Weaker Forms of Soft Nano Open Sets

S. S. Benchalli1, P.G. Patil1, Nivedita S. Kabbur2 and J. Pradeepkumar3

Basavaraj M. Ittanagi and Mohan V

1Department of Mathematics, Karnatak University, Dharwad-580003, Karnataka, INDIA.
22Department of Mathematics, BVB College of Engineering and Technology, Hubli-580031, Karnataka, INDIA.
33Department of Mathematics, SKSVMACET, Laxmeshwar-582116, Karnataka, INDIA.
benchalliss@gmail.com nivedita_kabbur@yahoo.in pgpatil01@gmail.com

(Received on: November 1, Accepted: November 5, 2017)

ABSTRACT

In the present paper, the notion of weaker forms of soft nano open sets namely soft nano semi open, soft nano preopen, soft nano α-open and soft nano β-open sets in soft nano topological spaces are introduced and studied. The characterizations and the inter-relationship between these new classes of soft nano open sets are studied.

2000 Mathematics Subject Classifications: 54A05.

Keywords:soft nano topology, soft nano open set, soft nano closed set, soft nano semi open set, soft nano pre open set, soft nano α-open set, soft nano β-open set.

INTRODUCTION

In 1999, Molodtsov13 initiated soft set theory to deal with the problems arises from many fields like economics, engineering, medicine, environment, sociology, which depends on modeling of uncertain data. He proposed the fundamental results of the soft set theory. Further, the notion of soft topological spaces is introduced by Shabir and Naz17, which are defined over an initial universe with a fixed set of parameters. They introduced the basic notions such as soft open sets, soft closed sets, soft closure, soft separation axioms. The study of soft topological spaces is also continued by Cagman et al.14 and Sabir and Bashir15. Theoretical studies of soft sets is carried out by many researchers in1,2,3,4,5,6,9,10,11,16,18,19,20. Presently, work on the soft set theory is making progress rapidly. The notion of Nano topology was introduced by Thivagar12, which was defined in terms of approximations and boundary region of a subset of an universe using an equivalence relation on it. Based on that work, Benchalli et al.7 introduced the notion of soft nano topological spaces using soft set equivalence relation on the universal set. The notion of soft nano continuity is introduced and studied using soft nano topological spaces in8. In the present paper, the notion of weaker forms of soft nano open sets namely soft nano semi open, soft nano preopen, soft nano α-open and soft nano β-open sets in soft nano topological spaces are introduced and studied. The characterizations and the inter-relationship between these new classes of soft nano open sets are studied.

PRELIMINARIES

Now, some definitions and results by various authors are recalled in following.

Definition 2.1[11]: Let U be an initial universe and E be a set of parameters. Let P (U) denote the power set of U and A be a non-empty subset of E. A pair (F, A) is called a soft set over U, where F is a mapping given byF: A->P(U ). In other words, a soft set over U is a parameterized family of subsets of the universe U. For e ∈ A, F (e) may be considered as the set of e-approximate elements of the soft set (F, A). Clearly, a soft set need not be a set.

Definition 2.2[9]: For two soft sets (F, A) and (G, B) over a common universe U, we say that (F, A) is a soft subset of (G, B) if (i) A ⊂ B and (ii) for all e ∈ A, F (e) and G(e) are identical approximations.

Definition 2.3 [3]: Let (F, A) and (G, B) be two soft sets over U, then the cartesian product of (F, A) and (G, B) is defined as (F,A)X(G,B) = (H,AXB), where H : AXB ->P(UXU )and H (a,b) = F (a)xF (b). where (a,b) ∈ AXB.
i.e.(a,b)hi,hj )/where hi∈F (a)andhj∈G(b

Theorem 2.4[3]: Let (F, A) and (G, B) be two soft sets over a universe U . Then a soft set relation from (F, A) to (G, B) is a soft subset of (F, A)X (G, B). In otherwords, a soft set relation from (F, A) to (G, B) is of the form (H1 , S), where S ⊆ AX B and H1 (a,b) = H (a,b) for all (a,b) ∈ S, where (H,AX B) = (F,A)X (G,B) as in the above definition.
In an equivalent way, we can define the soft set relation R on (F, A) in the parameterized form as follows;
if (F,A) = {F (a),F (b),...} then F (a)RF (b) ⇔ F (a)X F (b)∈R

