Bipolar neutrosophic soft contra generalized pre-continuous and contra generalized α-continuous mappings
1P. Arulpandy, 2M. Trinita Pricilla
1Department of Mathematics, KPR Institute of Engineering and Technology, Coimbatore, India
2Department of Mathematics, Nirmala College for Women, Coimbatore, India
arulpandy002@gmail.com1, abishai kennet@yahoo.in2
In this paper, we introduce and investigate the classes of continuous mappings in bipolar neutrosophic soft topological spaces such as bipolar neutrosophic soft contra generalized pre-continuous mappings and bipolar neutrosophic soft contra generalized α-continuous mappings. Further, we investigate some of its properties via theorems.
Keywords: Bipolar neutrosophic soft set; BNSCGP-continuous; BNSCG-closed set; BNSCGαcontinuous; BNSS-topology.
Zadeh5 introduced the concept of fuzzy sets in 1965. Initially, fuzzy sets were used in decision making problems; but nowadays it is applied in many science and engineering fields. Later, fuzzy topological spaces was introduced by Cheng9 in 1965. Atanassov10 developed some concepts in fuzzy sets and proposed intiutionistic fuzzy sets which is the extension of conventional Fuzzy sets. Smarandache1,2 introduced a theory of neutrosophic sets as the extension and development of fuzzy sets which includes three membership degrees namely, truth, indeterminacy and falsify memberships. Deli et al.8 extended the neutrosophic set and proposed bipolar neutrosophic sets. Later, many researchers were investigated and proposed various neutrosophic sets.4,7 Molodtsov3 introduced soft set theory in 1999. A.A.Salama et al.13 developed a new concept neutrosophic topology, i.e. the topology concepts were introduced for neutrosophic sets. Norman Levine11,12 introduced generalized closed set and some continuity mappings in point set topology in 1970. In this study, we have proposed bipolar neutrosophic soft contra generalized pre-continuous mapping and bipolar neutrosophic soft contra generalized α-continuous mappings.
This paper is organized as follows: Section 2 consists the required preliminary definitions for main concept. Section 3 consists bipolar neutrosophic soft generalized homeomorphism and prehomeomorphism. Section 4 consists bipolar neutrosophic soft contra generalized pre-continuous mapping. Section 5 consists bipolar neutrosophic soft contra generalized α-continuous mapping. Each section deals with theorems related to the proposed mappings and examples. Section 6 concludes the proposed work.
Definition 2.1. 1 Let X be a universe set. For every x ∈ X, the components u(x),v(x) and w(x) are truth, indeterminate and false degrees of x. Then the Neutrosophic set (NS) over X be defined as follows.
N = {u(x),v(x),w(x) : x ∈ X}
Here, u(x),v(x),w(x) ranges ı˙n the non-standard ı˙nterval ]−0,1+[ and theı˙r sum −0 ≤ u+v+w ≤ 3+. For scı˙entı˙fı˙c problems, we prefer standard ı˙nterval [0,1] ı˙nstead of non-standard ı˙nterval and ı˙t ı˙s called sı˙ngle-valued neutrosophı˙c set.
Definition 2.2. 4 For the universe set X and positive member values u+,v+,w+ : E → [0,1], negative member values u−,v−,w− : E → [−1,0], A bı˙polar neutrosophı˙c set (BNS) ı˙s defı˙ned by
Definition 2.3. 3 A sot set is a function which maps a parameter set to the power set of X. It is denoted by (f,E) and ı˙s defı˙ned by f : E → P(x)
Each member of X is parametrized with the parameter set E by the function f.
Definition 2.4. 4 A bı˙polar neutrosophı˙c soft set (BNSS) is the fusion of soft set and bipolar neutrosophic set and ı˙s defı˙ned as follows.
Here
.
Definition 2.5. 4 Let B1 and B2 be two BNSSs. Then for every x ∈ X,e ∈ E,
(i). (subset) B1 ⊆ B2 ı˙f and only ı˙f
h + + i h + + i h + + i
uB1(x) ≤ uB2(x) , vB1(x) ≥ vB2(x) , wB1(x) ≥ wB2(x) {B1 ⊆ B2} =
hu−B1(x) ≥ u−B2(x)i,hvB−1(x) ≤ vB−2(x)i,hwB−1(x) ≤ wB−2(x)i
.
(ii). (equal) B1 = B2 if and only if B1 ⊆ B2 and B2 ⊆ B1.
(iii). (complement)
.
(iv). (union)
. Here
,
(v). (intersection)
. Here
,
(vii). (complete) 
Definition 2.6. A bipolar neutrosophic soft topology (BNST) on X is a collection τ of bipolar neutrosophic soft sets (BNSS) in X satisfying the following conditions:
1). ϕB,1B ∈ τB
2). Si∈n Bi ∈ τB for each Bi ∈ τB
3). Bi ∩ Bj ∈ τB for any Bi,Bj ∈ τB
The paı˙r (X,τB)ı˙s called BNSS-topological space. The members of τB are called bı˙polar neutrosophı˙c soft open sets (BNOS) and theı˙r complements are called bipolar neutrosophic soft closed sets (BNCS).
