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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI ...

International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.

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International Journal of Mathematics and Statistics Invention (IJMSI) International Journal of Mathematics and Statistics Invention (IJMSI) Document Transcript

  • International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 1 Issue 1 ǁ August. 2013ǁ PP-31-37 On Nano Forms Of Weakly Open Sets M. Lellis Thivagar , Carmel Richard 1 1 2 School of Mathematics, Madurai Kamaraj University, Madurai-625021, Tamil Nadu, India 2 Department of Mathematics, Lady Doak College, Madurai - 625 002, Tamil Nadu, India ABSTRACT : The purpose of this paper is to define and study certain weak forms of nano-open sets namely, nano  -open sets, nano semi-open sets and nano pre-open sets. Various forms of nano  -open sets and nano semi-open sets corresponding to different cases of approximations are also derived. KEYWORDS: Nanotopology,nano-open sets,nano interior, nano closure, nano  -open sets, nano semi-open sets, nano pre-open sets, nano regular open sets. 2010 AMS Subject Classification:54B05 I. INTRODUCTION Njastad [5], Levine [2] and Mashhour et al [3] respectively introduced the notions of  - open, semi-open and pre-open sets.Since then these concepts have been widelyinvestigated. It was made clear that each  -open set is semi-open and pre-open but the converse of each is not true. Njastad has shown that the   family  of  - open sets is a topology on X satisfying    . The families SO(X,  ) of all semi–open sets and PO(X,  ) of all preopen sets in (X,  )are not topologies. It was proved that both SO(X,  ) and PO(X,  ) are closed under arbitrary unions but not under finite intersection. Lellis Thivagar et al [1] introduced a nano topological space with respect to a subset X of an universe which is defined in terms of lower and upper approximations of X. The elements of a nano topological space are called the nano-open sets. He has also studied nano closure and nano interior of a set. In this paper certain weak forms of nano-open sets such as nano  -open sets, nano semi-open sets and nano pre-open sets are established. Various forms of nano  open sets and nano semi-open sets under various cases of approximations sre also derived. A brief study of nano regular open sets is also made. II. PRELIMINARIES Definition 2.1 A subset A of a space ( X ,  ) is called (i) semi-open [2] if A  Cl ( Int ( A )) . (ii) pre open [3] if A  Int ( Cl ( A )) . (iii)  -open [4] if A  Int ( Cl ( Int ( A ))) . (iv) regular open [4] if A = Int ( Cl ( A )) . Definition 2.2 [6] Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U named as the indiscernibility relation. Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair ( U , R ) is said to be the approximation space. Let X  U . (i) The lower approximation of X with respect to R is the set of all objects, which can be for certain classified as X with respect to R and its is denoted by L R ( X ) . That is, L R ( X ) =  {R ( x) : R ( x)  X } , where R(x) xU denotes the equivalence class determined by x. (ii) The upper approximation of X with respect to R is the set of all objects, which can be possibly classified as X with respect to R and it is denoted by U R ( X ) . That is, U R ( X ) =  {R ( x) : R ( x)  X  } xU (iii) The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor www.ijmsi.org 31 | P a g e
