New Operators Using Neutrosophic δ-Open Set

A. Vadivel1,∗, C. John Sundar2

1PG and Research Department of Mathematics, Government Arts College (Autonomous), Karur - 639 005,

India.

1,2Department of Mathematics, Annamalai University, Annamalai Nagar - 608 002, India.

avmaths@gmail.com1, johnphdau@hotmail.com2

Abstract

In this paper, we introduce some new operators called neutrosophic δ frontier, neutrosophic δ border and neutrosophic δ exterior with the help of neutrosophic δ-open sets in neutrosophic topological space. Also, we discuss the important properties of them and the relations between them.

Keywords: neutrosophic open set, neutrosophic δ-open set, neutrosophic δ frontier, neutrosophic δ border, neutrosophic δ exterior

AMS (2000) subject classification: 03E72, 54A05, 54A40.

1          Introduction

In 1965, the idea of fuzzy set (briefly, fs) gives a degree of membership function was first introduced by Zadeh.19 In 1968, the concept of fuzzy topological space (briefly, fts) was introduced by Chang.4 In 1983, the next stage of fuzzy set was developed by Atanassov3 which gives a degree of membership and a degree of non-membership functions named as intutionistic fuzzy set (briefly, Ifs). In 1997, Coker5 introduced the concept of intutionistic fuzzy topological space (briefly, Ifts) in intutionistic fuzzy set. In 2005, the concept of neutrosophic crisp set and neutrosophic set (briefly, Nss) was investigated by Smaradache.9,14,15 After the introduction of neutrosophic set, there are many fields of mathematics and various applications.1,7,8,13 In 2012, Salama and Alblowi10 defined neutrosophic topological space (briefly, Nsts) and many of its applications in.11,12 The neutrosophic closed sets and neutrosophic continuous functions were introduced by Salama et al.12 in 2014. Saha16 defined δ-open sets in topological spaces. Vadivel et al. in17,18 introduced δ-open sets and their maps in a neutrosophic topological space. P. Iswarya and K. Bageerathi,6 studied neutrosophic frontier and semi-frontier in neutrosophic topological spaces.

In this paper we introduce neutrosophic δ frontier, neutrosophic δ border and neutrosophic δ exterior and discuss their properties in Nsts’s.

2          Preliminaries

Definition 2.1. 10 Let Y be a non-empty set. A neutrosophic set (briefly, Nss) L is an object having the form L = {⟨y,µL(y),σL(y), νL(y)⟩ : y ∈ Y } where µL → [0,1] denote the degree of membership function, σL → [0,1] denote the degree of indeterminacy function and νL → [0,1] denote the degree of non-membership function respectively of each element y ∈ Y to the set L and 0 ≤ µL(y)+σL(y)+νL(y) ≤ 3 for each y ∈ Y .

Remark 2.2. 10 A Nss L = {⟨y,µL(y),σL(y),νL(y)⟩ : y ∈ Y } can be identified to an ordered triple ⟨y,µL(y),σL(y),νL(y)⟩ in [0,1] on Y .

Definition 2.3. 10 Let Y be a non-empty set and the Nss’s L and M in the form L = {⟨y,µL(y),σL(y), νL(y)⟩ : y ∈ Y }, M = {⟨y,µM(y),σM(y),νM(y)⟩ : y ∈ Y }, then

(i)        0N = ⟨y,0,0,1⟩ and 1N = ⟨y,1,1,0⟩,

(ii)       L ⊆ M iff µL(y) ≤ µM(y), σL(y) ≤ σM(y) & νL(y) ≥ νM(y) : y ∈ Y ,

(iii)     L = M iff L ⊆ M and M ⊆ L,

(iv)     1N − L = {⟨y,νL(y),1 − σL(y),µL(y)⟩ : y ∈ Y } = Lc,

(v)       L ∪ M = {⟨y,max(µL(y),µM(y)),max(σL(y),σM(y)),min(νL(y),νM(y))⟩ : y ∈ Y },

(vi)     L ∩ M = {⟨y,min(µL(y),µM(y)),min(σL(y),σM(y)),max(νL(y),νM(y))⟩ : y ∈ Y }.

Definition 2.4. 10 A neutrosophic topology (briefly, Nst) on a non-empty set Y is a family ΨN of neutrosophic subsets of Y satisfying

(i)        0N, 1N ∈ ΨN.

(ii)       L1 ∩ L2 ∈ ΨN for any L1,L2 ∈ ΨN.

(iii)     S Lx ∈ ΨN, ∀ Lx : x ∈ X ⊆ ΨN.

Then (Y,ΨN) is called a neutrosophic topological space (briefly, Nsts) in Y . The ΨN elements are called neutrosophic open sets (briefly, Nsos) in Y . A Nss C is called a neutrosophic closed sets (briefly, Nscs) iff its complement Cc is Nsos.