Definition 2.7. Let (X,τB) be a BNST and
be
BNSS in X. Then the bipolar neutrosophic soft interior and bipolar neutrosophic soft closure are defined by
Remark 2.8. Let B be BNS of a BNTS(X,τ). Then
1. BNαcl(B) = B ∪ BNcl(BNint(BNcl(B)))
2. BNαint(B) = B ∩ BNint(BNcl(BNint(B)))
Remark 2.9. Following relations hold for any BNS set B ∈ (X,τ).
1. BNcl(Bc) = (BNint(B))c and BNint(Bc) = (BNcl(B))c.
2. BNcl(B) is a BNCS and BNint(B) is a BNOS in X.
3. B is BNCS in X ı˙f and only ı˙f BNcl(B) = B.
4. B is BNOS in X ı˙f and only ı˙f BNint(B) = B.
Definition 2.10. A BNSset B in BNSTS(X,τ) ı˙s saı˙d to be
1). Bipolar neutrosophı˙c soft semı˙ closed set (BNSCS) if BNint(BNcl(B)) ⊆ B,
2). Bipolar neutrosophic soft pre-closed set (BNPCS) if BNcl(BNint(B)) ⊆ B,
3). Bipolar neutrosophic soft α-closed set (BNαCS) if BNcl(BNint(BNcl(B))) ⊆ B,
4). Bipolar neutrosophic soft regular closed set (BNRCS) if B = BNcl(BNint(B))
Definition 2.11. Let (X,τ1B,E) and (Y,τ2B,E′) be two bipolar neutrosophic soft topologı˙cal spaces. For every bipolar neutrosophic closed set B of (Y,τ2B,E′), a map ψ : ((X,τ1B,E) → (Y,τB2,E′) is said to be,
1. Bı˙polar neutrosophı˙c soft contı˙nuous (BNS-contı˙nuous) ı˙f ψ−1(B) ı˙s bı˙polar neutrosophı˙c soft closed set ı˙n (X,τ1B,E).
2. Bı˙polar neutrosophı˙c soft semı˙-contı˙nuous (BNSS-contı˙nuous) ı˙f ψ−1(B)ı˙s bı˙polar neutrosophı˙c soft semı˙-closed set ı˙n (X,τ1B,E).
3. Bı˙polar neutrosophı˙c soft pre-contı˙nuous (BNSP-contı˙nuous) ı˙f ψ−1(B)ı˙s bı˙polar neutrosophı˙c soft pre-closed set ı˙n (X,τ1B,E).
4. Bı˙polar neutrosophı˙c soft α-contı˙nuous (BNSα-contı˙nuous) ı˙f ψ−1(B) ı˙s bı˙polar neutrosophı˙c soft α-closed set ı˙n (X,τ1B,E).
5. Bı˙polar neutrosophı˙c soft regular contı˙nuous (BNSR-contı˙nuous) ı˙f ψ−1(B)ı˙s bı˙polar neutrosophı˙c soft regular closed set ı˙n (X,τ1B,E).
6. Bı˙polar neutrosophı˙c soft generalı˙zed contı˙nuous (BNSG-contı˙nuous) ı˙f ψ−1(B) ı˙s bı˙polar neutrosophı˙c soft generalı˙zed closed set ı˙n (X,τ1B,E).
7. Bı˙polar neutrosophı˙c soft generalı˙zed semı˙-contı˙nuous (BNSGS-contı˙nuous) ı˙f ψ−1(B) ı˙s bı˙polar neutrosophı˙c soft generalı˙zed semı˙-closed set ı˙n (X,τ1B,E).
8. Bı˙polar neutrosophı˙c soft generalı˙zed α-contı˙nuous (BNSα-contı˙nuous) ı˙f ψ−1(B) ı˙s bı˙polar neutrosophı˙c soft generalı˙zed α-closed set ı˙n (X,τ1B,E).
Definition 2.12. A map ψ : ((X,τ1B,E) → (Y,τ2B,E′) ı˙s saı˙d to be bı˙polar neutrosophı˙c soft generalı˙zed pre ı˙rresolute (BNSGP-ı˙rresolute) mappı˙ng ı˙f ψ−1(B) ı˙s a BNSGPCS ı˙n (X,τ1B,E) for every BNSGPCS B ı˙n (Y,τ2B,E).
Definition 2.13. Let (X,τB,E) be a BNSGT. The bı˙polar neutrosophı˙c soft generalı˙zed pre closure (BNSGPcl(B)) for any BNS B is defined as follows.
BNSGPcl(B) = ∩{K| K is a BNSGPCS in X and B ∈ K}. If B is BNSGPCS, then BNSGPcl(B) = B.
