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Neutrosophic Sets and Systems, 18/2017
University of New Mexico
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R. Narmada Devi, 2 R. Dhavaseelan, 3 S. Jafari
of Mathematics, Vel Tech Rangarajan Dr.Sagunthala R and D Institute of Science and Technology,Chennai,Tamil Nadu, India. E-mail :narmadadevi23@gmail.com
2 3
Department of Mathematics, Sona College of Technology, Salem-636005, Tamil Nadu, India. E-mail: dhavaseelan.r@gmail.com
Department of Mathematics, College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark. E-mail: jafaripersia@gmail.com
Abstract: In this paper we introduce the concept of a new class of an ordered neutrosophic bitopological spaces. Besides giving some interesting properties of these spaces. We also prove analogues of
Uryshon’s lemma and Tietze extension theorem in an ordered neutrosophic bitopological spaces.
Keywords:Ordered neutrosophic bitopological space; lower(resp.upper) pairwise neutrosophic Gδ -ι-locally T1 -ordered space; pairwise neutrosophic Gδ -ιlocally T1 -ordered space; pairwise neutrosophic Gδ -ι-locally T2 -ordered space; weakly pairwise neutrosophic Gδ -ι-locally T2 -ordered space; almost pairwise neutrosophic Gδ -ι-locally T2 -ordered space and strongly pairwise neutrosophic Gδ -ι-locally normally ordered space.
1
Introduction and Preliminaries
The concept of fuzzy sets was introduced by Zadeh [17]. Fuzzy sets have applications in many ďŹ elds such as information theory [15] and control theory [16]. The theory of fuzzy topological spaces was introduced and developed by Chang [7]. Atanassov [2] introduced and studied intuitionistic fuzzy sets. On the other hand, Coker [8] introduced the notions of an intuitionistic fuzzy topological space and some other related concepts. The concept of an intuitionistic fuzzy Îą-closed set was introduced by B. Krsteshka and E. Ekici [5]. G. Balasubramanian [3] was introduced the concept of fuzzy Gδ set. Ganster and Reilly used locally closed sets [10] to deďŹ ne LC-continuity and LC-irresoluteness. The concept of an ordered fuzzy topological space was introduced and developed by A. K. Katsaras [11]. Later G. Balasubmanian [4] introduced and studied the concepts of an ordered L-fuzzy bitopological spaces. F. Smarandache [[13], [14] introduced the concepts of neutrosophy and neutrosophic set. supF = fsup , infF = finf The concepts of neutrosophic crisp set and neutrosophic crisp n − sup = tsup + isup + fsup topological space were introduced by A. A. Salama and S. A. n−inf = tinf +iinf +finf . T,I,F are neutrosophic components. Alblowi [12]. DeďŹ nition 1.3. [13, 14] Let X be a nonempty ďŹ xed set. A In this paper, we introduce the concepts of pairwise neutroneutrosophic set [briey NS] A is an object having the form sophic Gδ -Îą-locally T1 -ordered space, pairwise neutrosophic A = { x, ÎźA (x), ĎƒA (x), ÎłA (x) : x ∈ X} where ÎźA (x), ĎƒA (x) Gδ -Îą-locally T2 -ordered space, weakly pairwise neutrosophic and ÎłA (x) which represents the degree of membership function Gδ -Îą-locally T2 -ordered space, almost pairwise neutrosophic (namely ÎźA (x)), the degree of indeterminacy (namely ĎƒA (x)) Gδ -Îą-locally T2 -ordered space and strongly pairwise neutroand the degree of nonmembership (namely ÎłA (x)) respectively sophic Gδ -Îą-locally normally ordered space. Some interesting of each element x ∈ X to the set A. propositions are discussed. Urysohn’s lemma and Tietze extension theorem of an strongly pairwise neutrosophic Gδ -Îą-locally Remark 1.1. [13, 14] normally ordered space are studied and established. (1) A neutrosophic set A = { x, Îź (x), Ďƒ (x), Îł (x) : x ∈ A
DeďŹ nition 1.1. [7] Let X be a nonempty set and A ⊂ X. The characteristic function of A is denoted and deďŹ ned by χA (x) =
1 if x ∈ A 0 if x ∈ A
A
A
X} can be identiďŹ ed to an ordered triple ÎźA , ĎƒA , ÎłA in ]0− , 1+ [ on X.