Definition 2.5[3]: Let R be a relation on (F, A), then

  1. R is reflexive if H1 (a,a) ∈ R,∀a ∈A
  2. ER is symmetric if H1 (a,b) ∈ R ⇒ H1 (b,a) ∈ R∀(a,b) ∈ AX A
  3. R is transitive if H1(a,b) ∈ R,H1 (b,c) ∈ R ⇒ H1 (a,c) ∈ R,∀a,b,c ∈ A

Definition 2.6Definition 2.6 [3]: A soft set relation R on a soft set (F, A) is called an equivalence relation if it is reflexive, symmetric and transitive.

Example 2.7[3]: Consider a soft set (F, A) over U , where U = {C1 ,C2 ,C3 ,C4 },A = {m1 ,m2 } and F (m1 ) = {C1 ,C3 },F (m2 ) = {C2 ,C4 }. Consider a relation R defined on (F, A) as {F (m1 )X F (m2 ),F (m2 )X F (m1 ),F (m1 )X F (m1 ),F (m2 )X F (m2 )}. This relation is a soft set equivalence relation.

Definition 2.8 [3]: Let (F, A) be a soft set, then equivalence class of F (a) denoted by [F (a)] is defined as [F (a)] = {F (b):F (b) RF (a)}.

Definition 2.9[3]: The inverse of a soft set relation R denoted by R-1 is defined by R-1 = {(F (b)xF (a)) :F (a) RF (b)}.

Theorem 2.10 [18] Letss(U )A andss(V)B be families of soft sets. For a function fpu:ss(U )A ->ss(V)B the following statements are true.

  1. fpu-1((G,B)')=(fpu-1(G,B))' for any soft set (G,B) in ss(VB.
  2. fpupu-1 ((G,B))) ⊆ (G,B) for any soft set (G, B) in ss(VB .
  3. (F,A)⊆ f-1pu(fpu(F,A)) for any soft set (F, A) inss(UA

Definition 2.10 [7]: Let U be a non-empty finite set of objects called the universe and E be a set of parameters. Let R be a soft equivalence relation on U . The triplet (U,R,E) is said to be the soft approximation space. LetX⊆ U .
(i) The soft lower approximation of X with respect to R and set of parameters E, is the set of all objects, which can be for certain classified as X with respect to R and it is denoted by (LR(X),E). i.e. (LR(X),E) = ∪{R(x):R(x) ⊆X, where R(x) denotes the equivalence class determined by x∈ U .
(ii) The soft upper approximation of X with respect to R and set of parameters E, is the set of all objects, which can be possibly classified as X with respect to R and it is denoted by (UR(X),E). i.e. (UR(X),E) = ∪{R(x):R(x) ∩X=φ} (iii) The soft boundary region of X with respect to R and set of parameters E, is the set of all objects, which can be classified neither inside X nor as outside X with respect to R and is denoted by (BR(X),E). That is (BR(X),E)= (UR(X),E)-(LR(X),E

Definition 2.11 [7]: Let U be a non-empty universal set and E be a set of parameters. Let R be a soft equivalence relation on U. LetX⊆ U .
Let (τR(X),U,E) = {U,φ,(LR(X),E),(UR(X),E),(BR(X),E)}. Then, (τR(X),U,E) is a soft topology on (U,E), called as the soft nano topology with respect to X . Elements of the soft nano topology are known as the soft nano open sets and (τR(X),U,E) is called soft nano topological space. The complements of soft nano open sets are called as soft nano closed sets in (τR(X),U,E).

Definition 2.12 [7]: If (τR(X),U,E) is a soft nano topological space with respect to X and E, whereX⊆ U and if (A,E) ⊆ (U,E), then the soft nano interior of (A, E) is defined as the union of all soft nano open subsets of (A, E) and it is denoted byNint(A,E). i.e.Nint(A,E) is the largest soft nano open subset of (A, E).