  • On Nano Forms Of Weakly… as not-X with respect to R and it is denoted by B R ( X ) . That is, B R ( X ) = U R ( X )  L R ( X ) . Property 2.3 [6] If ( U , R) is an approximation space and X, Y  U , then (i) LR ( X )  X  U (ii) L R ( ) = U R R (X ) . (  ) =  and L R ( U ) = U R (U ) = U (iii) U R ( X  Y ) = U R ( X )  U R ( Y ) (iv) U R ( X  Y )  U R ( X )  U R ( Y ) (v) L R ( X  Y )  L R ( X )  L R (Y ) (vi) L R ( X  Y ) = L R ( X )  L R ( Y ) (vii) L R ( X )  L R ( Y ) and U R ( X )  U R ( Y ) whenever X  Y c (viii) U R ( X ) = [ L R ( X )] c c and L R ( X ) = [U R ( X )] c (ix) U R U R ( X ) = L R U R ( X ) = U R ( X ) (x) L R L R ( X ) = U R L R ( X ) = L R ( X ) Definition 2.4 [1] : Let U be the universe, R be an equivalence relation on U and  R ( X ) = { U ,  , L R ( X ), U R ( X ), B R ( X )} where X  U . Then by propetry 2.3,  R ( X ) satisfies the following axioms: (i) U and    R ( X ) . (ii) The union of the elements of any subcollection of  R ( X ) is in  R ( X ) . (iii) The intersection of the elements of any finite subcollection of  R ( X ) is in  R ( X ) . That is,  R ( X ) is a topology on U called the nanotopology on U with respect to X. We call (U ,  R ( X ) ) as the nanotopological space. The elements of  R ( X ) are called as nano-open sets. Remark 2.5 [1] If  R ( X ) is the nano topology on U with respect to X, then the set B = { U , L R ( X ), B R ( X )} is the basis for  R ( X ) . Definition 2.6 [1] If ( U ,  R ( X )) is a nano topological space with respect to X where X  U and if A  U , then the nano interior of A is defined as the union of all nano-open subsets of A and it is denoted by N Int(A). That is, N Int(A) is the largest nano-open subset of A. The nano closure of A is defined as the intersection of all nano closed sets containing A and it is denoted by N Cl(A). That is, N Cl(A) is the smallest nano closed set containing A. Definition 2.7 [1] A nano topological space ( U ,  R ( X )) is said to be extremally disconnected, if the nano closure of each nano-open set is nano-open. III. NANO   OPEN SETS Throughout this paper ( U ,  R ( X )) is a nano topological space with respect to X where X  U, R is an equivalence relation on U , U / R denotes the family of equivalence classes of U by R. Definition 3.1 Let ( U ,  R ( X )) be a nano topological space and A  U . Then A is said to be (i) nano semi-open if A  N Cl ( N Int ( A )) (ii) nano pre-open if A  N Int ( N Cl ( A )) (iii) nano  -open if A  N Int ( NCl ( NInt ( A ))  NSO( U , X ), NPO ( U ,X) and  R (X) respectively denote the families of all nano semi-open, nano pre-open www.ijmsi.org 32 | P a g e
  • On Nano Forms Of Weakly… and nano  -open subsets of U . Definition 3.2 Let ( U ,  R ( X )) be a nanotopological space and A  U . A is said to be nano  -closed (respectively, nano semi- closed, nano pre-closed), if its complement is nano  -open (nano semi-open, nano pre-open). Example 3.3 Let U = { a , b , c , d } with U / R = {{ a }, { c }, { b , d }} and X = { a , b } . Then the nano topology,  R ( X ) = { U ,  , { a }, { a , b , d }, { b , d }} . The nano closed sets are U ,  , { b , c , d }, { c } and { a , c } . Then, N SO ( U , X ) = {U ,  , { a }, { a , c }, { a , b , d }, { b , c , d }} , N PO ( U , X ) = {U ,  , { a }, { b }, { d }, { a , b } ,  { a , d } , { b , d }, { a , b , c }, { a , b , d }, { a , c , d }} and  R ( X ) = {U ,  , { a }, { b , d }, { a , b , d }} . We note that, N SO ( U , X ) does not form a topology on U , since { a , c } and { b , c , d }  N SO ( U , X ) but { a , c }  { b , c , d } = { c }  N SO ( U , X ) . Similarly, N PO ( U , X ) is not a topology on U , since  { a , b , c }  { a , c , d } = { a , c }  N PO ( U , X ) , even though { a , b , c } and { a , c , d }  N PO ( U , X ) . But   the sets of  R ( X ) form a topology on U . Also, we note that { a , c }  N SO ( U , X ) but is not in N PO ( U , X ) and { a , b }  N PO ( U , X ) but does not belong to N SO ( U , X ) . That is, N SO ( U , X ) and N PO ( U , X ) are independent. Theorem 3.4 If A is nano-open in ( U ,  R ( X )) , then it is nano  -open in U . Proof: Since A is nano-open in U , N IntA = A . Then N Cl ( N IntA ) = N Cl ( A )  A . That is A  N Cl ( N IntA ) . Therefore, N Int ( A )  N Int ( N Cl ( N Int ( A ))) . That is, A  N Int ( N Cl ( N Int ( A ))) . Thus, A is nano  -open.  Theorem 3.5  R ( X )  N SO ( U , X ) in a nano topological spce ( U ,  R ( X )) .  Proof: If A   R ( X ) , A  N Int ( N Cl ( N Int ( A )))  N Cl ( N Int ( A )) and hence A  N SO ( U , X ) . Remark 3.6 The converse of the above theorem is not true. In example 3.3, {a,c} and {b,c,d} and nano semiopen but are not nano  -open in U .  Theorem 3.7  R ( X )  N PO ( U , X ) in a nano topological space ( U ,  R ( X )) .  Proof: If A   R ( X ) , A  N Int ( N Cl ( N Int ( A ))) . Since N Int ( A )  A , N Int ( N Cl ( N Int ( A )))  N Int ( N Cl ( A )) . That is, A  N Int ( N Cl ( A )) . Therefore, A  N PO ( U , X ) .  That is,  R ( X )  N PO ( U , X ) . Remark 3.8 The converse of the above theorem is not true. In example 3.3, the set {b} is nano pre-open but is not nano  -open in U .  Theorem 3.9  R ( X ) = N SO ( U , X )  N PO ( U , X ) .  Proof: If A   R ( X ) , then A  N SO ( U , X ) and A  N PO ( U , X ) by theorems 3.5 and 3.7 and hence  A  N SO ( U , X )  N PO ( U , X ) . That is  R ( X )  N SO ( U , X )  N PO ( U , X ) . Conversely, if A  N SO ( U , X )  N PO ( U , X ) , then A  N Cl ( N Int ( A )) and A  N Int ( N Cl ( A )) . Therefore, N Int ( N Cl ( A ))  N Int ( N Cl ( N Cl ( N Int ( A )))) = N Int ( N Cl ( N Int ( A ))) . That is, N Int ( N Cl ( A ))  N Int ( N Cl ( N Int ( A )))) . Also A  N Int ( N Cl ( A ))  N Int ( N Cl ( N Int ( A )))  implies that A  N Int ( N Cl ( N Int ( A ))) . That is, A   R ( X ) . Thus, N SO ( U , X )  N PO ( U , X )    R ( X ) . Therefore,   R ( X ) = N SO ( U , X )  N PO ( U , X ) . www.ijmsi.org 33 | P a g e
  • On Nano Forms Of Weakly… Theorem 3.10 : If, in a nano topological space ( U ,  R ( X )) , L R ( X ) = U R ( X ) = X , then U ,  , L R ( X )(= U R ( X )) and any set A  L R ( X ) are the only nano-  -open sets.in U . Proof: Since L R ( X ) = U R ( X ) = X , the nano topology,  R ( X ) = { U ,  , L R ( X )} . Since any nano-open set is nano-  -open, U ,  and L R ( X ) are nano  -open in U . If A  L R ( X ) , then N Int ( A ) =  , since  is the only nano-open subset of A. Therefore N Cl ( N Int ( A ))) =  and hence A is not nano  -open. If A  L R ( X ) , L R ( X ) is the largest nano-open subset of A and hence, N