Definition 2.5. 10 Let (Y,ΨN) be Nsts on Y and L be an Nss on Y , then the neutrosophic interior of L

(briefly, Nsint(L)) and the neutrosophic closure of L (briefly, Nscl(L)) are defined as

Nsint(L) = [{I : I ⊆ L and I is a Nsos in Y }

Nscl(L) = \{J : L ⊆ J and J is a Nscs in Y }.

Definition 2.6. 2 Let (Y,ΨN) be Nsts on Y and L be an Nss on Y . Then L is said to be a neutrosophic regular open set (briefly, Nsros ) if L = Nsint(Nscl(L)).

The complement of a Nsros is called a neutrosophic regular closed set (briefly, Nsrcs) in Y .

Definition 2.7. 17 A set K is said to be a neutrosophic

(i)      δ interior of G (briefly, Nsδint(K)) is defined by Nsδint(K) = S{B : B ⊆ K and B is a Nsros in Y }.

(ii)     δ closure of K (briefly, Nsδcl(K)) is defined by Nsδcl(K) = T{J : K ⊆ J and J is a Nsrcs in Y }.

Definition 2.8. 17 A set L is said to be a neutrosophic δ-open set (briefly, Nsδos) if L = Nsδint(L). The complement of an Nsδos is called a neutrosophic δ closed set (briefly, Nsδcs) in Y .

Proposition 2.9. 17 The Neutrosophic δ-interior operator satisfies

(i)         Nsδint(K) ⊆ K.

(ii)        K ⊆ M ⇒ Nsδint(K) ⊆ Nsδint(M).

(iii)      Nsδint(K ∩ M) = Nsδint(K) ∩ Nsδint(M).

(iv)      Nsδint(K) is the greatest Nsδos containing K.

(v)        Nsδint(K) = K iff K is an Nsδos.

(vi)      Nsδint(Nsδint(K)) = Nsδint(K).

(vii)     (1Ns − Nsδint(K)) = Nsδcl(1Ns − K).

Proposition 2.10. 17 The Neutrosophic δ-closure operator satisfies

(i)            K ⊆ Nsδcl(K).

(ii)          K ⊆ M ⇒ Nsδcl(K) ⊆ Nsδcl(M).

(iii)         Nsδcl(K ∪ M) = Nsδcl(K) ∪ Nsδcl(M).

(iv)         Nsδcl(K) is the smallest Nsδcs ⊆ K.

(v)          Nsδcl(K) = K iff K is an Nsδc set.

(vi)         Nsδcl(Nsδcl(K)) = Nsδcl(K).

(vii)       (1Ns − Nsδcl(K)) = Nsδint(1Ns − K).

(viii)      y ∈ Nsδcl(K) iff K ∩ C for every Nsδos C containing y.

(ix)         Nsδcl(S ∩ T) ⊆ Nsδcl(S) ∩ Nsδcl(T).

Proposition 2.11. 17 The statements are hold for Nsts. Every Nsδos (resp. Nsδcs) is a Nsos (resp. Nscs). But not converse.

3            Neutrosophic δ frontier

In this section, we introduce neutrosophic δ frontier and discuss their properties in neutrosophic topological spaces.

Definition 3.1. Let (Y,ΨN) be a Nsts. Let A be a neutrosophic subset of Y . Then the neutrosophic (resp.

δ) frontier of a neutrosophic subset A were denoted by NsFr(A) (resp. NsδFr(A)) and were defined by NsFr(A) = Nscl(A) ∩ Nscl(Ac) (resp. NsδFr(A) = Nsδcl(A) ∩ Nsδcl(Ac)). Example 3.2. Let Y = {l,m,n} and define Ns’s Y1,Y2 & Y3 in Y are

,

Then we have τN = {0N,Y1,Y2,1N}. Let, then

.

.

Remark 3.3. For a neutrosophic subset A of Y , NsFr(A) (resp. NsδFr(A)) is Nsc (resp. Nsδc).

Theorem 3.4. For a neutrosophic subset A in Nsts (Y,ΨN),

(i)      NsFr(A) = NsFr(Ac).

(ii)     NsδFr(A) = NsδFr(Ac).

Proof. (i) Let A be a neutrosophic subset in Nsts (Y,ΨN). Then by Definition 3.1 NsFr(A) = Nscl(A) ∩ Nscl(Ac) = Nscl(Ac) ∩ Nscl(A) = Nscl(Ac) ∩ (Nscl(Ac)c). Again by Definition 3.1 this is equal to NsFr(Ac). Hence NsFr(A) = NsFr(Ac).