(2) For the sake of simplicity, we shall use the symbol A = ÎźA , ĎƒA , ÎłA for the neutrosophic set A = { x, ÎźA (x), ĎƒA (x), ÎłA (x) : x ∈ X}.
DeďŹ nition 1.2. [13, 14] Let T,I,F be real standard or non standard DeďŹ nition 1.4. [12] Let X be a nonempty set and the neutrosubsets of ]0− , 1+ [, with supT = tsup , infT = tinf sophic sets A and B in the form supI = isup , infI = iinf
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Neutrosophic Sets and Systems, 18/2017
A = { x, ÎźA (x), ĎƒA (x), ÎłA (x) : x { x, ÎźB (x), ĎƒB (x), ÎłB (x) : x ∈ X}. Then
∈
X}, B
=
(a) A ⊆ B iff ÎźA (x) ≤ ÎźB (x), ĎƒA (x) ≤ ĎƒB (x) and ÎłA (x) ≼ ÎłB (x) for all x ∈ X; (b) A = B iff A ⊆ B and B ⊆ A; (c) AÂŻ = { x, ÎłA (x), ĎƒA (x), ÎźA (x) : x ∈ X}; [Complement of A]
(b) A ⊆ B and A ⊆ C ⇒ A ⊆ B ∊ C, (c) A ⊆ C and B ⊆ C ⇒ A âˆŞ B ⊆ C, (d) A ⊆ B and B ⊆ C ⇒ A ⊆ C, (e) A âˆŞ B = A ∊ B, (f) A ∊ B = A âˆŞ B, (g) A ⊆ B ⇒ B ⊆ A,
(d) A ∊ B = { x, ÎźA (x) ∧ ÎźB (x), ĎƒA (x) ∧ ĎƒB (x), ÎłA (x) ∨ ÎłB (x) : x ∈ X};
(h) (A) = A,
(e) A âˆŞ B = { x, ÎźA (x) ∨ ÎźB (x), ĎƒA (x) ∨ ĎƒB (x), ÎłA (x) ∧ ÎłB (x) : x ∈ X};
(j) 0N = 1N .
(f) [ ]A = { x, ÎźA (x), ĎƒA (x), 1 − ÎźA (x) : x ∈ X}; (g) A = { x, 1 − ÎłA (x), ĎƒA (x), ÎłA (x) : x ∈ X}.
(i) 1N = 0N ,
Now we shall deďŹ ne the image and preimage of neutrosophic sets. Let X and Y be two nonempty sets and f : X → Y be a function.
DeďŹ nition 1.5. [12] Let {Ai : i ∈ J} be an arbitrary family of DeďŹ nition 1.9. [9] neutrosophic sets in X. Then (a) If B = { y, ÎźB (y), ĎƒB (y), ÎłB (y) : y ∈ Y } is a neutro (a) Ai = { x, âˆ§ÎźAi (x), âˆ§ĎƒAi (x), âˆ¨ÎłAi (x) : x ∈ X}; sophic set in Y, then the preimage of B under f, denoted by f −1 (B), is the neutrosophic set in X deďŹ ned by (b) Ai = { x, âˆ¨ÎźAi (x), âˆ¨ĎƒAi (x), âˆ§ÎłAi (x) : x ∈ X}. f −1 (B) = { x, f −1 (ÎźB )(x), f −1 (ĎƒB )(x), f −1 (ÎłB )(x) : x ∈ X}. Since our main purpose is to construct the tools for developing neutrosophic topological spaces, we must introduce the neutrosophic sets 0N and 1N in X as follows: DeďŹ nition 1.6. [12] 0N = { x, 0, 0, 1 : x ∈ X} and 1N = { x, 1, 1, 0 : x ∈ X}. DeďŹ nition 1.7. [9] A neutrosophic topology (NT) on a nonempty set X is a family T of neutrosophic sets in X satisfying the following axioms: (i) 0N , 1N ∈ T , (ii) G1 ∊ G2 ∈ T for any G1 , G2 ∈ T , (iii) âˆŞGi ∈ T for arbitrary family {Gi | i ∈ Λ} ⊆ T .