Definition 2.13 [7]: The soft nano closure of (A, E) is defined as the intersection of all soft nano closed sets containing (A, E) and it is denoted byNcl(A,E). i.e.Ncl(A,E) is the smallest soft nano closed set containing (A, E).

Definition 2.14 [8]: Let (τR(X),U,E) and (τR' (Y),V,K) be soft nano topological spaces. Then a mapping f: (τR(X),U,E) -> (τR'(y),V,k ) is said to be soft nano continuous over U if the inverse image of each soft nano open set in V is soft nano open in U .

Definition 2.15 [8] A function f: (τR(X)U,E) -> (τR'(Y) ),V,K) is said to be a soft nano homeomorphism if

  1. f is one-one and onto
  2. f is soft nano continuous and
  3. f is soft nano open

WEAKER FORMS OF SOFT NANO OPEN SETS

Definition 3.1: Let (τR(X),U,E) be a soft nano topological space and (G, E) be any soft set over U .Then, (G, E) is said to be

  1. soft nano semi-open if (G,E)⊆Ncl(Nint(G,E))
  2. soft nano pre-open if (G,E)⊆Nint(Ncl(G,E))
  3. soft nano α-open if (G,E)⊆Nint(Ncl(Nint(G,E)))

Throughout this paper, let SNSO(U,E),SNP(U,E),SNα(U,E) denotes the family of all soft nano semi-open, soft nano pre-open and soft nano α-open sets over U with respect to R and E.

Definition 3.2:Let (τR(X),U,E) be a soft nano topological space and (G, E) be any soft set over U . Then, (G, E) is said to be soft nano semi-closed(soft nano pre-closed, soft nano α-closed respectively) if its soft complement is soft semi open(soft nano pre-open, soft nano α-closed respectively).

Example 3.3: Let U= {x, y, z, w}, E = {m1 , m2 , m3 } and (A, E) = {(m1 , {x}), (m2, {z}), (m3 , {y, z})} be a soft set over U . Let soft equivalence relation R on U is defined as R = {F (m1 )X F (m2 ), F (m2 )X F (m1 ), F (m1 )X F (m1 ), F (m2 )X F (m2 ), F (m3 )X F (m3 )}.
Now, soft equivalence classes are [F (m1 )] = {F (b)/F (b)RF (a)} = {F (m1 ), F (m2 )} =
[F (m2 )], [F (m3 )] = {F (m3 )}. Let x={x,y}⊆U and
U/R = {F (m1 ), F (m2 ), F (m3 )} = {{x}, {z}, {y, z}}
Then,(LR(X),E) = {(m1 ,{x}),(m2 ,{x}),(m3 ,{x})}
(UR(X),E) = {(m1 ,{x,y,z}),(m2 ,{x,y,z}),(m3 ,{x,y,z})}, (BR(X),E) ={(m1 ,{y,z}),(m2 ,{y,z}),(m3 ,{y,z})}
Then, soft nano topology is (τR(X),U,E) ={U,φ,(LR(X),E),(UR(X),E),(BR(X),E)}
Here, soft nano closed sets are U,φ, (LR(X),E)' = {(m1 ,{y,z,w}),(m2 ,{y,z,w}),(m3 ,{y,z,w})}, UR(X),E)' = {(m1 ,{w}),(m2 ,{w}),(m3 ,{w})},
(BR(X),E)' = {(m1 ,{x,w}),(m2 ,{x,w}),(m3 ,{x,w})}
Then, family of soft nano semi-open sets is SNSO (U,E) = {U,φ,(A1 ,E),(A2 ,E),(A3 ,E),(A4 ,E)}
Where (A1 , E) = {(m1 , {x}), (m2 , {x}), (m3 , {x})}
(A2 , E) = {(m1 , {x, w}), (m2 , {x, w}), (m3 , {x, w})}
(A3 , E) = {(m1 , {y, z}), (m2 , {y, z}), (m3 , {y, z})}
(A4 , E) = {(m1 , {x, y, z}), (m2 , {x, y, z}), (m3 , {x, y, z})}
The family of all soft nano pre-open sets isSNPO‚(U,E) = {U,φ,(B1 ,E),(B2 ,E),(B3 ,E),...(B9 ,E)}
Where (B1 , E) = {(m1 , {x}), (m2 , {x}), (m3 , {x})}
(B2 , E) = {(m1 , {y}), (m2 , {y}), (m3 , {y})}
(B3 , E) = {(m1 , {z}), (m2 , {z}), (m3 , {z})}
(B4 , E) = {(m1 , {x, y}), (m2 , {x, y}), (m3 , {x, y})}
(B5 , E) = {(m1 , {x, z}), (m2 , {x, z}), (m3 , {x, z})}
(B6 , E) = {(m1 , {y, z}), (m2 , {y, z}), (m3 , {y, z})}
(B7 , E) = {(m1 , {x, y, z}), (m2 , {x, y, z}), (m3 , {x, y, z})}
(B8 , E) = {(m1 , {x, z, w}), (m2 , {x, z, w}), (m3 , {x, z, w})}
(B9 , E) = {(m1 , {x, y, w}), (m2 , {x, y, w}), (m3 , {x, y, w})}
And the family of all soft nanoα-open sets is
SNαO(U,E) = {U,φ,(A1 ,E),(A2 ,E),(A3 ,E)} Where (C1 , E) = {(m1 , {x}), (m2 , {x}), (m3 , {x})} (C2 , E) = {(m1 , {y, z}), (m2 , {y, z}), (m3 , {y, z})} (C3 , E) = {(m1 , {x, y, z}), (m2 , {x, y, z}), (m3 , {x, y, z})}