Int ( N Cl ( N Int ( A ))) = N Int ( N Cl ( L R ( X ))) = N Int ( B R ( X ) ) = N Int ( U ) , since B R ( X ) =  . Therefore, C N Int ( N Cl ( N Int ( A ))) = U and hence, A  N Int ( N Cl ( N Int ( A ))) . Therefore, A is nano  -open. Thus U ,  , L R ( X ) and any set A  L R ( X ) are the only nano  -open sets in U , if L R ( X ) = U R ( X ) . Theorem 3.11 : U ,  , U R ( X ) and any set A  U R ( X ) are the only nano  -open sets in a nanotopological space ( U ,  R ( X )) if L R ( X ) =  . Proof: Since L R ( X ) =  , B R ( X ) = U R ( X ) . Therefore,  R ( X ) = { U ,  , U R ( X )} and the members of  R ( X ) are nano  -open in U . Let A  U R ( X ) . Then N Int ( A ) =  and hence N Int ( N Cl ( N Int ( A ))) =  . Therefore A is not nano   open in U . If A  U R ( X ) , then U R ( X ) is the largest nano-open subsetof A (unless, U R ( X ) = U , in case of which U and  are the only nano-open sets in U ). Therefore, N Int ( N Cl ( N Int ( A ))) = N Int ( N Cl (U R ( X ) ) = N Int ( U ) and hence A  N Int ( N Cl ( N Int ( A ))) . Thus, any set A  U R ( X ) is nano  -open in U . Hence, U ,  , U R ( X ) and any superset of U R ( X ) are the only nano  -open sets in U Theorem 3.12 : If U R ( X ) = U and L R ( X )   , in a nano topological space (U ,  R ( X )) , then U ,  , L R ( X ) and B R ( X ) are the only nano  -open sets in U . Proof: Since U R ( X ) = U and L R ( X )   , the nano-open sets in U are U ,  , L R ( X ) and B R ( X ) and hence they are nano  -open also. If A =  ,then A is nano  -open. Therefore, let A   . When A  L R ( X ) , N Int ( A ) =  , since the largest open subset of A is  and hence A  N Int ( N Cl ( N Int ( A ))) , unless A is  . That is, A is not nano  -open in U . When L R ( X )  A , N Int ( A ) = L R ( X ) and therefore, N Int ( N Cl ( N Int ( A ))) = N Int ( N Cl ( L R ( X ))) C = N Int ( B R ( X ) ) = N Int ( L R ( X )) = L R ( X )  A . That is, A  N Int ( N Cl ( N Int ( A ))) . Therefore, A is not nano  -open in U . Similarly, it can be shown that any set A  B R ( X ) and A  B R ( X ) are not nano  -open in U . If A has atleast one element each of L R ( X ) and B R ( X ) , then N Int ( A ) =  and hence A is not nano  open in U . Hence, U ,  , L R ( X ) and B R ( X ) are the only nano  -open sets in U when U R ( X ) = U and LR ( X )   .  Corollary 3.13 :  R ( X ) =  R ( X ) , if U R ( X ) = U . Theorem 3.14 : Let L R ( X )  U R ( X ) where L R ( X )   and U R ( X )  U in a nano topological space ( U ,  R ( X )) . Then U ,  , L R ( X ) , B R ( X ) , U R ( X ) and any set A  U sets in U R ( X ) are the only nano  -open . Proof: The nano topology on U is given by  R ( X ) = { U ,  , L R ( X ), B R ( X ) , U R ( X )} and hence U ,  , L R ( X ) , B R ( X ) and U R ( X ) are nano  -open in U . Let A  U such that A  U www.ijmsi.org R ( X ) . Then 34 | P a g e
  • On Nano Forms Of Weakly… N Int ( A ) = U R ( X ) and therefore, N Int ( N Cl (U R ( X ))) = N Int ( U ) = U . Hence, A  N Int ( N Cl ( N Int ( A )) . Therefore, any A  U R ( X ) is nano  -open in U . When A  L R ( X ) , N Int ( A ) =  and hence N ( Int ( N Cl ( N Int ( A ))) =  . Therefore, A is not nano  -open in U . When A  B R ( X ) , N Int ( A ) =  and hence A is not nano  -open in U . When A  U R ( X ) such that A is neither a subset of L R ( X ) nor a subset of B R ( X ) , N Int ( A ) =  and hence A is not