(ii) Let A be a neutrosophic subset in Nsts (Y,ΨN). Then by Definition 3.1 NsδFr(A) = Nsδcl(A) ∩ Nsδcl(Ac) = Nsδcl(Ac) ∩ Nsδcl(A) = Nsδcl(Ac) ∩ (Nsδcl(Ac)c). Again by Definition 3.1 this is equal to NsδFr(Ac). Hence NsδFr(A) = NsδFr(Ac).  

Theorem 3.5. Let A be a neutrosophic subset in Nsts (Y,ΨN). Then

(i)      NsFr(A) = Nscl(A) − Nsint(A).

(ii)     NsδFr(A) = Nsδcl(A) − Nsδint(A).

Proof. (i) Let A be a neutrosophic subset in Nsts (Y,ΨN). By Theorem 2.10 (vii), (Nscl(Ac))c = Nsint(A) and by Definition 3.1, NsFr(A) = Nscl(A) ∩ (Nscl(Ac)) = Nscl(A) ∩ (Ns int(Ac))c. By using A − B = A ∩ Bc, NsFr(A) = Nscl(A) − Nsint(A). Hence NsFr(A) = Nscl(A) − Nsint(A).

(ii) Let A be a neutrosophic subset in Nsts (Y,ΨN). By Theorem 2.10 (vii), (Nsδcl(Ac))c = Nsδint(A) and by Definition 3.1, NsδFr(A) = Nsδcl(A) ∩ (Nsδcl(Ac)) = Nsδcl(A) ∩ (Nsδ int(Ac))c. By using A−B = A∩Bc, NsδFr(A) = Nsδcl(A)−Nsδint(A). Hence NsδFr(A) = Nsδcl(A)−Nsδint(A).

Theorem 3.6. A neutrosophic subset A is Nsc (resp. Nsδc) set in Y if and only if NsFr(A) ⊆ A (resp. NsδFr(A) ⊆ A).


be a Nsδc set in the Nsts (Y,ΨN). Then by Definition 3.1, NsδFr(A) = Nsδcl(A) ∩ Nsδcl(Ac) ⊆ Nsδcl(A). By using Theorem 2.10 (v), Nsδcl(A) = A. Hence NsδFr(A) ⊆ A, if A is Nsδc in Y .

Conversely, Assume that, NsδFr(A) ⊆ A. Then Nsδcl(A) − Nsδint(A) ⊆ A. Since Nsδint(A) ⊆ A, we conclude that Nsδcl(A) = A and hence A is Nsδc. The proof of the other is similar.        

Theorem 3.7. If A is a Nso (resp. Nsδo) set in Y , then NsFr(A) ⊆ Ac (resp. NsδFr(A) ⊆ Ac).

Proof. Let A be a Nsδo set in the Nsts (Y,ΨN). By Definition 2.8, Ac is Nsδc set in Y . By Theorem 3.6, NsδFr(Ac) ⊆ Ac and by Theorem 3.6, we get NsδFr(A) ⊆ Ac.

      The proof of the other is similar.                                                                                                                             

Theorem 3.8. Let A ⊆ B and B be any Nsc (resp. Nsδc) set in Y . Then NsFr(A) ⊆ B (resp. NsδFr(A) ⊆

B).

Proof. By Theorem 2.10 (ii), A ⊆ B, Nsδcl(A) ⊆ Nsδcl(B). By Definition 3.1, NsδFr(A) = Nsδcl(A) ∩ Nsδcl(Ac) ⊆ Nsδcl(B) ∩ Nsδcl(Ac) ⊆ Nsδcl(B). Then by Proposition 2.10 (v), this is equal to B. Hence NsδFr(A) ⊆ B.

      The proof of the other is similar.                                                                                                                             

Theorem 3.9. Let A be a neutrosophic subset in the Nsts (Y,ΨN). Then (NsFr(A))c = Nsint(A) ∪ Nsint(Ac) (resp. (NsδFr(A))c = Nsδint(A) ∪ Nsδint(Ac)).

Proof. Let A be a neutrosophic subset in the Nsts (Y,ΨN). Then by Definition 3.1, (NsδFr(A))c = (Nsδcl(A) ∩ Nsδcl(Ac))c = ((Nsδcl(A))c ∪ (Nsδcl(Ac))c. By Theorem 2.10 (vii), which is equal to Nsδint(Ac) ∪ Nsδint(A). Hence (NsδFr(A))c = Nsδint(A) ∪ Nsδint(Ac). The proof of the other is similar.             

Theorem 3.10. For a neutrosophic subset A in the Nsts (Y,ΨN), then NsδFr(A) ⊆ NsFr(A).