Here note that,SNSO(U,E) does not form a soft topology over U , since (A2 ,E A3 ,E) ∈/ SNSO (U,E). Also, SNPO‚(U,E) does not form a soft topology over U , since (B8 ,E) B9 ,E) ∈/SNPO‚(U,E). But,SN(U,E) forms a soft topology over U.

Theorem 3.4 : If (G, E) is a soft nano open set in (τR(X),U,E) then it is soft nanoα-open in (τR(X),U,E).

Proof : Let (G, E) be a soft nano open set. Then,Nint(G,E)=(G,E). Hence,Ncl(Nint(G,E))=Ncl(G,E)⊇(G,E).Which implies (G,E)⊆Ncl(Nint(G,E)) impliesNint(G,E)⊆Nint(Ncl(Nint(G,E))). Then, (G,E)⊆Nint(Ncl(Nint(G,E))). Therefore(G, E) is soft nanoα-open.

Theorem 3.5: SN S O(U,E) ⊆SNSO(U,E) in a soft nano topological space (τR(X),U,E).

Proof: Let (G,E)∈SNαO(U,E). Then, by definition, we have (G,E)⊆Nint(Ncl(Nint(G,E))) which implies (G,E)⊆Ncl(Nint(G,E)), sinceNint(G,E)⊆(G,E). Therefore, (G, E) is soft nano semi-open set and (G,E) ∈SNSO(U,E). Hence,SNαO(U,E) ⊆SNSO(U,E).

Remark 3.6: The converse of theorem 3.5 is not true. Because in example 3.3, (A2 , E) and (A3 , E) are soft nano semi-open sets but not soft nanoα-open.

Theorem 3.7: SNαO(U,E)⊆SNPO(U,E) in a soft nano topological space (τR(X),U,E).

Proof : Let (G,E) ⊆SNSO(U,E). Then, (G,E) ⊆Nint(Ncl(Nint(G,E))) implies (G,E)⊆Nint(Ncl(G,E)) since,Nint(G,E) ⊆ (G,E). Then, (G,E) ∈SNPO(U,E). Hence, SNαO(U,E) ⊆SNOƒ O(U,E).

Remark 3.8: The converse of theorem 3.7 is not true. Because in example 3.3, (B2, E) is soft nano pre-open but not soft nanoα-open.

Theorem 3.9: SNαO(U,E) = SNSO(U,E) ∩SNPO(U,E).