nano  -open in U . Thus, U ,  , L R ( X ), B R ( X ), U IV. R ( X ) and any set A  U R ( X ) are the only nano  -open sets in U . FORMS OF NANO SEMI-OPEN SETS AND NANO REGULAR OPEN SETS In this section, we derive forms of nano semi-open sets and nano regular open sets depending on various combinations of approximations. Remark 4.1 U ,  are obviously nano semi-open, since N Cl ( N Int ( U )) = U and N Cl ( N Int (  )) =  Theorem 4.2 If, in a nano topological space ( U ,  R ( X )) , U R ( X ) = L R ( X ) , then  and sets A such that A  L R ( X ) are the only nano semi-open subsets of U Proof:  R ( X ) = { U ,  , L R ( X )} .  is obviously nano semi-open. If A is a non- empty subset of U and A  L R ( X ) , then N Cl ( N Int ( A )) = N Cl (  ) =  . Therefore, A is not nano semi-open, if A  L R ( X ) . If A  LR ( X ) , then N Cl ( N Int ( A )) = N Cl ( L R ( X )) = U , since LR ( X ) = U R (X ) . Therefore, A  N Cl ( N Int ( A )) and hence A is nano semi-open. Thus  and sets containing L R ( X ) are the only nano semi-open sets in U , if L R ( X ) = U R ( X ) . Theorem 4.3 If L R ( X ) =  and U R ( X )  U , then only those sets contianing U R ( X ) are the nano semiopen sets in U . Proof:  R ( X ) = { U ,  , U R ( X )} . Let A be a non-empty subset of U . If A  U R ( X ) , then N Cl ( N Int ( A )) = N Cl (  ) =  and hence A  N Cl ( N Int ( A )) . Therefore, A is not nano semi-open in U . If A  U R ( X ) , then N Cl ( N Int ( A )) = N Cl (U R ( X )) = U and hence A  N Cl ( N Int ( U )) . Therefore, A is nano semi-open in U . Thus, only the sets A such that A  U R ( X ) are the only nano semi-open sets in U . Theorem 4.4 If U R ( X ) = U is a nano topological space, then U ,  , L R ( X ) and B R ( X ) are the only nano semi-open sets in U . Proof:  R ( X ) = { U ,  , L R ( X ), B R ( X )} . Let A be a non-empty subset of U . If A  L R ( X ) , then N Cl ( N Int ( A )) =  and hence A is not nano semi-open in U . If A = LR ( X ) , then N Cl ( N Int ( A )) = N Cl ( L R ( X )) = L R ( X ) and hence, A  N Cl ( N Int ( A )) . Therefore, A is nano semi- open in U . If A  LR ( X ) , then N Cl ( N Int ( A )) = N Cl ( L R ( X )) = L R ( X ) . Therefore, A  N Cl ( N Int ( A )) and hence A is not nano semi-open in U . If A  B R ( X ) , N Cl ( N Int ( A )) =  and hence A is not nano semi-open in U . If A = B R ( X ) , then N Cl ( N Int ( A )) = N Cl ( B R ( X )) = B R ( X ) and hence A  N Cl ( N Int ( A )) . Therefore, A is nano semi-open in U . If A  BR (X ) , then N Cl ( N Int ( A )) = N Cl ( B R ( X )) = B R ( X )  A and hence A is not nano semi-open in U . If A has atleast one element of L R ( X ) and atleast one element of B R ( X ) , then N Cl ( N Int ( A )) = N Cl ( ) =  and hence A is not nano semi-open in U . Thus, U ,  , L R ( X ) and B R ( X ) are the only nano semi-open sets in U , if U R ( X ) = U and L R ( X )   . If L R ( X ) =  , U and  are the only nano semi-open sets in U , since U www.ijmsi.org 35 | P a g e