(b) If A = { x, ÎźA (x), ĎƒA (x), ÎłA (x) : x ∈ X} is a neutrosophic set in X,then the image of A under f, denoted by f (A), is the neutrosophic set in Y deďŹ ned by f (A) = { y, f (ÎźA )(y), f (ĎƒA )(y), (1 − f (1 − ÎłA ))(y) : y ∈ Y }. where supx∈f −1 (y) ÎźA (x), if f −1 (y) = ∅, f (ÎźA )(y) = 0, otherwise, supx∈f −1 (y) ĎƒA (x), if f −1 (y) = ∅, f (ĎƒA )(y) = 0, otherwise, inf x∈f −1 (y) ÎłA (x), if f −1 (y) = ∅, (1 − f (1 − ÎłA ))(y) = 1, otherwise,
In this case the ordered pair (X, T ) or simply X is called a neutrosophic topological space (NTS) and each neutrosophic set in For the sake of simplicity, let us use the symbol f− (ÎłA ) for T is called a neutrosophic open set (NOS). The complement A 1 − f (1 − Îł ). A of a NOS A in X is called a neutrosophic closed set (NCS) in X. Corollary 1.2. [9] Let A , Ai (i ∈ J) be neutrosophic sets in DeďŹ nition 1.8. [9] Let A be a neutrosophic set in a neutrosophic X, B, Bi (i ∈ K) be neutrosophic sets in Y and f : X → Y a topological space X. Then function. Then N int(A) = {G | G is a neutrosophic open set in X and (a) A1 ⊆ A2 ⇒ f (A1 ) ⊆ f (A2 ), G ⊆ A} is called the neutrosophic interior of A; N cl(A) = {G | G is a neutrosophic closed set in X and (b) B1 ⊆ B2 ⇒ f −1 (B1 ) ⊆ f −1 (B2 ), G ⊇ A} is called the neutrosophic closure of A. (c) A ⊆ f −1 (f (A)) { If f is injective,then A = f −1 (f (A)) } , Corollary 1.1. [9] Let A,B,C be neutrosophic sets in X. Then the basic properties of inclusion and complementation: (d) f (f −1 (B)) ⊆ B { If f is surjective,then f (f −1 (B)) = B }, (a) A ⊆ B and C ⊆ D ⇒ AâˆŞ C ⊆ B âˆŞ D and A∊ C ⊆ B ∊ D, (e) f −1 ( Bj ) = f −1 (Bj ),
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(f) f −1 ( Bj ) = f −1 (Bj ), (g) f ( Ai ) = f (Ai ), (h) f ( Ai ) ⊆ f (Ai ) { If f is injective,then f ( Ai ) = f (Ai )}, (i) f −1 (1N ) = 1N , (j) f −1 (0N ) = 0N , (k) f (1N ) = 1N , if f is surjective, (l) f (0N ) = 0N , (m) f (A) ⊆ f (A), if f is surjective,
Neutrosophic Sets and Systems, 18/2017
DeďŹ nition 2.3. Let (X, T ) be a neutrosophic topological space. Let A = x, ÎźA , ĎƒA , ÎłA be a neutrosophic set in a neutrosophic topological space (X, T ). The neutrosophic Gδ -Îą-locally closure of A is denoted and deďŹ ned by N Gδ -Îą-lcl(A)= {B:B = x, ÎźB , ĎƒB , ÎłB is a neutrosophic Gδ -Îą-locally closed set in X and A ⊆ B}. DeďŹ nition 2.4. Let (X, T ) be a neutrosophic topological space. Let A = x, ÎźA , ĎƒA , ÎłA be a neutrosophic set in a neutrosophic topological space (X, T ). The neutrosophic Gδ -Îą-locally interior of A is denoted and deďŹ ned by N Gδ -Îą-lint(A)= {B:B = x, ÎźB , ĎƒB , ÎłB is a neutrosophic Gδ -Îą-locally open set in X and B ⊆ A}.