Definition 2.14. Let (X,τ1) and (Y,τ2) be any two topologı˙cal spaces. A map f : (X,τ1) → (Y,τ2) ı˙s saı˙d to be contra contı˙nuous ı˙f f−1(V ) is closed set in (X,τ1) for every open set V in (Y,τ2).
Definition 2.15. Let f be a bijective mapping from a topological space (X,τ1) into a topological space (Y,τ2). Then f ı˙s saı˙d to be generalı˙zed homeomorphı˙sm ı˙f f and f−1 are generalized continuous mappings.
Definition 2.16. A map f : (X,τ1) → (Y,τ2) is said to be generalized pre closed if f(V ) is generalized pre closed set in (Y,τ2) for every closed set V in (X,τ1).
Definition 3.1. Let ψ be a bijective mappı˙ng from a BNST (X,τ1,E) into a BNST (Y,τ2,E). Then ψ ı˙s saı˙d to be
1. Bı˙polar neutrosophı˙c soft homeomorphism (BNS-homeomorphism) if ψ and ψ−1 are BNScontinuous mapping.
2. Bipolar neutrosophic soft pre homeomorphism (BNSP-homeomorphı˙sm) ı˙f ψ and ψ−1 are BNSP-contı˙nuous mapping.
3. Bipolar neutrosophı˙c soft generalized homeomorphism (BNSG-homeomorphism) if ψ and ψ−1 are BNSG-continuous mapping.
Definition 3.2. A map ψ : (X,τ1,E) → (Y,τ2,E) is saı˙d to be Bipolar neutrosophic soft generalized pre closed if ψ(V ) is bipolar neutrosophic soft generalized pre closed set ı˙n (Y,τ2,E) for every BNS-closed set V in (X,τ1,E).
Definition 3.3. A bijection mappı˙ng ψ : (X,τ1,E) → (Y,τ2,E) ı˙s called a Bipolar neutrosophic soft generalized pre-homeomorphism (BNSGP-homeomorphism) if ψ and ψ−1 are BNSGPcontinuous mapping.
Example 3.4. Let X = {a,b},Y = {u,v},E = {e1,e2}. Then
Then τ1 = {0,B1,1} and τ2 = {0,B2,1} are BNSTs on X and Y respectı˙vely. Now we defı˙ne a bı˙jectı˙ve mappı˙ng ψ : (X,τ1,E) → (Y,τ2,E) by ψ(a) = u and ψ(b) = v with . Then ψ is a BNSGP-contı˙nuous mappı˙ng and ψ−1 is also a BNSGP-contı˙nuous mappı˙ng. Therefore ψ is a BNSGP homeomorphı˙sm.
Theorem 3.5. Every BNS-homeomorphism is a BNSGP-homeomorphism but not conversely.
Proof. Let ψ : (X,τ1,E) → (Y,τ2,E) be a BNS-homeomorphı˙sm. Then ψ and ψ−1 are BNScontinuous mapping. This gives, ψ and ψ−1 are BNSGP-continuous mapping. Therefore, ψ ı˙s a BNSGP-homeomorphı˙sm. 
Example 3.6. Let X = {a,b},Y = {u,v},E = {e1,e2}. Then
Then τ1 = {0,B1,1} and τ2 = {0,B2,1} are BNSTs on X and Y respectı˙vely. Now we define a bijective mapping ψ : (X,τ1,E) → (Y,τ2,E) by ψ(a) = u and ψ(b) = v. Then ψ is a BNSGPhomeomorphı˙sm but not BNS-homeomorphı˙sm sinceψ and ψ−1 are not BNS-continuous mapping.
Theorem 3.7. Every BNSP-homeomorphism is a BNSGP-homeomorphism but not conversely.
Proof. Let ψ : (X,τ1,E) → (Y,τ2,E) be a BNSP-homeomorphı˙sm. Then ψ and ψ−1 are BNSPcontinuous mapping. This gives, ψ and ψ−1 are BNSGP-continuous mapping. Therefore, ψ is a BNSGP-homeomorphı˙sm.
Example 3.8. Let X = {a,b},Y = {u,v},E = {e1,e2}. Then
Then τ1 = {0,B1,1} and τ2 = {0,B2,1} are BNSTs on X and Y respectı˙vely. Now we define a bijective mapping ψ : (X,τ1,E) → (Y,τ2,E) by ψ(a) = u and ψ(b) = v. Then ψ is a BNSGPhomeomorphism but not BNSP-homeomorphism since ψ and ψ−1 are not BNSP-continuous mapping.
Theorem 3.9. Let ψ : (X,τ1,E) → (Y,τ2,E) be a bijective mapping. If ψ is a BNSGP continuous mapping, then the following statements are equivalent.
1. ψ is a BNSGP closed mapping.
2. ψ is a BNSGP open mapping.
3. ψ is a BNSGP homeomorphism.
Proof. (1) → (2): Let ψ : (X,τ1,E) → (Y,τ2,E) be a bijective mapping and let ψ be a BNSGPclosed mapping. This gives, ψ−1 : (Y,τ2,E) → (X,τ1,E) is a BNSGP-continuous mapping. This means, every BNOS in X is a BNSGPOS in Y . Hence ψ is a BNSGP-open mapping.
