ becomes nano topology on( and called nano -topology denoted by .  2.8. Definition :- [ Revathy and Ilango, 2015]         Let  be a nano topological  space  with respect  to   and if   , then the nano  -interior  of    is  defined  as  the union of all  nano  -open  subsets  of    and  it  is  denoted  by  .  Also , the nano -closure of  is defined  as the intersection of all nano  -closed sets containing   and it is denoted by . Also , the nano -closure of  is defined as the intersection of all nano -closed sets containing   and it is denoted by . For any subset  of a nano topological space  , the following are satisfied: (i)  . (ii)[  need not to be a nano -closed for all(  . (iii){  for all  . (iv)[  for all  . (v) If( , then[  for all  . (vi)  is a nano -open for all  . 2.9. Remark :- [Lellis and Carmel,  2013] (i) Every nano open subset of  is nano -open.  (ii) Every nano closed subset of  is nano -closed. (iii) If[  ], then .         The converse (i , ii) of the remark above is not true as the following example shows:    Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٧٣2.10. Example:          Let(   , with   and  .    , the nano closed sets are  and  .  Clearly ,   .   2.11. Theorem:             If  and  be a nano  -open sets in a space  , then  it is clear  nano -open in  by Proof: Since  and  are -open sets,(  and  .  Then: , we have  is nano -open in(  .   2.12. Theorem:            If  and  be a (nano open) nano  -open sets respectively in an -space  respectively, then  is nano -open set in  . Proof: Since  be a nano open set and  be a nano  -open , we have  is nano closed and  is nano -closed in . By remark (2.9.ii) , then  and  are nano - closed sets in . Then  i.e.  is nano -closed set . This implies that  is nano -open . Similarity, if  and  be a (nano closed) nano  -closed sets respectively in an -space above , we have(   is nano -closed set in  .  2.13. Corollary:       If  and  be nano open in an( -space  , then(  is nano -open set in . Proof:  Follows from theorem (2.12) .  2.14. Definition:         A nano topological  space is said to be nano  -normal , if for each  pair  of disjoint nano -closed sets(  and(  of(  )  , there are disjoint  nano -open  sets ) and  such that(  and(  ).  2.15. Theorem:           A nano topological space is nano -normal if and only if for every  nano   -closed set(  in( and each nano -open sets in( ) such that (  , there is a nano -open set(  )such that [ . Proof: Straightforward.   Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٧٤2.16. Definition:    A family   of subsets of a nano topological space   is said to be a nano - locally finite family, if for  each  point  , there is nano  -open set  of    such that  the  set  is finite .  2.17. Definition:        Let  be a nano topological space and be a cover of  , then  is called a nano -refinement of   if, and only if ,it’s a cover of   and for each , there is  such that . 3. Nano -Covering  Dimension:         In this section we will introduce necessary definition and every nano topological space is an - space. Let  be a set  and be a non-empty family of subsets of , the order of is the largest integer    such that( )for some( ) members ( of    [Pears R. 1975].   3.1. Definition:          The  nano  -covering dimension  of a nano  topological space  is  the least positive integer  such that every finite nano -open cover of  has a nano -open refinement of order  .  We shall denote the nano -covering dimension of by .   (i) ( , whenever (  ,   (ii)( if and only if  , but it is no  .  (iii)( if and only if   for no   .  3.2. Example:         Let(  and  where   with the nano topology   , clearly that   and(   3.3. Theorem:         Let be a nano topological space , if  , then it is nano -normal. Proof: Let(  be two nano -closed sets in , with( . Then   is a nano ( -open cover) of(  , since( then it is has nano -open refinement of  order  .  Hence  there are nano  -open sets  and    such that(   with ( , so that , .Thus (  and ( ) , this is complete prove. 3.4. Theorem:         Let be a nano topological space ,  be a nano open set of , then . Proof:  Suppose ( . Let   be a nano  -open  cover  of  . Now for each   , ,  where is  nano ( -open) set in) ). Since  , then a nano -open cover of  has a nano -open refinement  in  of order  .  Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٧٥Let  be a nano  -open refinement of  of order  So  .        Recall that a family   is a reduce of a family( , if  for each(   [ Dugundji , 1966]. 3.5. Theorem:         Let  be a nano topological space , the following statements are equivalent: (i)( . (ii) Every finite nano -open cover of can be reduced to a nano -open cover of of order .  Proof:  Suppose that(  . Let( be a nano -open cover of( . Then( has a nano -open refinement (  which it is of (order  . ,   such that ( , . Let  thus (  is a nano -open reduction of  which it is nano -open cover of( of order  . , suppose that the condition is holds and  be a finite nano -open covering of . Since every  nano -open reduction  of  which it is nano -open covering of   is nano -open refinement. Thus  has a nano -open refinement which it is of order . The other result holds immediately.  3.6. Corollary:         Let  be a nano topological space, the following statements are equivalent: (i) Every finite nano -open cover of has a nano -open refinement of ( of order .  (ii) Every finite nano -open cover of can be reduced to a nano -open cover of order . Proof: It is easy to show that . Now apply theorem (3.5). 