DeďŹ nition 2.5. Let X be a nonempty set and x ∈ X a ďŹ xed element in X. If r, t ∈ I0 = (0, 1] and s ∈ I1 = [0, 1) are DeďŹ nition 1.10. [1] A neutrosophic set A in a neutrosophic ďŹ xed real numbers such that 0 < r + t + s < 3, then xr,t,s = topological space (X, T ) is called a neutrosophic Îą-open set x, r, t, s is called a neutrosophic point (briey NP) in X, where (N ÎąOS) if A ⊆ N int(N cl(N int(A))). r denotes the degree of membership of xr,t,s , t denotes the degree of indeterminacy and s denotes the degree of nonmembership of and x ∈ X the support of xr,t,s . x 2 Ordered neutrosophic Gδ -Îą-locally bitopologi- r,t,s The neutrosophic point xr,t,s is contained in the neutrosophic cal Spaces A(xr,t,s ∈ A) if and only if r < ÎźA (x), t < ĎƒA (x), s > ÎłA (x). (n) f −1 (B) = f −1 (B).
In this section, the concepts of a neutrosophic Gδ set, neutrosophic ι-closed set, neutrosophic Gδ -ι-locally closed set, upper pairwise neutrosophic Gδ -ι-locally T1 -ordered space, lower pairwise neutrosophic Gδ -ι-locally T1 -ordered space, pairwise neutrosophic Gδ -ι-locally T1 -ordered space, pairwise neutrosophic Gδ -ι-locally T2 -ordered space, weakly pairwise neutrosophic Gδ -ι-locally T2 -ordered space, almost pairwise neutrosophic Gδ -ι-locally T2 -ordered space and strongly pairwise neutrosophic Gδ -ι-locally normally ordered space are introduced. Some basic properties and characterizations are discussed. Urysohn’s lemma and Tietze extension theorem of an strongly pairwise neutrosophic Gδ -ι-locally normally ordered space are studied and established.
DeďŹ nition 2.6. A neutrosophic set A = x, ÎźA , ĎƒA , ÎłA in a neutrosophic topological space (X, T ) is said to be a neutrosophic neighbourhood of a neotrosophic point xr,t,s , x ∈ X, if there exists a neutrosophic open set B = x, ÎźB , ĎƒB , ÎłB with xr,t,s ⊆ B ⊆ A.
DeďŹ nition 2.2. Let (X, T ) be a neutrosophic topological space. Let A = x, ÎźA , ĎƒA , ÎłA be a neutrosophic set on a neutrosophic topological space (X, T ). Then A is said be a neutrosophic Gδ Îą-locally closed set (in short,N Gδ -Îą-lcs) if A = B ∊ C, where B is a neutrosophic Gδ set and C is an neutrosophic Îą-closed set. The complement of a neutrosophic Gδ -Îą-locally closed set is said to be a neutrosophic Gδ -Îą-locally open set (in short,N Gδ Îą-los).
(ii) decreasing neutrosophic set if x ≤ y implies A(x) ⊇ A(y).That is, ÎźA (x) ≼ ÎźA (y) ,ĎƒA (x) ≼ ĎƒA (y)and ÎłA (x) ≤ ÎłA (y).
DeďŹ nition 2.7. A neutrosophic set A = x, ÎźA , ĎƒA , ÎłA in a neutrosophic topological space (X, T ) is said to be a neutrosophic Gδ -Îą-locally neighbourhood of a neutrosophic point xr,t,s , x ∈ X, if there exists a neutrosophic Gδ -Îą-locally open set B = x, ÎźB , ĎƒB , ÎłB with xr,t,s ⊆ B ⊆ A.
Notation 2.1. In what follows, we denote neutrosophic neighbourhood A of a in X by neutrosphic neighbourhood A of a neutrsophic point ar,t,s for a ∈ X. DeďŹ nition 2.1. Let (X, T ) be a neutrosophic topological space. Let A = x, ÎźA , ĎƒA , ÎłA be a neutrosophic set of a neutrosophic DeďŹ nition 2.8. A neutrosophic set A = x, ÎźA , ĎƒA , ÎłA in a partopological space X. Then A is said to be a neutrosophic Gδ ∞ tially ordered set (X, ≤) is said to be an set (briey N Gδ S) if A = i=1 Ai , where each Ai ∈ T and Ai = x, ÎźAi , ĎƒAi , ÎłAi . (i) increasing neutrosophic set if x ≤ y implies A(x) ⊆ The complement of neutrosophic Gδ set is said to be a neutroA(y).That is, sophic FĎƒ set(briey N FĎƒ S). ÎźA (x) ≤ ÎźA (y), ĎƒA (x) ≤ ĎƒA (y) and ÎłA (x) ≼ ÎłA (y).