(2) → (3): Let ψ : (X,τ1,E) → (Y,τ2,E) be a bijective mapping and let ψ be a BNSGP-open mapping. This gives, ψ−1 : (Y,τ2,E) → (X,τ1,E) is a BNSGP-continuous mapping. Hence ψ and ψ−1 are BNSGP-homeomorphism.
(3) → (1): Let ψ is a BNSGP homeomorphism. This means, ψ and ψ−1 are BNSGP contı˙nuous mappı˙ng. Since every BNCS ı˙n X ı˙s a BNSGPCS in Y , then ψ is a BNSGP-closed mapping. 
Remark 3.10. The composı˙tı˙on of two BNSGP homeomorphı˙sm need not be a BNSGP homeomorphism ı˙n general.
Example 3.11. Let X = {a,b},Y = {u,v},Z = {p,q},E = {e1,e2}. Then
Let τ1 = {0,B1,1}, τ2 = {0,B2,1} and τ3 = {0,B3,1} are BNSTs on X, Y and Z respectively. Now we define a bı˙jectı˙ve mapping ψ : (X,τ1,E) → (Y,τ2,E) by ψ(a) = u and ψ(b) = v and ϕ : (Y,τ2,E) → (Z,τ3,E) by ϕ(x) = p and ϕ(y) = q. Then ψ and ψ−1 are BNSGPcontı˙nuous mappings, also ϕ and ϕ−1 are BNSGP-contı˙nuous mappings. Hence ψ and ϕ are BNSGPhomeomorphism. But the mapping ψ◦ϕ : (X,τ1,E) → (Z,τ3,E) is not a BNSGP-homeomorphism since ψ ◦ ϕ is not a BNSGP-continuous mapping.
Definition 4.1. A map ψ : (X,τ1,E) → (Y,τ2,E) ı˙s saı˙d to be a Bipolar neutrosophı˙c soft contra generalized pre-continuous (BNSCGP-continuous) mappı˙ng if ψ−1(B) ı˙s a BNSGPCS ı˙n (X,τ1,E) for every BNOS B in (Y,τ2,E).
Example 4.2. Let X = {a,b},Y = {u,v},E = {e1,e2}. Then
Then τ1 = {0,B1,1} and τ2 = {0,B2,1} are BNSTs on X and Y respectı˙vely. Now we define a bijective mapping ψ : (X,τ1,E) → (Y,τ2,E) by ψ(a) = u and ψ(b) = v. Then ψ is a BNSCGPcontı˙nuous mappı˙ng.
Theorem 4.3. Every BNSC-continuous mapping is a BNSCGP-continuous mapping but not conversely.
Proof. Let ψ : (X,τ1,E) → (Y,τ2,E) be a BNSC-contı˙nuous mappı˙ng. Let B be a BNOS ı˙n Y . Then ψ−1(B) ı˙s a BNCS ı˙n X. Sı˙nce every BNCS is a BNSGPCS, ψ−1(B) is a BNSGPCS ı˙n X. Hence, ψ is a BNSCGP contı˙nuous mappı˙ng. 
Example 4.4. Let X = {a,b},Y = {u,v},E = {e1,e2}. Then
Then τ1 = {0,B1,1} and τ2 = {0,B2,1} are BNSTs on X and Y respectı˙vely. Now we define a bijective mapping ψ : (X,τ1,E) → (Y,τ2,E) by ψ(a) = u and ψ(b) = v. Then ψ is a BNSCGPcontı˙nuous mappı˙ng but not a BNSC-contı˙nuous mappı˙ng.
Theorem 4.5. Every BNSCα-continuous mapping is a BNSCGP continuous mapping but not conversely.
Proof. Let ψ : (X,τ1,E) → (Y,τ2,E) be a BNSCα-contı˙nuous mappı˙ng. Let B be a BNOS in Y . Then ψ−1(B) is a BNSαCS ı˙n X. Sı˙nce every BNSαCS is a BNGPCS, ψ−1(B) is a BNGPCS in X. Hence, ψ ı˙s a BNSCGP-contı˙nuous mappı˙ng. 
Example 4.6. Let X = {a,b},Y = {u,v},E = {e1,e2}. Then
Then τ1 = {0,B1,1} and τ2 = {0,B2,1} are BNSTs on X and Y respectı˙vely. Now we define a bijective mapping ψ : (X,τ1,E) → (Y,τ2,E) by ψ(a) = u and ψ(b) = v. Then ψ is a BNSCGPcontı˙nuous mappı˙ng but not a BNSCα-contı˙nuous mappı˙ng.
Theorem 4.7. Every BNSCP-continuous mapping is a BNSCGP continuous mapping but not conversely.