Proof : Let (G,E) ∈SN αO(U,E). Then, (G,E) ∈SNSO(U,E) and (G,E) ∈ SNPO(U,E) by theorems 3.5 and 3.7. Hence, (G,E) ∈SNSO(U,E)∩SNPO(U,E). Therefore,SNαO(U,E) ⊆SNSO(U,E) ∩SNPO,(U,E). Now, suppose that (G,E) ∈SNSO(U,E) ∩SNPO(U,E). Then, (G,E) ⊆Ncl(Nint(G,E))) and (G,E)⊆Nint(Ncl(G,E)). Therefore,Nint(Ncl(G,E))⊆Nint(Ncl(Nint(G,E)))= Nint(Ncl(Nint(G,E))), sinceNcl(Ncl(G,E))=Ncl(G,E). Then, Nint(Ncl(G,E))⊆Nint(Ncl(Nint(G,E))). Also, (G,E) ⊆Nint(Ncl(G,E)) ⊆Nint(Ncl(Nint(G,E))) which implies (G,E)⊆Nint(Ncl(Nint(G,E))). Hence, (G,E) ∈SN αO(U,E).Thus,SNSO(U,E) ∩SNPO(U,E) ⊆SNαO(U,E). Therefore,SNαO(U,E) =SNSO(U,E) ∩SNPO(U,E).

Theorem 3.10: In a soft nano topological space (τR(X),U,E) if (LR(X),E) = (UR(X),E) then U,φ,(LR(X),E) = (UR(X),E) and any soft set (G,E) ⊃ (LR(X),E) are the only soft nanoα-open sets.

Proof:By assumption, (LR(X),E) = (UR(X),E) then the soft nano topology is (τR(X),U,E) = {U,φ,(LR(X),E)}. Since from theorem 3.4, any soft nano open set is soft nanoααopen, we have U,φ,(LR(X),E) are the soft nanoα-open sets in (τR(X),U,E). Suppose that (G,E) ⊂ (LR(X),E) thenNint(G,E)=φ becauseφ is the only soft nano open set contained in (G, E). Therefore,Nint(Ncl(Nint(G,E)))=φ and hence (G, E) is not a soft nanoα-open set. Now, suppose that (G,E)⊃(LR(X),E). Then, (LR(X),E) is the largest soft nano open set contained in (G, E) and hence
Nint(Ncl(Nint(G,E)))=Nint(Ncl((LR(X),E)))=Nint((BR(X),E)') =Nint((U,E))=(U,E),
since (BR(X),E) =φ. Thus,Nint(Ncl(Nint(G,E)))= (U,E) and hence (G,E)⊆Nint(Ncl(Nint(G,E))). Therefore, (G, E) is a soft nanoα-open set. Thus, U,φ,(LR(X),E) and any soft set (G,E) ⊃ (LR(X),E) are the only soft nanoα-open sets in (τR(X),U,E) if (LR(X),E) = (UR(X),E).

Theorem 3.11: In a soft nano topological space (τR(X),U,E) if (LR(X),E) =φ then U,φ,(UR(X),E) and any soft set (G,E) ⊃ (UR(X),E) are the only soft nanoα-open sets.

Proof:Proof : Suppose that (LR(X),E)=φ,(BR(X),E)=(UR(X),E). Then, (τR(X),U,E) = {U,φ,(UR(X),E)} and hence U,φ,(UR(X),E) are the soft nanoα-open sets. Let (G,E) ⊂ (UR(X),E) thenNint(G,E) =φ, sinceφ is the only soft nano open set contained in (G, E). Hence,Nint(Ncl(Nint(G,E)))=φ. Therefore, (G, E) is not a soft nanoα-open set. Now, let (G,E) ⊃ (UR(X),E) then (UR(X),E) is the largest soft nano open set contained in (G,E). Then,Nint(Ncl(Nint(G,E)))=Nint(Ncl((UR(x),E)))=Nint(U,E)=(U,E). Thus,
(G,E)⊆Nint(Ncl(Nint(G,E))). Therefore, (G, E) is a soft nanoα-open set if (G,E) ⊃ (UR(X),E). Hence, U,φ,(UR(X),E) and any soft set (G, E) containing (UR(X),E) are the only soft nanoα-open sets if (LR(X),E) =φ.