DeďŹ nition 2.9. An ordered neutrosophic bitopological space is a neutrosophic bitopological space (X, Ď„1 , Ď„2 , ≤) (where Ď„1 and Ď„2 are neutrosophic topologies on X ) equipped with a partial order ≤.
R. Narmada Devi, R. Dhavaseelan, S. Jafari, On Separation Axioms in an Ordered Neutrosophic Bitopological Space
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Definition 2.10. An ordered neutrosophic bitopological space (X, τ1 , τ2 , ≤) is said to be an upper pairwise neutrosophic T1 ordered space if a, b ∈ X such that a b, there exists a decreasing τ1 neutrosophic neighbourhood (or) an decreasing τ2 neutrosophic neighbourhood A of b such that A = x, μA , σA , γA is not a neutrosophic neighbourhood of a.
(ii) For each a, b ∈ X such that a b, there exists an increasing (resp. decreasing) τ1 neutrosophic Gδ -α-locally open set(or) an increasing (resp.decreasing) τ2 neutrosophic Gδ α-locally open set A = x, μA , σA , γA such that A(a) > 0 (resp. A(b) > 0) and A is not a neutrosophic Gδ -α-locally neighbourhood of b (resp.a).
Definition 2.11. An ordered neutrosophic bitopological space (X, τ1 , τ2 , ≤) is said to be a lower pairwise neutrosophic T1 ordered space if a, b ∈ X such that a b, there exists an increasing τ1 neutrosophic neighbourhood (or) an increasing τ2 neutrosophic neighbourhood A of a such that A = x, μA , σA , γA is not a neutrosophic neighbourhood of b.
Proof: (i)⇒(ii) Let X be a lower pairwise neutrosophic Gδ -α-locally T1 -ordered space. Let a, b ∈ X such that a b. There exists an increasing τ1 neutrosophic Gδ -α-locally neighbourhood (or) an increasing τ2 neutrosophic Gδ -α-locally neighbourhood A of a such that A is not a neutrosophic Gδ -α-locally neighbourhood of b. It follows that there exists a τi neutrosophic Gδ -α-locally open set (i = 1(or)2), Ai = x, μAi , σAi , γAi with Ai ⊆ A and Ai (a) = A(a) > 0. As A is an increasing neutrosophic set, A(a) > A(b) and since A is not a neutrosophic Gδ -α-locally neighbourhood of b, Ai (b) < A(b) implies Ai (a) = A(a) > A(b) ≥ Ai (b). This shows that Ai is an increasing neutrosophic set and Ai is not a neutrosophic Gδ -αlocally neighbourhood of b, since A is not a neutrosophic Gδ -αlocally neighbourhood of b. (ii)⇒(i) Since A1 is an increasing τ1 neutrosophic Gδ -αlocally open set (or) increasing τ2 neutrosophic Gδ -α-locally open set. Now, A1 is a neutrosophic Gδ -α-locally neighbourhood of a with A1 (a) > 0. By (ii), A1 is not a neutrosophic Gδ -α-locally neighbourhood of b. This implies, X is a lower pairwise neutrosophic Gδ -α-locally T1 -ordered space.
Example 2.1. Let X = {1, 2} with a partial order relation ≤. Let τ1 = {0N , 1N , A} and τ2 = {0N , 1N , B} where A = (0.3, 0.3, 0.5), (0.7, 0.7, 0.4) and B = (0.5, 0.5, 0.5), (0.5, 0.5, 0.5) be any two topologies on X. Then (X, τ1 , τ2 , ≤) is an ordered neutrosophic bitopological space. Let 1(0.25,0.3,0.5) and 2(0.25,0.25,0.35) be any two neutrosophic points on X. For 1(0.25,0.3,0.5) 2(0.25,0.25,0.35) , there exists an increasing τ1 neutrosphic neighbourhood A of 1(0.25,0.3,0.5) such that A is not neutrosophic neighbourhood of 2(0.25,0.25,0.35) . Therefore (X, τ1 , τ2 , ≤) is a lower pairwise neutrosophic T1 -ordered space.