Proof. Let A be a neutrosophic subset in the Nsts (Y,ΨN). Then by Proposition 2.11, Nsδcl(A) ⊇ Nscl(A) and Nscl(Ac) ⊆ Nscl(Ac). By Definition 3.1, NsδFr(A) = Nsδcl(A)∩Nsδcl(Ac) ⊆ Nscl(A)∩Nscl(Ac), this is equal to NsFr(A). Hence NsδFr(A) ⊆ NsFr(A).

Theorem 3.11. For a neutrosophic subset A in the Nsts (Y,ΨN), Nscl(NsFr(A)) ⊆ NsFr(A) (resp. Nsδcl(NsδFr(A)) ⊆ NsδFr(A)).

Proof. Let A be the neutrosophic subset in the Nsts (Y,ΨN). Then by Definition 3.1, Nsδcl(NsδFr(A)) = Nsδcl(Nsδcl(A) ∩ (Nsδcl(Ac))) ⊆ (Nsδcl(Nsδcl(A))) ∩ (Nsδcl(Nsδcl(Ac))). By Theorem 2.10 (vi), Nsδcl(NsδFr(A)) = Nsδcl(A) ∩ (Nsδcl(Ac)). By Definition 3.1, this is equal to NsδFr(A). The proof of the other is similar.   

Theorem 3.12. For a neutrosophic subset A in the Nsts (Y,ΨN), NsFr(Nsint(A)) ⊆ NsFr(A) (resp. NsδFr(Nsδint(A)) ⊆ NsδFr(A)).

Proof. Let A be the neutrosophic subset in the Nsts (Y,ΨN). Then

NsδFr(Nsδint(A)) =Nsδcl(Nsint(A)) ∩ (Nsδcl(Nsδint(A))c)[by Definition 3.1]

=Nsδcl(Nsδint(A)) ∩ (Nsδcl(Nsδcl(Ac)))[by Theorem 2.9 (vii)]

=Nsδcl(Nsδint(A)) ∩ (Nsδcl(Ac))[by Theorem 2.10 (vi)]

⊆Nsδcl(A) ∩ Nsδcl(Ac)[by Theorem 2.9 (i)] =NsδFr(A)[by Definition 3.1].

Hence NsδFr(Nsδint(A)) ⊆ (NsδFr(A)). The proof of the other is similar.               

Theorem 3.13. For a neutrosophic subset A in the Nsts (Y,ΨN), then NsFr(Nscl(A)) ⊆ NsFr(A) (resp. NsδFr(Nsδcl(A)) ⊆ NsδFr(A)).

be a neutrosophic subset in the Nsts (Y,ΨN). Then

NsδFr(Nsδcl(A)) =Nsδcl(Nsδcl(A)) ∩ (Nsδcl(Nsδcl(A))c)[by Definition 3.1]

=Nsδcl(A) ∩ (Nsδcl(Nsδint(Ac)))[by Theorems 2.10 (vii) and 2.10 (ii) & (vi)]

⊆Nsδcl(A) ∩ Nsδcl(Ac)[by Theorem 2.9 (i)]

=NsδFr(A)[by Definition 3.1]

Hence NsδFr(Nsδcl(A)) ⊆ NsδFr(A). The proof of the other is similar.    

Theorem 3.14. Let A be a neutrosophic subset in the Nsts (Y,ΨN). Then Nsint(A) ⊆ A−NsFr(A) (resp. Nsδint(A) ⊆ A − NsδFr(A)).

Proof. Let A be a neutrosophic subset in the Nsts (Y,ΨN). Now by Definition 3.1,

A − NsδFr(A) =A ∩ (NsδFr(A))c

=A ∩ [Nsδcl(A) ∩ Nsδcl(Ac)]c

=A ∩ [Nsδint(Ac) ∪ Nsδint(A)]

=[A ∩ Nsδint(Ac)] ∪ [A ∩ Nsδint(A)] =[A ∩ Nsδint(Ac)] ∪ Nsδint(A) ⊇ Nsδint(A)

Hence Nsδint(A) ⊆ A − NsδFr(A).

      The proof of the other is similar.                                                                                                                             

Remark 3.15. In general topology, the following conditions are hold:

(i)        NsFr(A) ∩ Nsint(A) = 0N (resp. NsδFr(A) ∩ Nsδint(A) = 0N),

(ii)       Nsint(A) ∪ NsFr(A) = Nscl(A) (resp. Nsδint(A) ∪ NsδFr(A) = Nsδcl(A)),

(iii)     Nsint(A) ∪ Nsint(Ac) ∪ NsFr(A) = 1N (resp. Nsδint(A) ∪ Nsδint(Ac) ∪ NsδFr(A) = 1N).

But the neutrosophic topology, we give counter-examples to show that the condition of neutrosophic subset of the above remark may not be hold in general.