3.7. Theorem:         Let    It is known I guess be a nano topological space , the following statements are equivalent: (i)( . (ii) Every nano -open cover of( with   sets has a reduction which a nano -open cover of   with sets and empty intersection .  Proof:  Suppose that( , then every finite nano -open cover of   has a (nano -open) reduction which cover of( order  by theorem (3.5) . The number of the elements of this cover is ( , then the number its reduction is also( with( order . The intersection is not empty for at most (  elements , i.e. the intersection of all( with (order  is empty .  let  be a finite nano -open cover of(  .  We  shall prove that this cover has a reduction of order . Suppose that order  , i.e.( such that . Put (  . Then   be  a  finite  nano  -open  cover  of    with    sets has a reduction  which it is a nano -open cover of and empty intersection .  is a nano -open reduction of  of order  . The intersection of the first sets of this cover is not empty intersection . But the intersection of all the sets of this cover has empty intersection . By continuous Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٧٦this operation finite numbers at time , we will get the reduction with sets and empty intersection . This completes the proof .  3.8. Definition:        Let  and be a nano  topological spaces . Then a mapping(  ) is  said  to be a nano  -continuous on(  , if the inverse image of every nano  -open set in    is nano( -open in .   3.9. Definition:       A nano( -continuous mapping(  ) is said to have a vague order at most   , if for each finite nano -open )cover of ,there is a finite (nano -open) cover  of  such that :   (i) [ for all(  ). (ii)(   holds for at most( values of  and for every . 3.10. Theorem:      Let(  )and  be two a( nano -continuous), onto mappings, if the nano α-vague order of  is at most , then the nano  α- vague order of  is at most . Proof:   Suppose the condition is holds and let [   be  a finite  nano  -open cover of ( . Then there is   be a finite nano -open  cover of   such that(    ) and ( )  for at most(  values of(   . Then(  is a finite nano -open cover of( , where  .Suppose that, which is a contradiction. 3.11. Theorem:          Let (  and (  be two a (nano -continuous), onto mappings and(  is a( nano -open) , if the( nano α- vague order) of  is at most (  , then the nano α- vague order of( ) is at most(  . Proof: suppose that the( nano  α- vague) order of  is at most n. We shall prove that the nano  α- vague order of g is at most n . Let (  be a finite nano (  α- open covering) of . Then there is (  be a finite nano  (α- open covering )of( U) such that , and  hold at most n+1 values of I and for every . Now since f is nano -open ,then the collection  (  is a finite nano -open covering of  satisfying:  (  . To prove  for at most n+1 vales of i and for every  . Suppose for some  there is  such that  for n+2 values of I that is  then (  for n+2 values of i .This means that  hold for at most n+2 values of i which is contradiction with theorem (3.10). Thus the nano α- vague order of  is at most  . 3.12. Theorem:         Let  be a nano topological space, the following statements are equivalent: (i)( . (ii) The( nano α- vague) order of the identity mapping(  of  is at most  .   Journal of Babylon University/Pure and Applied Sciences/ Vol.(26), No.(3), 2018   ٧٧(iii) The ( nano α- vague) order of every( nano -continuous) onto mapping with range(  is at most(  .   Proof:  (i ii)  the following result follows immediately from definition (3.9)  and we obtain  (ii iii) by theorem (3.10) . Also , (iii ii) by theorem (3.11) . Now, we prove that  (ii i) .  Let(   be a nano -open cover of . Since  is nano α- vague order at most  . Then(  be a nano -open cover of  such that : •(  .   •(  . i.e. order(  , we get the desired result .  3.13. Theorem:        Let(    and ( ) be a nano -continuous , onto  mapping with(  contains at most( points of )  for all ( , then the( nano α- vague) order of(  is at most .  Proof:  Let(   be a finite( nano -open cover) of( . Then (  is a finite( nano -open) cover of( . Since ( ) , then there exist be a disjoint nano -open refinement which cover of(  with   .  Now since( contains at most( points of( , then  for at most ) values of  for all( .        Reference: Alias, B. K. and Haji, M., 2011," -Covering Dimension ", Int. J. Pure Appl. Sci.  Tech, 3(1).   Caldas, M., Jafari, S., Georgion, D. N., 2003,"Characterization of low separation axioms via -open sets  and -closure operator", Bol. Soc. Paran. Math., 21 1/2.        Dugundji J., 1966, " Topology " , Allyn and Bacon Inc., Boston. Lellis,  M. T. and Carmel, R., 2013, " On Nano forms of Weakly open sets ", Int.  J. Math.  S., Vol. 1, No. 1. Lellis, M. T.,and Carmel,R.,2013,"On Nano Continuity", Math. Theory. and Modeling , Vol. 3, No.7. Najastad,O.,"A note on some applications of -open sets",2013, Pacific J.Math.Vol.2. Pears R.,1975," Dimension Theory of General Spaces " ,  Camb. , Univ. Press  Camb.   Revathy,  A.  and  Ilango G., 2015, " On Nano -open sets  " , Int.  J.  of  Engineering  Contemporary ,Math. and S. , Vol. 1 , No. 2.      

On vague Order and Decomposition Mapping by -Open Sets Using Nano

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On vague Order and Decomposition Mapping by -Open Sets Using Nano

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