  • On Nano Forms Of Weakly… and  are the only sets in U which are nano-open and nano-closed. Theorem 4.5 If L R ( X )  U R ( X ) where L R ( X )   and U R ( X )  U , then U ,  , L R ( X ) , B R ( X ) , C sets containing U R ( X ) , L R ( X )  B and B R ( X )  B where B  (U R ( X )) are the only nano semiopen sets in U . Proof:  R ( X ) = { U ,  , L R ( X ), U R ( X ), B R ( X )} . Let A be a non-empty, proper subset of U . If A  L R ( X ) , then N int ( A ) =  and hence, N cl ( N int ( A )) =  . Therefore, A is not nano semi-open in U . C If A = L R ( X ) , then N cl ( N int ( A )) = N cl ( L R ( X )) = L R ( X )  [U R ( X )] and hence A  N cl ( N int ( A )) . Therefore, L R ( X ) is nano semi-open in U . If A  B R ( X ) , then N int ( A ) =  and hence A is not nano semi-open in U . If A = B R ( X ) , then N cl ( N int ( A )) = N cl ( B R ( X )) = B R ( X )  [U R ( X )] C and hence A  N cl ( N int ( A )) . Therefore, B R ( X ) is nano semi-open in U . Since L R ( X ) and B R ( X ) are nano semi-open, L R ( X )  B R ( X ) = U R ( X ) is also nano semi-open in U . Let A  U R ( X ) such that A has atleast one element each of L R ( X ) and B R ( X ) . Then N int ( A ) =  or L R ( X ) or B R ( X ) and consequently, N cl ( N int ( A )) =  or L R ( X )  [U R ( X )] B R ( X )  [U A  U R R ( X )] C C or and hence A ÚN cl ( N int ( A )) . Therefore, A is not nano semi-open in U . If ( X ) , then N cl ( N int ( A )) = N cl (U R ( X )) = U and hence A  N cl ( N int ( A )) . Therefore, A is nano semi-open. If A has a single element each of L R ( X ) and B R ( X ) and atleast one element of [U R ( X )] C , then N int ( A ) =  . Then, N cl ( N int ( A )) =  and hence A is not nano semi-open in U . C Similarly, when A has a single element of L R ( X ) and atleast one element of [U R ( X )] , or a single element C of B R ( X ) and atleast one element of [U R ( X )] then N cl ( N int ( A )) =  and hence A is not nano semiC open in U . When A = L R ( X )  B where B  [U R ( X )] , then N cl ( N int ( A )) = N cl ( L R ( X )) = L R ( X )  [U B  (U R ( X )) R C ( X )] C  A . Therefore, A is nano semi-open in U . Similarly, if A = B R ( X )  B where , then N cl ( N int ( A )) = B R ( X )  [U R ( X )] C  A . Therefore, A is nano semi-open in U . Thus, U ,  , L R ( X ) , U R ( X ) , B R ( X ) , any set containing U R ( X ) , L R ( X )  B and B R ( X )  B where B  [U R ( X )] C are the only nano semi-open sets in U . Theorem 4.6 If A and B are nano semi-open in U , then A  B is also nano semi-open in U . Proof: If A and B are nano semi-open in U , then A  N Cl ( N Int ( A )) and B  N Cl ( N Int ( A )) . Consider A  B  N Cl ( N Int ( A ))  N Cl ( N Int ( B )) = N Cl ( N Int ( A )  N Int ( B ))  N Cl ( N Int ( A  B )) and hence A  B is nano semi-open. Remark 4.7 If A and B are nano semi-open in U , then A  B is not nano semi-open in U . For example, let U = { a , b , c , d } with U / R = {{ a }, { c }, { b , d }} and X = { a , b } . Then  R ( X ) = { U ,  , { a }, { a , b , d }, { b , d }} . The nano semi-open sets in U are U ,  , { a } , { a , c } , { b , d } , { a , b , d } , { b , c , d } . If A = { a , c } and B = { b , c , d } , then A and B are nano semi-open but A  B = { c } is not nano semi-open in U . Definition 4.8 A subset A of a nano topological space (U ,  R ( X )) is nano-regular open in U, if N Int (N Cl ( A )) = A . Example 4.9 Let U = { x , y , z } and U / R = {{ x }, { y , z }} . Let X = { x , z } . Then the nano topology on U with respect to X is given by  R ( X ) = {U ,  , { x }, { y , z }} . The nano closed sets are U ,  , { y , z }, { x } . Also, N Int (N Cl ( A )) = A for A = U ,  , { x } and { y , z } and hence these sets are nano regular open in U. www.ijmsi.org 36 | P a g e