Theorem3.12: In a soft nano topological space (τR(X),U,E) if (UR(X),E) = (U,E) and (LR(X),E) =φ then U,φ,(LR(X),E) and (BR(X),E) are the only soft nanoα-open sets.

Proof: Suppose that (UR(X),E) = (U,E) and (LR(X),E) =φ then the soft nano open sets are U,φ,(LR(X),E),(BR(X),E) and these are also soft nanoα- open sets. If (G,E) =φ then (G,E) is a soft nanoα-open set. If (G,E) =φ and (G,E) ⊂ (LR(X),E) thenNint(G,E)=φ, becauseφ is the only largest soft nano open set contained in (G, E). Hence, (G,E) is not a proper subset ofNint(Ncl(Nint(G,E))) and therefore (G, E) is not a soft nanoα-open set. Now, if (LR(X),E) ⊂ (G,E), we haveNint(G,E)=(LR(X),E), since (LR(X),E) is the largest soft nano open set contained in (G,E). Therefore,Nint(Ncl(Nint(G,E)))=Nint(Ncl((LR(X),E)))=Nint((BR(X),E)′ )=Nint((LR(X),E))=(LR(X),E) ⊂(G,E). Thus, (G, E) is not a proper subset ofNint(Ncl(Nint(G,E))). Therefore, (G,E) is not a soft nanoα-open set. Similarly, we can verify that any soft set (G,E) ⊂ (BR(X),E) and (G,E) ⊃ (BR(X),E) are not soft nanoα-open. Hence, U,φ,(LR(X),E),(BR(X),E) are the only soft nanoα-open sets in (τR(X),U,E) if (UR(X),E) = (U,E) and (LR(X),E) =φ.

Theorem 3.13: In a soft nano topological space (τR(X),U,E), the setφ and the soft set (G, E) such that (G,E) ⊇ (LR(X),E) are the only soft nano semi-open sets if (UR(X),E) = (LR(X),E).

Proof: Consider (τR(X),U,E) = {U,φ,(LR(X),E)}, since (LR(X),E) = (UR(X),E). Then,φ is obviously a soft nano semi-open sinceNcl(Nint(φ))=φ. If (G, E) is any soft set such that (G,E) ⊂ (LR(X),E) thenNcl(Nint(G,E))=Ncl(φ)=φ. Thus, (G, E) is not a soft nano semi-open if (G,E) ⊂ (LR(X),E). Now, suppose that (G,E) ⊇ (LR(X),E) then,
Ncl(Nint(G,E))=Ncl((LR(X),E))=(U,E). Therefore, (G,E)⊆Ncl(Nint(G,E)) and thus (G, E) is a soft nano semi-open set. Hence,φ and the soft set (G,E) ⊃ (LR(X),E) are the only soft nano semi-open sets in (τR(X),U,E) if (LR(X),E) = (UR(X),E).

Theorem 3.14: In a soft nano topological space (τR(X),U,E), if (LR(X),E) =φ and (UR(X),E) = (U,E) then the soft sets containing (UR(X),E) are the only soft nano semi-open sets.

Proof : Consider, (ττττR(X),U,E) = {U,φ,(UR(X),E)}. Let (G, E) be any soft set over U . If (G,E) ⊂(UR(X),E), thenNcl(Nint(G,E))=Ncl(φ)=φ and hence (G, E) is not a soft nano semi-open set. Now, suppose that (G,E) ⊃ (UR(X),E) thenNcl(Nint(G,E))=Ncl((UR(X),E)=(U,E) and hence (G,E)⊆Ncl(Nint(G,E)). Therefore, (G, E) is a soft nano semi-open set. Hence, the only set (G, E) containing (UR(X),E) is soft nano seni-open set over U .