Proof. Let ψ : (X,τ1,E) → (Y,τ2,E) be a BNSCP-contı˙nuous mappı˙ng. Let B be a BNOS ı˙n Y . Then ψ−1(B) ı˙s a BNSPCS in X. Sı˙nce every BNSPCS is a BNSGPCS, ψ−1(B) is a BNSGPCS in X. Hence, ψ is a BNSCGP-contı˙nuous mappı˙ng.
Example 4.8. Let X = {a,b},Y = {u,v},E = {e1,e2}. Then
Then τ1 = {0,B1,1} and τ2 = {0,B2,1} are BNSTs on X and Y respectı˙vely. Now we define a bijective mapping ψ : (X,τ1,E) → (Y,τ2,E) by ψ(a) = u and ψ(b) = v. Then ψ ı˙s a BNSCGPcontı˙nuous mappı˙ng but not a BNSCP-contı˙nuous mappı˙ng.
Theorem 4.9. Let ψ : (X,τ1,E) → (Y,τ2,E) be a mapping. Then the following statements are equivalent.
1. ψ is a BNSCGP continuous mapping.
2. ψ−1(B) is a BNSGPOS in X for every BNCS B in Y .
Proof. (1) → (2): Let B be a BNCS ı˙n Y . Then Bc is a BNOS ı˙n Y . By statement, ψ−1(Bc) is a BNSGPCS in X. Hence ψ−1(B) is a BNSGPOS in X.
(2) → (1): Let B be a BNOS in Y . Then Bc is a BNCS in Y . By statement, ψ−1(Bc) is a BNSGPOS in X. Hence ψ−1(B) is a BNSGPCS in X. Thus ψ is a BNSCGP-contı˙nuous mapping.
Theorem 4.10. Let ψ : (X,τ1,E) → (Y,τ2,E) be a mapping. If one of the following properties hold:
1. ψ(BNSPcl(A)) ⊆ BNSint(ψ(A)) for each A in X.
2. BNSPcl(ψ−1(B)) ⊆ ψ−1(BNSint(B)) for each B in Y .
3. ψ−1(BNScl(B)) ⊆ BNSPint(ψ−1(B)) for each B in Y .
Then ψ is a BNSCGP-continuous mapping.
Proof. (1) → (2): Let B be a BNS in Y . Then ψ−1(B) is a BNS in X. By statement, we have ψ(BNSPcl(ψ−1(B))) ⊆ BNSint(ψ(ψ−1(B))) ⊆ BNSint(B). Now BNSPcl(ψ−1(B)) ⊆ ψ−1(ψ(BNSPcl(ψ−1(B)))) ⊆ ψ−1(BNSint(B)).
(2) → (1): By taking the complement in (2). Suppose that (3) holds. Let B be a BNCS in Y . Then BNcl(B) = B. By the assumption, ψ−1(B) = ψ−1(BNcl(B)) ⊆ BNSPint(ψ−1(B)). But BNSPint(ψ−1(B)) ⊆ ψ−1(B), hence BNSPint(ψ−1(B)) = ψ−1(B). Thı˙s ı˙mplı˙es ψ−1(B) ı˙s a BNSPOS in X and hence ψ−1(B) ı˙s a BNSGPOS ı˙n X. Thus ψ ı˙s a BNSCGP-contı˙nuous mapping. 
Theorem 4.11. Let ψ : (X,τ1,E) → (Y,τ2,E) be a bijective mapping. If one of the following properties hold:
1. ψ−1(BNcl(B)) ⊆ BNint(BNSPcl(ψ−1(B))) for each B in Y .
2. BNcl(BNSPint(ψ−1(B))) ⊆ ψ−1(BNint(B)) for each B in Y .
3. ψ(BNcl(BNSPint(A))) ⊆ BNint(ψ(A)) for each A in X.
4. ψ(BNcl(A)) ⊆ BNint(ψ(A)) for each BNPOS A in X.
Then ψ is a BNSCGP-continuous mapping.
Proof. (1) → (2): It is obvious, by taking the complement in (1).
(2) → (3): Let A be a BNS in X. Put B = ψ(A) in Y . Thı˙s ı˙mplı˙es A = ψ−1(ψ(A)) = ψ−1(B) in X. Now BNcl(BNSPint(A)) = BNcl(BNSPint(ψ−1(B))) ⊆ ψ−1(BNint(B)) by the hypothesis. Therefore, ψ(BNcl(BNSPint(A))) ⊆ ψ(ψ−1(BNint(B))) = BNint(B) = BNint(ψ(A)).
(3) → (4): Let A be a BNSPOS in X. Then BNSPint(A) = A. By hypothesı˙s, ψ(BNcl(BNSPint(A))) ⊆ BNint(ψ(A)).
Therefore, ψ(BNcl(A)) = ψ(BNcl(BNSPint(A))) ⊆ BNint(ψ(A)).
