NANO bT CLOSED SET IN NANO TOPOLOGICAL SPACES

In this paper, we introduce a new class of set namely n a n o bT -closed sets in nano topological space. Wealsodiscussedsomeproperties ofnanobTclosedset.


INTRODUCTION
Nano topological space was introduced (Lellis Thivagar and Camel Richard, 2013a, b) with respect to a subset X of a universe which is defined in terms of lower and upper approximations of X. He also established certain weak forms of nano open sets such as nano α -open sets, nano semi-open sets and nano pre open sets. b-open sets in topological spaces was introduced and studied (Andrijevic,1996). Several properties of a new type of sets called supra T-closed set and supra Tcontinuous maps was studied (Arockiarani and Trintia Pricilla, 2011). Also a new class of bT -That is UR(X) = : The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor as not-X with respect to R and it is denoted by BR(X).

Property
I f ( U , R ) i s a n a p p r o x i m a t i o n s p a c e a n d X , Y U, then (i) LR(X)  X  UR(X). (ii) L (ϕ) = U (ϕ) = ϕ and L (U) = U (U) = U closed set in supra topological spaces was R R R R introduced and studied (Krishnaveni and Vigneshwaran, 2013). In this paper, we introduced a new class of set called nano bT -closed sets and study its basic properties.

Definition
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. Then U is divided into disjoint equivalence classes. 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.
The lower approximation of X with respect to R is the set of all objects, which can be for certainly classified as X with respect to R and it is denoted by LR(X).That is LR(X) = : ( )X ,where R(x) denotes the equivalence class determined by x U.
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 UR(X).
(ii) The union of the elements of any subcollection of τ R (X) is in τ R (X) .

(iii)
The intersection of the elements of any finite subcollection of τ R (X) is in τ R (X).
That is, τ R (X) is a topology on U called the nano topology on U with respect to X. We call (U, τ R (X)) as the nano topological space. The elements of τ R (X) are called as nano open sets.

Definition
If (U, τ R (X)) is a nano topological space with respect to X where X U and i f 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 Nint( A). That is, Nint( 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 Ncl( A).
That is, Ncl( A) is the smallest nano closed set containing A.

Definition
A set A of X is called generalized b-closed set (simply gb−closed) if bcl (A)  U whenever A U and U is open. The complement of generalized bclosed set is generalized b-open set.

Definition
Let (X,µ) is a supra topological spaces. A subset A of (X,µ) is called T µ -closed set if bcl µ (A) U whenever A  U and U is g µ b -open in (X,µ). The complement of T µ -closed set is called T µ -open set.

Definition
A subset A of a topological space (X,τ) is called regular open if A = cl(int(A)). The complement of regular open set is called regular closed set .

Definition
A subset A of a topological space (X,τ) is called generalized b-regular closed set if bcl (A)  U and whenever A  U and U is regular open of (X,τ). The complement of generalized b-regular closed set is called generalized b-regular open set.

Definition
A subset A of a supra topological space (X, µ) is called bT µ -closed set (Krishnaveni and Vigneshwaran, 2013) if bcl µ (A)U whenever A  U and U is T µ -open in (X, µ).

Definition
L e t ( U , τ R ( X ) ) b e a n a n o t o p o l o g i c a l s p a c e . A subset A of (U,τ R(X)) is called nano T -closed set if Nbcl(A) U whenever A U and U is nano gb-open in ( U , τ R ( X ) ) .

Theorem
Every nano closed set is nano bT closed.

Proof Let A  U and U is nano T-open set.Since
A is nano closed then Ncl(A) = A  U . We know that Nbcl (A)  N cl (A) U , implies Nbcl(A) U . Therefore A is nano bT -closed.
The converse of the above theorem need not be true as seen from the following example.

Theorem
Every nano b closed set is nano bT closed.

Proof Let A  U and U is nano T-open set.
Since A is nano b closed then Nbcl (A)  U . Therefore A is nano bT -closed.

Theorem
Every nano bT -closed set is nano gbclosed set.

Proof Let A  U and U is nano open set. We know that every nano open set is nano T -open set , then U is nano T-open set. Since A is nano
bT -closed set, we have Nbcl (A)  U. Therefore A is nano gb-closed set.

Theorem
Every N a n o bT-closed set is nano gbrclosed set.

Proof Let A  U and U is nano regular open set. We know that every nano regular open set is nano T-open set, then U is nano T-open set. Since
A is nano bT -closed set, we have Nbcl(A) U . Therefore A is nano gbr-closed set.

Theorem
The union of two nano bT -closed set is nano bT-closed set.
Proof Let A and B two nano bT -closed set. Let A B  G, where G is nano T -open. Since A and B are nano bT-closed sets. Therefore Nbcl(A)  Nbcl (B)  G. Thus N bcl (A B )  G. Hence AB is Nano bT-closed set.

Theorem
Let A be n a n o bT -closed set of (U,X). Then Nbcl (A) -A does not contain any non empty nano T-closed set.
Proof: Necessity Let A be n a n o bT -closed set. suppose F  is a n a Hence Nbcl(A)  U c =  and N bcl(A) U.Therefore A is nano bT -closed.

Theorem
If A is nano bT -closed set in a supra topological space (U,X) and AB Nbcl (A) then B is also nano bT-closed set.
Proof Let U be nano T-open in set (U,X) such that B  U. Since A B  AU and since A is nano bT -closed set in (U,X). Nbcl (A)  U, since B  Nbcl(A). Then N bcl(B) U. Therefore B is also nano bT -closed set in (U,X)

Theorem
Let A be nano bT -closed set then A is nano b-closed if Nbcl(A)-A is nano T-closed.
Proof Let A be nan o bT-closed set. If A is nano b -closed, we have N bcl (A) -A = , which is nano Tclosed. Conversely, let N bcl(A)-A is nano bT -closed. Then by the theorem 3.13, Nbcl (A) -A does not contain any non empty nano T-closed and N bcl (A)-A=. Hence A is nano b -closed.

A subset AX is nano bT -open iff F  N bint(A) whenever F is nano T -closed and F A.
Proof Let A be n a n o bT -open set and suppose F A, where F is nano T-closed. Then X-A is nano bT -closed set contained in the n a n o Topen set X-F. Hence N bcl (X-A ) X-F. Thus F N bint(A). Conversely, if F is nano T -closed set with F Nbint(A) and FA , then X-Nbint (A)  X − F . This implies that Nbcl(X-A)  X-F. Hence X-A is nano bT -closed. Therefore A is nano bT -open set.

Theorem
If B is nano T-open and n a n o bT -closed set in X, then B is nano b-closed.
Proof Since B is nano T-open and n a n o bTclosed then Nbcl (B)  B, but B N bcl(B). Therefore B=Nbcl(B).Hence B is nano b -closed.

Corollary
If B is nano open and n a n o bT -closed set in X. Then B is nano b-closed.

Theorem
Let A be nano g b-open and nano bTclosed set. Then A F is nanoT-closed whenever F is nano b-closed.
Proof Let A be nano g b-open and n a n o bTclosed set then Nbcl (A) A and also A Nbcl (A). Therefore N bcl (A) = A. Hence A is nano b-closed. Since F is nano b-closed. Therefore A F is nano b -closed in X. Hence A F is nano T -closed in X.
From the above theorem and example we have the following diagram nano closed  nano b closed  nano bT-closed  nano gb-closed  nano gbr-closed