Theorem 3.15: If (G, E) and (H, E) are two soft nano semi-open sets in soft nano topological space (τττR(X),U,E) the (G,E) ∪ (H,E) is also a soft nano semi-open set in (&atu;R(X),U,E).

Proof : Let (G, E) and (H, E) be two soft nano semi open sets in soft nano topological space (ττR(X),U,E). Then, (G,E)⊆Ncl(Nint(G,E)) and (H,E)⊆Ncl(Nint(H,E)). Now,
(G,E)∪(H,E)⊆Ncl(Nint(G,E))∪Ncl(Nint(H,E))⊆Ncl(Nint(G,E)∪Nint(H,E))⊆Ncl(Nint((G,E)∪(H,E))).Thus, (G,E) ∪ (H,E) is a soft nano semi open set.

Definition 3.16: A soft set (G, E) of a soft nano topological space (τR(X),U,E) is said to be soft nano β-open if (G,E)⊆Ncl(Nint(Ncl(G,E))). The set of all soft nano β-open sets is denoted bySNβO(U,E).

Proposition 3.17: Every soft nano open set is soft nano β-open in soft nano topological space (τR(X),U,E).

Proof : Let (G, E) be a soft nano open set in (τR(X),U,E). Then,Nint(G,E)=(G,E) and (G,E)⊆Ncl(G,E) is always true. Then, (G,E)⊆Ncl(Nint(G,E))⊆Ncl(Nint(Ncl(G,E))).Therefore, (G,E)⊆Ncl(Nint(Ncl(G,E))). Hence, (G,E) is a soft nano β-open.

Proposition 3.18: If (G, E) is soft nano semi-open in (ττR(X),U,E) then it is soft nano β-open.

Proof : Let (G, E) be a soft nano semi-open in (τR(X),U,E) . Then, (G,E)⊆Ncl(Nint(G,E))⊆Ncl(Nint(Ncl(G,E))) since (G,E) ⊆Ncl(G,E) is al- ways true. Hence, (G, E) is soft nano β-open.

Proposition 3.19: Every soft nano pre-open set soft nano β-open in (τR(X),U,E).

Proof : Let (G, E) be a soft nano pre-open set. Then, (G,E)⊆Nint(Ncl(G,E))⊆Ncl(Nint(Ncl(G,E))). Hence, (G, E) is a soft nano β-open set.

Proposition 3.20: Every soft nanoα-open set is soft nano β-open in (τR(X),U,E).

Proof : Let (G, E) be a soft nanoα-open set. We have,Nint(G,E)⊆(G,E) impliesNcl(Nint(G,E))⊆Ncl(G,E).
Thus,Nint(Ncl(Nint(G,E)))⊆Nint(Ncl(G,E))⊆Ncl(Nint(Ncl(G,E))). Hence, (G, E) is a soft nano β-open in (τR(X),U,E).

Remark 3.21: The converse of the propositions 3.17, 3.18, 3.19, 3.20 need not be true always.

Example 3.22: Let U = {a,b,c,d}, E = {m1 , m2 , m3 } and (A, E) = {(m1 , {a, b}), (m2 , {c}), (m3 , {d})} be a soft set over U. Let soft equivalence relation R on U is defined as R = {F (m1) X F (m2 ), F (m2 )X F (m1 ), F (m1 )X F (m1 ), F (m2 )X F (m2 ), F (m3 )X F (m3 )}.
Now, soft equivalence classes are [F (m1 )] = {F (b)/F (b)RF (a)} = {F (m1 ), F (m2 )} = [F (m2 )], [F (m3 )] = {F (m3 )}. Let X = {a, c} ⊆ U and U/R = {F (m1 ), F (m2 ), F (m3 )} = {{a, b}, {c}, {d}} Then,(LR (X ), E) = {(m1 , {c}), (m2 , {c}), (m3 , {c})}
(UR (X ), E) = {(m1 , {a, b, c}), (m2 , {a, b, c}), (m3 , {a, b, c})}, (BR (X ), E) = {(m1 , {a, b}), (m2 , {a, b}), (m3 , {a, b})}
Then, soft nano topology is (τR(X),U,E) = {U,φ,(LR(X),E),(UR(X),E),(BR(X),
E Here the soft set (G, E) = {(m1 , {a, d}), (m2 , {a, d}), (m3 , {a, d})} is a soft nano β-open but not a soft nano open, soft nano semi-open, soft nano pre-open and soft nanoα-open.