Example 3.16. In Example 3.2, Let, then

, .

,

,

Example 3.17. In Example 3.2, Let, then

,

,

Theorem 3.18. Let A and B be neutrosophic subsets in the Nsts (Y,ΨN). Then NsFr(A∪B) ⊆ NsFr(A)∪ NsFr(B) (resp. NsδFr(A ∪ B) ⊆ NsδFr(A) ∪ NsδFr(B)).

and B be neutrosophic subsets in the Nsts (Y,ΨN). Then

NsδFr(A ∪ B) =Nsδcl(A ∪ B) ∩ Nsδcl(A ∪ B)c[by Definition 3.1]

=Nsδcl(A ∪ B) ∩ Nsδcl(Ac ∩ Bc)

⊆(Nsδcl(A) ∪ Nsδcl(B) ∩ ((Nsδcl(Ac))) ∩ (Nsδcl(Bc))[by Theorem 2.10 (iii) & (ix)]

=[(Nsδcl(A) ∪ (Nsδcl(B)) ∩ (Nsδcl(Ac)))] ∩ [(Nsδcl(A) ∪ (Nsδcl(B)) ∩ (Nsδcl(Bc)))]

=[(Nsδcl(A) ∩ Nsδcl(Ac)) ∪ ((Nsδcl(B) ∩ (Nsδcl(Ac))))] ∩ [(Nsδcl(A) ∩ (Nsδcl(Bc))) ∪ ((Nsδcl(B) ∩ (Nsδcl(Bc))))]

=[NsδFr(A) ∪ (Nsδcl(B)) ∩ (Nsδcl(Ac))] ∩ [(Nsδcl(A) ∩ (Nsδcl(Bc))) ∪ (NsδFr(B))]

[by Definition 3.1]

=(NsδFr(A) ∪ NsδFr(B)) ∩ [(Nsδcl(B) ∩ (Nsδcl(Ac))) ∪ ((Nsδcl(A) ∩ Nsδcl(Bc)))] ⊆NsδFr(A) ∪ NsδFr(B).

Hence, NsδFr(A ∪ B) ⊆ NsδFr(A) ∪ NsδFr(B).

      The proof of the other is similar.                                                                                                                             

Note 1. The following example shows that

(i)      NsFr(A ∩ B) ̸⊆ NsFr(A) ∪ NsFr(B) and NsFr(A) ∩ NsF(B) ̸⊆ NsFr(A ∩ B).

(ii)     NsδFr(A ∩ B) ̸⊆ NsδFr(A) ∪ NsδFr(B) and NsδFr(A) ∩ NsδF(B) ̸⊆ NsδFr(A ∩ B).

Theorem 3.19. For any neutrosophic subsets A and B in the Nsts (Y,ΨN), NsFr(A ∩ B) ⊆ (NsFr(A) ∩

(Nscl(B))) ∪ (NsFr(B) ∩ Nscl(A)) (resp. NsδFr(A ∩ B) ⊆ (NsδFr(A) ∩ (Nsδcl(B))) ∪ (NsδFr(B) ∩

Nsδcl(A))).

Proof. Let A and B be neutrosophic subsets in the Nsts (Y,ΨN). Then

NsδFr(A ∩ B) =Nsδcl(A ∩ B) ∩ (Nsδcl(A ∩ B)c)[by Definition 3.1]

=Nsδcl(A ∩ B) ∩ (Nsδcl(Ac ∪ Bc))

⊆(Nsδcl(A) ∩ Nsδcl(B)) ∩ (Nsδcl(Ac) ∪ Nsδcl(Bc))[by Theorem 2.10 (iii) & (ix)]

=[(Nsδcl(A) ∩ Nsδcl(B)) ∩ Nsδcl(Ac)] ∪ [(Nsδcl(A) ∩ Nsδcl(B)) ∩ Nsδcl(Bc)] =(NsδFr(A) ∩ Nsδcl(B)) ∪ (NsδFr(B) ∩ Nsδcl(A))[by Definition 3.1].

Hence NsδFr(A ∩ B) ⊆ ((NsδFr(A) ∩ (Nsδcl(B))) ∪ (NsδFr(B) ∩ (Nsδcl(A)))). The proof of the other is similar.   

Corollary 3.20. For any neutrosophic subsets A and B in the Nsts (Y,ΨN), NsFr(A ∩ B) ⊆ NsFr(A) ∪ NsFr(B) (resp. NsδFr(A ∩ B) ⊆ NsδFr(A) ∪ NsδFr(B)).

