This document defines and studies various weak forms of nano-open sets, including nano α-open sets, nano semi-open sets, and nano pre-open sets. It introduces these concepts in a nano topological space based on lower and upper approximations. Some key results shown are: 1) every nano-open set is nano α-open, 2) the family of nano α-open sets is contained within the families of nano semi-open and nano pre-open sets, and 3) the nano α-open sets equal the intersection of nano semi-open and nano pre-open sets. Examples are provided to illustrate these concepts.

1. 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
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2. 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
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3. 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 ) .
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4. 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
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R
( X ) . Then
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5. 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
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6. 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.
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36 | P a g e

7. 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.
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[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
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-sets in topologicalspaces, Tamkang J.Math.,16 (1985), 7-11
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