Proposition 3.23: If (G, E) and (H, E) are two soft nano β-open sets in soft nano topological space (τR(X),U,E) then (G,E) ∪ (H,E) is also soft nano β- open set in (τR(X),U,E).

Proof : Let (G, E) and (H, E) be soft nano β-open sets. Then, (G,E)⊆Ncl(Nint(Ncl(G,E))) and (H,E)⊆Ncl(Nint(Ncl(H,E))).
Then, (G,E)∪(H,E)⊆Ncl(Nint(Ncl(G,E)))∪Ncl(Nint(Ncl(G,E))) =Ncl[(Ncl(G,E))∪Nint(Ncl(H,E))] ⊆Ncl[Nint[Ncl(G,E)∪Ncl(H,E)]] =Ncl[Nint[Ncl((G,E)∪(H,E))]].
Therefore, (G,E)∪(H,E) ⊆Ncl(Nint(Ncl((G,E)∪(H,E)))). Hence, (G,E)∪(H,E) is a soft nano β-open set.

Remark 3.24: If (G, E) and (H, E) are two soft nano β-open sets then (G,E)∩ (H,E) need not be soft nano β-open.
In example 3.22, the soft sets (G, E) = {(m1 , {a, d}), (m2 , {a, d}), (m3 , {a, d})} and (H, E) = {(m1 , {b, d}), (m2 , {b, d}), (m3 , {b, d})} are soft nano β-open sets but (G,E)∩(H,E) = {(m1 , {d}), (m2 , {d}), (m3 , {d})} is not a soft nano β-open set.

Proposition 3.25: If (G, E) is a soft set over U with respect to parameter set E and (H, E) is a soft nano pre-open over U such that (H,E)⊆(G,E)⊆Ncl(Nint(H,E)) then (G, E) is a soft nano β-open set over U .

Proof: Suppose that (H, E) is a soft nano pre-open set, we have (H,E)⊆Nint(Ncl(H,E)). Now, given that (G,E)⊆Ncl(Nint(H,E)).
Thus, (G,E) ⊆Ncl(Nint(H,E))
Ncl(Nint(Nint(Ncl(H,E)))) (from definition of soft nano pre-open) ⊆Ncl(Nint(Ncl(H,E))) (sinceNint(Nint(H,E))=Nint(H,E)) ⊆Ncl(Nint(Ncl(G,E))) (since(H,E)⊆(G,E))
Hence, (G,E)⊆Ncl(Nint(Ncl(G,E))). Thus, (G,E) is a soft nano β-open set.

REFERENCES

  1. Abdulkadir.A and Halis.A, Some notes on soft topological spaces, Neural Computers and Applications, Vol.21,113-119 (2012).
  2. K.V.Babita, J.J.Sunil, Soft set relations and functions, Computers and Mathematics with Applications, Vol.60, 1840-1849 (2010).
  3. Benchalli.S.S., P.G.Patil, Nivedita Kabbur, On some weaker forms of soft closed sets in soft topological spaces, International Journal of Applied Mathematics, Vol.28(3), 223-235 (2015).
  4. Benchali.S.S., P.G.Patil, Nivedita Kabbur, On soft γ-operations in soft topological spaces, Journal of new theory, Vol.6, 20-32 (2015).
  5. Benchalli.S.S., P.G.Patil, Nivedita Kabbur, On soft γ-compact and soft γ-normal spaces in soft topological spaces, International Journal of Scientific and Innovative Mathematical Research, Vol. 3(1), 213-219 (2015).
  6. Benchalli.S.S., P.G.Patil, Nivedita Kabbur, Soft γ-connected spaces in soft topological spaces, Mathematical Sciences International Research Journal, Vol.4(2), 332-335 (2015).
map1 map2 map3