Suppose (4) holds: Let A be a BNOS in Y . Then ψ−1(A) is a BNOS in X and BNSPint(ψ−1(A)) is a BNSPOS in X. Hence, by hypothesis, ψ(BNcl(BNSPint(ψ−1(A)))) ⊆ BNint(ψ(BNSPint(ψ−1(A)))) ⊆ BNint(ψ(ψ−1(A))) = BNint(A) ⊆ A.
Therefore BNcl(BNSPint(ψ−1(A))) = ψ−1(ψ(BNcl(BNSPint(ψ−1(A))))) ⊆ ψ−1(A). Now, BNcl(BNint(ψ−1(A))) ⊆ BNcl(BNSPint(ψ−1(A))) ⊆ ψ−1(A). Thı˙s ı˙mplı˙es ψ−1(A) is a BNSPCS ı˙n X and hence a BNSGPCS ı˙n X. Thus ψ is a BNSCGP-contı˙nuous mappı˙ng. 
Theorem 4.12. Let ψ : (X,τ1,E) → (Y,τ2,E) be a bijective mapping. Then ψ is a BNSCGP continuous mapping if BNcl(ψ(A)) ⊆ ψ(BNSPint(A)) for every BNS A in X.
Proof. Let A be a BNCS in Y . Then BNcl(A) = A and ψ−1(A) is a BNS in X. By hypothesis, BNcl(ψ(ψ−1(A))) ⊆ ψ(BNSPint(ψ−1(A))). Since ψ is onto, ψ(ψ−1(A)) = A. Therefore A = BNcl(A) = BNcl(ψ(psi−1(A))) ⊆ ψ(BNSPint(ψ−1(A))).
Now ψ−1(A) ⊆ ψ−1(ψ(BNSPint(ψ−1(A)))) = BNSPint(ψ−1(A)) ⊆ ψ−1(A). Hence ψ−1(A) is a BNSPOS in X and hence BNSGPOS in X. Thus ψ is a BNSCGP-contı˙nuous mappı˙ng. 
Theorem 4.13. 1. If ψ : (X,τ1,E) → (Y,τ2,E) is a BNSCGP-continuous mapping and ϕ : (Y,τ2,E) → (Z,τ3,E) is a continuous mapping, then ϕ ◦ ψ : (X,τ1,E) → (Z,τ3,E) is a BNSCGP-continuous mapping.
2. If ψ : (X,τ1,E) → (Y,τ2,E) is a BNSCGP-continuous mapping and ϕ : (Y,τ2,E) → (Z,τ3,E) is a BNSC-continuous mapping, then ϕ◦ψ : (X,τ1,E) → (Z,τ3,E) is a BNSGPcontinuous mapping.
3. If ψ : (X,τ1,E) → (Y,τ2,E) is a BNSGP-irresolute mapping and ϕ : (Y,τ2,E) → (Z,τ3,E) is a BNSCGP-contı˙nuous mappı˙ng, then ϕ ◦ ψ : (X,τ1,E) → (Z,τ3,E) is a BNSCGP-contı˙nuous mappı˙ng.
Proof. 1. Let A be BNOS ı˙n Z. Then ϕ−1(A) is a BNOS ı˙n Y , by hypothesı˙s. Since ψ is a BNSCGP-contı˙nuous mapping, ψ−1(ϕ−1(A)) is a BNSGPCS ı˙n X. Hence ϕ ◦ ψ is a BNSCGP-contı˙nuous mapping.
2. Let A be BNOS in Z. Then ϕ−1(A) is a BNCS in Y , by hypothesı˙s. Since ψ is a BNSCGPcontı˙nuous mappı˙ng, ψ−1(ϕ−1(A)) is a BNSGPOS in X. Hence ϕ◦ψ is a BNSGP-contı˙nuous mapping.
3. Let A be BNOS ı˙n Z. Then ϕ−1(A) is a BNSGPCS in Y , by hypothesı˙s. Since ψ is a BNSGP-contı˙nuous mappı˙ng, ψ−1(ϕ−1(A)) is a BNSGPCS ı˙n X. Hence ϕ◦ψ is a BNSCGPcontı˙nuous mappı˙ng.
Theorem 4.14. A mapping ψ : (X,τ1,E) → (Y,τ2,E) is a BNSCGP-continuous mapping if ψ−1(BNSPcl(B)) ⊆ BNint(ψ−1(B)) for every BNS B in Y .
Proof. Let B be a BNCS in Y . Then BNcl(B) = B. Since every BNCS is a BNSPCS, this implies BNSPcl(B) = B. By hypothesı˙s, ψ−1(B) = ψ−1(BNSPcl(B)) ⊆ BNint(ψ−1(B)) ⊆ ψ−1(B). This ı˙mplı˙es ψ−1(B) is a BNOS in X. Therefore ψ is a BNSC-contı˙nuous mapping, since every BNSC-continuous mapping is a BNSCGP-contı˙nuous mapping, ψ is a BNSCGP-continuous mapping.
Theorem 4.15. A BNS-continuous mapping, ψ : (X,τ1,E) → (Y,τ2,E) is a BNSCGP continuous mapping if BNSGPO(X) = BNSGPC(X).