Proof. Let A and B be neutrosophic subsets in the Nsts (Y,ΨN). Then

NsδFr(A ∩ B) =Nsδcl(A ∩ B) ∩ (Nsδcl(A ∩ B)c)[by Definition 3.1]

=Nsδcl(A ∩ B) ∩ (Nsδcl(Ac ∪ Bc)

⊆(Nsδcl(A) ∩ Nsδcl(B)) ∩ (Nsδcl(Ac) ∪ Nsδcl(Bc))[by Theorem 2.10 (iii) & (ix)]

=(Nsδcl(A) ∩ Nsδcl(B)) ∩ (Nsδcl(Ac) ∪ (Nsδcl(A) ∩ Nsδcl(B)) ∩ (Nsδcl(Bc)))

=(NsδFr(A) ∩ Nsδcl(B)) ∪ (Nsδcl(A) ∩ NsδFr(B))[by Definition 3.1] ⊆NsδFr(A) ∪ (NsδFr(B).

Hence NsδFr(A ∩ B) ⊆ NsδFr(A) ∪ NsδFr(B).

      The proof of the other is similar.                                                                                                                             

Theorem 3.21. For any neutrosophic subset A in the Nsts (Y,ΨN),

(i)          (a) NsFr(NsFr(A)) ⊆ NsFr(A),

(b)    NsFr(NsFr(NsFr(A))) ⊆ NsFr(NsδFr(A)).

(ii)         (a) NsδFr(NsδFr(A)) ⊆ NsδFr(A),


(b)    NsδFr(NsδFr(NsδFr(A))) ⊆ NsδFr(NsδFr(A)).

Proof. (ii) (a) Let A be a neutrosophic subset in the Nsts (Y,ΨN). Then

NsδFr(NsδFr(A)) =Nsδcl(NsδFr(A)) ∩ Nsδcl(NsδFr(A)c)[by Definition 3.1]

=Nsδcl(Nsδcl(A) ∩ (Nsδcl(Ac)) ∩ (Nsδcl(Nsδcl(A)) ∩ (Nsδcl(Ac))c))

[by Definition 3.1]

⊆(Nsδcl(Nsδcl(A)) ∩ (Nsδcl(Nsδcl(Ac))) ∩ (Nsδcl(Nsδint(Ac))) ∪ (Nsδint(A)))

[by Theorem 2.10 (vi) & (ix)] =(Nsδcl(A) ∩ (Nsδcl(Ac)) ∩ (Nsδcl(Nsδint(A) ∪ Nsδint(A))))

[by Theorem 2.10 (vi)]

⊆Nsδcl(A) ∩ Nsδcl(Ac)

=NsδFr(A)[by Definition 3.1].

Therefore NsδFr(NsδFr(A)) ⊆ NsδFr(A).

(b) Again, NsδFr(NsδFr(NsδFr(A))) ⊆ NsδFr(NsδFr(A)). The proof of the other is similar.          

4          Neutrosophic δ border and neutrosophic δ exterior

In this section, we introduce the neutrosophic δ border, neutrosophic δ exterior using neutrosophic δ open sets and their properties are discussed in Nsts’s.

Definition 4.1. Let A be a neutrosophic subset of Nsts (Y,ΨN). Then the set NsBr(A) = A ∩ Nsint(A) (resp. NsδBr(A) = A∩Nsδint(A)) is called the neutrosophic (resp. δ) border of A (briefly, NsBr(A) (resp.

NsδBr(A)))

Example 4.2. In Example 3.2, Let, then

.

.

Theorem 4.3. If a subset A of Y is Nsc (resp. Nsδc), then NsBr(A) = NsFr(A) (resp. NsδBr(A) = NsδFr(A)).

Proof. Let A be a Nsδc subset of Y . Then by Theorem 2.10 (vii), Nsδcl(A) = A. Now, NsδFr(A) = Nsδcl(A) − Nsδint(A) = A − Nsδint(A) = NsδBr(A).

      The proof of the other is similar.                                                                                                                             

Theorem 4.4. For a neutrosophic subset A of Y , A = Nsint(A) ∪ NsBr(A) (resp. A = Nsδint(A) ∪ NsδBr(A)).

Proof. Let x(α,β,γ) ∈ A. If x(α,β,γ) ∈ Nsδint(A), then the result is obvious. If x(α,β,γ) Nsδint(A), then by the definition of NsδBr(A),x(α,β,γ) ∈ NsδBr(A). Hence x(α,β,γ) ∈ Nsδint(A) ∪ NsδBr(A) and so A ⊆ Nsδint(A) ∪ NsδBr(A). On the other hand, since Nsδint(A) ⊆ A and NsδBr(A) ⊆ A, we have Nsδint(A) ∪ NsδBr(A) ⊆ A.