  • On Nano Forms Of Weakly… Theorem 4.10 Any nano regular open set is nano-open. Proof: If A is nano regular open in (U ,  R ( X )), A = N Int ( N Cl ( A )) . Then N Int ( A ) = N Int ( N Int ( N Cl ( A ))) = N Int ( N Cl ( A )) = A . That is, A nano-open in U. Remark 4.11 The converse of the above theorem is not true. For example, let U = { a , b , c , d , e } with U / R = {{ a , b }, { c , e }, { d }} . Let X = { a , d } . Then  R ( X ) = { U ,  , { d }, { a , b , d }, { a , b }} and the nano closed sets are U ,  , { a , b , c , e }, { c , e }, { c , d , e } . The nano regula open sets are U ,  , { d } and {a,b}. Thus, we note that {a,b,d} is nano-open but is not nano regular open. Also, we note that the nano regular open sets do not form a topology, since { d }  { a , b } = { a , b , d } is not nano regular open, even though {d} and {a,b} are nano regular. Theorem 4.12 In a nano topological space ( U ,  R ( X )) ,if L R ( X )  U R ( X ) , then the only nano regular open sets are U ,  , L R ( X ) and B R ( X ) . Proof: The only nano-open sets ( U ,  R ( X )) are U ,  , L R ( X ), U R ( X ) and B R ( X ) and hence the only nano closed sets in U are U ,  , [ L R ( X )] , [U R ( X )] and [ B R ( X )] C C LR (X ) and L R ( X )  L R ( X C C C which are respectively U ,  , U R ( X C ), ). C C Case 1: Let A = L R ( X ) .Then N CL ( A ) = [ B R ( X )] . Therefore, N Int ( N Cl ( A )) = N Int [ B R ( X )] = [N Cl ( B R ( X ))] C C = [( L R ( X )) ] C = L R ( X ) = A . Therefore, A = L R ( X ) is nano-regular open. C C Case 2: Let A = B R ( X ) . Then N Cl ( A ) = [ L R ( X )] , Then N Int ( N Cl ( A )) = N Int [ L R ( X )] = [N Cl ( L R ( X ))] C C = [[ B R ( X )] ] C = B R ( X ) = A . That is, A = B R ( X ) is nano regular open. Case 3: If A = U R ( X ) , then N Cl ( A ) = U . Therefore, N Int ( N Cl ( A )) = N Int (U ) = U  A . That is, A =U R ( X ) is not nano regular open unless U R ( X ) = U . Case 4: Since N Int ( N Cl (U )) = U and N Int ( N Cl (  )) =  , U and  are nano regular open. Also any nano regular open set is nano-open. Thus, U ,  , L R ( X ) and B R ( X ) are the only nano regular open sets . Theorem 4.13 : In a nano topological space ( U ,  R ( X )) ,if L R ( X ) = U R ( X ) , then the only nano regular open sets are U and  . Proof: The nano- open sets in U are U ,  and L R ( X ) And N Int ( N Cl ( L R ( X ))) = U  L R ( X ) Therefore, L R ( X ) is not nano regular open. Thus, the only nano regular open sets are U and  . Corollary: If A and B are two nano regular open sets in a nano topological space, then A  B is also nano regular open. REFERENCES [1] [2] [3] M.Lellis Thivagar and Carmel Richard, Note on nano topological spaces,Communicated. N. Levine, Semi–open sets and semicontinuity in topological spaces, Amer.Math.Monthly 70 (1963), 36–41. A.S. Mashhour, M.E. Abd El-Monsef and S.N. El-Deeb, On pre-topological spaces, Bull.Math. de la Soc. R.S. de Roumanie 28(76) (1984), 39–45. [5] Miguel Caldas, A note on some applications of  -open sets, IJMMS, 2 (2003),125-130 O. Njastad, On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961–970. Z. Pawlak, Rough sets, International journal of computer and Information Sciences, 11(1982),341-356. [6] I.L.Reilly and M.K.Vamanamurthy, On [4]  -sets in topologicalspaces, Tamkang J.Math.,16 (1985), 7-11 www.ijmsi.org 37 | P a g e