Proof. Let B be a BNOS ı˙n Y . By hypothesı˙s, ψ−1(B) ı˙s a BNOS ı˙n X and hence is a BNSGPOS in X. Sı˙nce BNSGPO(X)=BNSGPC(X), ψ−1(B) is a BNSGPCS in X. Therefore, ψ is a BNSCGPcontı˙nuous mappı˙ng. 
ı˙n thı˙s sectı˙on we ı˙ntroduce bı˙polar neutrosophı˙c soft contra generalı˙zed α-contı˙nuous mappı˙ng and ı˙nvestı˙gate some of ı˙ts propertı˙es.
Definition 5.1. A bı˙polar neutrosophı˙c soft set B ı˙n (X,τ,E) ı˙s saı˙d to be a bipolar neutrosophı˙c soft generalı˙zed α-closed set (BNSGαCS) if BNαcl(B) ⊆ U whenever B ⊆ U is a BNαOS in (X,τ,E).
Definition 5.2. A mappı˙ng ψ : (X,τ1,E) → (Y,τ2,E) ı˙s saı˙d to be a bı˙polar neutrosophı˙c soft contra generalı˙zed α-contı˙nuous (BNSCGα-continuous) if ψ−1(B) is a bipolar neutrosophı˙c soft generalized α-closed set ı˙n (x,τ1,E) for every bipolar neutrosophı˙c soft open set B in (Y,τ2,E).
Definition 5.3. A mappı˙ng ψ : (X,τ1,E) → (Y,τ2,E) is said to be a bı˙polar neutrosophı˙c soft strongly generalı˙zed α-contı˙nuous (BNSSGα-continuous) if ψ−1(B) is a bipolar neutrosophı˙c soft open set in (x,τ1,E) for every bipolar neutrosophı˙c soft generalized α-open set B in (Y,τ2,E).
Definition 5.4. A mappı˙ng ψ : (X,τ1,E) → (Y,τ2,E) is said to be a bı˙polar neutrosophı˙c soft contra strongly generalı˙zed α-contı˙nuous (BNSCSGα-contı˙nuous) if ψ−1(B) is a bipolar neutrosophic soft closed set in (x,τ1,E) for every bı˙polar neutrosophı˙c soft generalized α-open set B in (Y,τ2,E).
Definition 5.5. A mappı˙ng ψ : (X,τ1,E) → (Y,τ2,E) is said to be a bı˙polar neutrosophı˙c soft contra generalized α-irresolute if ψ−1(B) is a bı˙polar neutrosophı˙c soft generalized α-closed set in (x,τ1,E) for every bı˙polar neutrosophı˙c soft generalı˙zed α-open set B in (Y,τ2,E).
Theorem 5.6. Every BNSC-continuous mapping is a BNSCGα-continuous mapping but converse not true.
Proof. Let ψ : (X,τ1,E) → (Y,τ2,E) be a BNSC-contı˙nuous mappı˙ng and let B be BNOS ı˙n Y . Then ψ−1(B) is a BNCS in X. Sı˙nce every BNCS is a BNSGαCS, ψ−1(B) is a BNSGαCS in (X,τ1,E). Hence ψ is a BNSGα-contı˙nuous mappı˙ng.
Example 5.7. Let X = {a,b},Y = {u,v},E = {e1,e2}. Then
Then τ1 = {0,B1,1} and τ2 = {0,B2,1} are BNSTs on X and Y respectively. Now we define a bijective mapping ψ : (X,τ1,E) → (Y,τ2,E) by ψ(a) = u and ψ(b) = v. Then ψ is a BNSCGαcontı˙nuous mappı˙ng but not a BNSC-contı˙nuous mappı˙ng.
Theorem 5.8. Every BNSCα-continuous mapping is a BNSGα-continuous mapping but converse not true.
Proof. Let ψ : (X,τ1,E) → (Y,τ2,E) be a BNSCα-contı˙nuous mapping and let B be BNOS in Y . Then ψ−1(B) is a BNαCS in X. Since every BNαCS is a BNSGαCS, ψ−1(B) is a BNSGαCS in (X,τ1,E). Hence ψ is a BNSGα-contı˙nuous mapping. 
Example 5.9. Let X = {a,b},Y = {u,v},E = {e1,e2}. Then
Then τ1 = {0,B1,1} and τ2 = {0,B2,1} are BNSTs on X and Y respectı˙vely. Now we define a bijective mapping ψ : (X,τ1,E) → (Y,τ2,E) by ψ(a) = u and ψ(b) = v. Then ψ is a BNSGαcontinuous mapping but not a BNSCα-contı˙nuous mapping.
Theorem 5.10. Every BNSCSGα-continuous mapping is a BNSCGα-continuous mapping but converse not true.