      The proof of the other is similar.                                                                                                                             

Theorem 4.5. For a neutrosophic subset A of Y , Nsint(A)∩NsBr(A) = 0N (resp. Nsδint(A)∩NsδBr(A) = 0N).

Proof. Suppose Nsδint(A) ∩ NsδBr(A) ̸= 0N. Let x(α,β,γ) ∈ Nsδint(A) ∩ NsδBr(A). Then x(α,β,γ) ∈ Nsδint(A) and x(α,β,γ) ∈ NsδBr(A). Since NsδBr(A) = A − Nsδint(A), then x(α,β,γ) ∈ A. But x(α,β,γ) ∈ Nsδint(A),x(α,β,γ) ∈ A. There is a contradiction. Hence Nsδint(A) ∩ NsδBr(A) = 0N. The proof of the other is similar.               

Theorem 4.6. For a neutrosophic subset A of Y , A is a Nso (resp. Nsδo) set if and only if NsBr(A) = 0N (resp. NsδBr(A) = 0N).

Proof. Necessity: Suppose A is Nsδo. Then by Theorem 2.9 (v), Nsδint(A) = A. Now, NsδBr(A) = A − Nsδint(A) = A − A = 0N.

Sufficiency: Suppose NsδBr(A) = 0N. This implies, A−Nsδint(A) = 0N. Therefore A = Nsδint(A) and hence A is Nsδo.

      The proof of the other is similar.                                                                                                                             

Corollary 4.7. For a Nsts, NsBr(0N) = 0N and NsBr(1N) = 0N (resp. NsδBr(0N) = 0N and NsδBr(1N) = 0N).

Proof. Since 0N and 1N are Nsδo, by Theorem 4.6, NsδBr(0N) = 0N and NsδBr(1N) = 0N.

      The proof of the other is similar.                                                                                                                             

Theorem 4.8. For a neutrosophic subset A of Y , NsBr(Nsint(A)) = 0N (resp. NsδBr(Nsδint(A)) = 0N).

Proof. By the definition of Nsδ border, NsδBr(Nsδint(A)) = Nsδint(A) − Nsδint(Nsδint(A)). By Theorem 2.9 (vi), Nsδint(Nsδint(A)) = Nsδint(A) and hence NsδBr(Nsδint(A)) = 0N. The proof of the other is similar.     

Theorem 4.9. For a neutrosophic subset A of Y , Nsint(NsBr(A)) = 0N (resp. Nsδint(NsδBr(A)) = 0N).

Proof. Let x(α,β,γ) ∈ Nsδint(NsδBr(A)). Since NsδBr(A) ⊆ A, by Theorem 2.9 (i), Nsδint(NsδBr( A)) ⊆ Nsδint(A). Hence x(α,β,γ) ∈ Nsδint(A). Since Nsδint(NsδBr(A)) ⊆ NsδBr(A),x(α,β,γ) ∈

NsδBr(A). Therefore x(α,β,γ) ∈ Nsδint(A) ∩ NsδBr(A),x(α,β,γ) = 0N. The proof of the other is similar.     

Theorem 4.10. For a neutrosophic subset A of Y , NsBr(NsBr(A)) = NsBr(A) (resp. NsδBr(NsδBr(A)) = NsδBr(A)).

Proof. By the definition of Nsδ border, NsδBr(NsδBr(A)) = NsδBr(A) − Nsδint(NsδBr(A)). By Theorem 4.9 Nsδint(NsδBr(A)) = 0N and hence NsδBr(NsδBr(A)) = NsδBr(A). The proof of the other is similar.   

Theorem 4.11. For a subset A of a space Y , the following statements are equivalent

(i)        A is Nso (resp. Nsδo).

(ii)       A = Nsint(A) (resp. A = Nsδint(A)).

(iii)     NsBr(A) = 0N (resp. NsδBr(A) = 0N).

Proof. (i) → (ii) Obvious from Theorem 2.9.

(ii)      → (iii). Suppose that A = Nsδint(A). Then by Definition, NsδBr(A) = Nsδint(A)−Nsδint(A) =

0N.

(iii)    → (i). Let NsδBr(A) = 0N. Then by Definition 4.1, A − Nsδint(A) = 0N and hence A =

Nsδint(A).

      The proof of the other is similar.                                                                                                                             

Theorem 4.12. Let A be a neutrosophic subset of Y . Then, NsBr(A) = A ∩ Nscl(Ac) (resp. NsδBr(A) = A ∩ Nsδcl(Ac)).

Proof. Since NsδBr(A) = A − Nsδint(A) and by Theorem 2.10, NsδBr(A) = A − (Nsδcl(Ac))c = A ∩ (Nsδcl (Ac)c) = A ∩ Nsδcl(Ac).