Proof. Let ψ : (X,τ1,E) → (Y,τ2,E) be a BNSCSGα-contı˙nuous mappı˙ng and let B be BNOS in Y . Sı˙nce every BNOS is a BNSGαOS, B is a BNSGαOS in Y . Also, sı˙nce ψ is a BNSCSGαcontı˙nuous mapping, ψ−1(B) is a BNCS in X. Since every BNCS is a BNSGαCS, ψ−1(B) is a BNSGαCS in (X,τ1,E). Hence ψ is a BNSGα-contı˙nuous mapping. 
Example 5.11. Let X = {a,b},Y = {u,v},E = {e1,e2}. Then
Then τ1 = {0,B1,1} and τ2 = {0,B2,1} are BNSTs on X and Y respectively. Now we define a bijective mapping ψ : (X,τ1,E) → (Y,τ2,E) by ψ(a) = u and ψ(b) = v. Then ψ is a BNSCGαcontinuous mapping but not a BNSCSGα-continuous mapping.
Theorem 5.12. Every BNSCSGα-continuous mapping is a BNSC-continuous mapping but converse not true.
Proof. Let ψ : (X,τ1,E) → (Y,τ2,E) be a BNSCSGα-contı˙nuous mappı˙ng and let B be BNOS in Y . Sı˙nce every BNOS is a BNSGαOS, B is a BNSGαOS in Y . Also, since ψ is a BNSCSGαcontı˙nuous mappı˙ng, ψ−1(B) is a BNCS in X. Hence ψ is a BNSC-contı˙nuous mappı˙ng.
Example 5.13. Let X = {a,b},Y = {u,v},E = {e1,e2}. Then
Then τ1 = {0,B1,1} and τ2 = {0,B2,1} are BNSTs on X and Y respectively. Now we define a bijective mapping ψ : (X,τ1,E) → (Y,τ2,E) by ψ(a) = u and ψ(b) = v. Then ψ is a BNSCcontinuous mapping but not a BNSCSGα-continuous mapping.
Theorem 5.14. 1. If ψ : (X,τ1,E) → (Y,τ2,E) is a BNSCGα-continuous mapping and ϕ : (Y,τ2,E) → (Z,τ3,E) is a BNS-continuous mapping, then ϕ ◦ ψ : (X,τ1,E) → (Z,τ3,E) is a BNSCGα-continuous mapping.
2. If ψ : (X,τ1,E) → (Y,τ2,E) is a BNSCGα-continuous mapping and ϕ : (Y,τ2,E) → (Z,τ3,E) is a BNSCα-continuous mapping, then ϕ◦ψ : (X,τ1,E) → (Z,τ3,E) is a BNSGαcontinuous mapping.
3. If ψ : (X,τ1,E) → (Y,τ2,E) is a BNSCGα-irresolute mapping and ϕ : (Y,τ2,E) → (Z,τ3,E) is a BNSCGα-continuous mapping, then ϕ ◦ ψ : (X,τ1,E) → (Z,τ3,E) is a BNSGα-continuous mapping.
4. If ψ : (X,τ1,E) → (Y,τ2,E) is a BNSGα-irresolute mapping and ϕ : (Y,τ2,E) → (Z,τ3,E) is a BNSCGα-continuous mapping, then ϕ ◦ ψ : (X,τ1,E) → (Z,τ3,E) is a BNSCGα-continuous mapping.
Proof. 1. Let B be a BNOS in Z. Since ϕ is a BNS-contı˙nuous mapping, ϕ−1(B) is BNOS in Y . Sı˙nce ψ is a BNSCα-contı˙nuous mapping, ψ−1(ϕ−1(B)) is a BNSGαCS in X. Hence, ϕ ◦ ψ is a BNSCGα-continuous mapping.
2. Let B be a BNOS in Z. Since ϕ is a BNSC-contı˙nuous mapping, ϕ−1(B) is BNCS in Y . Since ψ is a BNSCGα-contı˙nuous mapping, ψ−1(ϕ−1(B)) is a BNSGαOS in X. Hence, ϕ ◦ ψ is a BNSGα-contı˙nuous mapping.
3. Let B be a BNOS in Z. Since ϕ is a BNSCGα-contı˙nuous mapping, ϕ−1(B) is BNSGαCS in Y . Sı˙nce ψ is a BNSCGα-ı˙rresolute mapping, ψ−1(ϕ−1(B)) is a BNSGαOS in X. Hence, ϕ ◦ ψ is a BNSGα-continuous mapping.
4. Let B be a BNOS in Z. Since ϕ is a BNSCGα-contı˙nuous mappı˙ng, ϕ−1(B) is BNSGαCS in Y . Since ψ is a BNSGα-ı˙rresolute mapping, ψ−1(ϕ−1(B)) is a BNSGαCS in X. Hence, ϕ ◦ ψ is a BNSCGα-contı˙nuous mappı˙ng.
In this paper, we have introduced bipolar neutrosophic soft contra generalized pre-continuous mapping and bipolar neutrosophic soft contra generalized α-continuous mapping. Results are in this paper shows that preservation of topological structures such as closeness and openness by various continuity mappings.
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