      The proof of the other is similar.                                                                                                                             

Theorem 4.13. For a neutrosophic subset A of Y , NsBr(A) ⊆ NsFr(A) (resp. NsδBr(A) ⊆ NsδFr(A)).

Proof. Since A ⊆ Nsδcl(A), A − Nsδint(A) ⊆ Nsδcl(A) − Nsδint(A). That implies, NsδBr(A) ⊆ NsδFr(A).

      The proof of the other is similar.                                                                                                                             

Definition 4.14. Let A be a neutrosophic subset of a Nsts (Y,ΨN). The neutrosophic (resp. δ) interior of Ac is called the neutrosophic (resp. δ) exterior of A and it is denoted by NsExt(A) (resp. NsδExt(A)). That is, NsExt(A) = Nsint(Ac) (resp. NsδExt(A) = Nsδint(Ac)).

Example 4.15. In Example 3.2, Let, then

.

.

Theorem 4.16. For a neutrosophic subset A of Y , NsExt(A) = (Nscl(A))c (resp. NsδExt(A) = (Nsδ cl(A))c).

Proof. We know that, 1N − Nsδcl(A) = Nsδint(Ac), then NsδExt(A) = Nsδint(Ac) = (Nsδcl(A))c. The proof of the other is similar.

Theorem 4.17. For a neutrosophic subset A of Y , NsExt(NsExt(A)) = Nsint(Nscl(A)) ⊇ Nsint(A) (resp. NsδExt(NsδExt(A)) = Nsδint(Nsδcl(A)) ⊇ Nsδint(A)).

Proof. Now, NsδExt(NsδExt(A)) = NsδExt(Nsδint(Ac)) = Nsδint((Nsδint(Ac))c) = Nsδint(Ns δcl(A)) ⊇ Nsδint(A).

      The proof of the other is similar.                                                                                                                             

Theorem 4.18. For a neutrosophic subset A of Y , If A ⊆ B, then NsExt(B) ⊆ NsExt(A) (resp. Nsδ Ext(B) ⊆ NsδExt(A)).

Proof. Suppose A ⊆ B. Now, NsδExt(B) = Nsδint(Bc) ⊆ Nsδint(Ac) = NsδExt(A).

      The proof of the other is similar.                                                                                                                             

Theorem 4.19. For a neutrosophic subset A of Y , NsExt(1N) = 0N and NsExt(0N) = 1N (resp. Nsδ Ext(1N) = 0N and NsδExt(0N) = 1N).

Proof. Now, NsδExt(1N) = Nsδint((1N)c) = Nsδint(0N) and NsδExt(0N) = Nsδint((0N)c) = Nsδint(1N). Since 0N and 1N are Nsδo sets, then Nsδint(0N) = 0N and Nsδint(1N) = 1N. Hence

NsδExt(0N) = 1N and NsδExt(1N) = 0N.

      The proof of the other is similar.                                                                                                                             

Theorem 4.20. For a neutrosophic subset A of Y , NsExt(A) = NsExt((NsExt(A))c) (resp. NsδExt(A) = NsδExt((NsδExt(A))c)).

Proof. Now, NsδExt((NsδExt(A))c) = NsδExt((Nsδint(Ac))c) = Nsδint((((Nsδint(Ac))c))c) = Nsδ int(Nsδint(Ac)) = Nsδint(Ac) = NsδExt(A). The proof of the other is similar. 

Theorem 4.21. For a sub sets A and B of Y , the followings are valid.

(i)                 NsExt(A ∪ B) ⊆ NsExt(A) ∩ NsExt(B) (resp. NsδExt(A ∪ B) ⊆ NsδExt(A) ∩ NsδExt(B)). (ii) NsExt(A ∩ B) ⊇ NsExt(A) ∪ NsExt(B) (resp. NsδExt(A ∩ B) ⊇ NsδExt(A) ∪ NsδExt(B)).

Proof. (i) NsδExt(A ∪ B) = Nsδint((A ∪ B)c) = Nsδint((Ac) ∩ (Bc)) ⊆ Nsδcl(Ac) ∩ Nsδcl(Bc) = NsδExt(A) ∩ NsδExt(B).

(ii)               NsδExt(A ∩ B) = Nsδint((A ∩ B)c) = Nsδint((Ac) ∪ (Bc)) ⊇ Nsδcl(Ac) ∪ Nsδcl(Bc) = NsδExt(A) ∪ NsδExt(B).

      The proof of the other is similar.                                                                                                                             

5          Conclusions

So far, we have studied some new operators called neutrosophic δ frontier respective border and exterior with the help of neutrosophic δ-open sets in neutrosophic topological space. Also, we discussed the important properties of them and the relations between them. This can be extended to some weaker forms of neutrosophic open sets, in future.

Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest.

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