Definition 2.12. An ordered neutrosophic bitopological space (X, τ1 , τ2 , ≤) is said to be a pairwise neutrosophic T1 -ordered space if and only if it is both upper and lower pairwise neutroRemark 2.1. Similar proof holds for upper pairwise neutrosophic T1 -ordered space. sophic Gδ -α-locally T1 -ordered space. Definition 2.13. An ordered neutrosophic bitopological space (X, τ1 , τ2 , ≤) is said to be an upper pairwise neutrosophic Gδ - Proposition 2.2. If (X, τ1 , τ2 , ≤) is a lower (resp. upper) pairα-locally T1 -ordered space if a, b ∈ X such that a b, there wise neutrosophic Gδ -α-locally T1 -ordered space and τ1 ⊆ ∗ ∗ ∗ ∗ exists a decreasing τ1 neutrosophic Gδ -α-locally neighbourhood τ1 , τ2 ⊆ τ2 , then (X, τ1 , τ2 , ≤) is a lower (resp. (or) a decreasing τ2 neutrosophic Gδ -α-locally neighbourhood upper) pairwise neutrosophic Gδ -α-locally T1 -ordered space. A = x, μA , σA , γA of b such that A is not a neutrosophic Gδ - Proof: Let (X, τ1 , τ2 , ≤) be a lower pairwise neutrosophic Gδ -αα-locally neighbourhood of a. locally T1 -ordered space. Then if a, b ∈ X such that a b, there Definition 2.14. An ordered neutrosophic bitopological space exists an increasing τ1 neutrosophic Gδ -α-locally neighbourhood (X, τ1 , τ2 , ≤) is said to be a lower pairwise neutrosophic Gδ - (or) an increasing τ2 neutrosophic Gδ -α-locally neighbourhood α-locally T1 -ordered space if a, b ∈ X such that a b, there ex- A = x, μA , σA , γA of a such that A is not a neutrosophic Gδ -αists an increasing τ1 neutrosophic Gδ -α-locally neighbourhood locally neighbourhood of b. Since τ1 ⊆ τ1∗ and τ2 ⊆ τ2∗ . There(or) an increasing τ2 neutrosophic Gδ -α-locally neighbourhood fore, if a, b ∈ X such that a b, there exists an increasing τ1 ∗ A = x, μA , σA , γA of a such that A is not a neutrosophic Gδ - neutrosophic Gδ -α-locally neighbourhood (or) an increasing τ2 ∗ α-locally neighbourhood of b. neutrosophic Gδ -α-locally neighbourhood A = x, μA , σA , γA of a such that A is not a neutrosophic Gδ -α-locally neighbourDefinition 2.15. An ordered neutrosophic bitopological space hood of b. Thus (X, τ ∗ , τ ∗ , ≤) is a lower pairwise neutrosophic 1 2 (X, τ1 , τ2 , ≤) is said to be a pairwise neutrosophic Gδ -α-locally G -α-locally T -ordered space. δ 1 T1 -ordered space if and only if it is both upper and lower pairwise Remark 2.2. Similar proof holds for upper pairwise neutroneutrosophic Gδ -α-locally T1 -ordered space. sophic Gδ -α-locally T1 -ordered space. Proposition 2.1. For an ordered neutrosophic bitopological Definition 2.16. An ordered neutrosophic bitopological space space (X, τ1 , τ2 , ≤) the following are equivalent (X, τ1 , τ2 , ≤) is said to be a pairwise neutrosophic T2 -ordered (i) X is a lower (resp. upper) pairwise neutrosophic Gδ -α- space if for a, b ∈ X with a b, there exist a neutrosophic open locally T1 -ordered space. sets A = x, μA , σA , γA and B = x, μB , σB , γB such that
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A is an increasing Ď„i neutrosophic neighbourhood of a, B is a decreasing Ď„j neutrosophic neighbourhood of b (i, j = 1, 2 and i = j) and A ∊ B = 0N .
DeďŹ nition 2.21. Let A and B be any two neutrosophic sets of a nonempty set X. Then a neutrosophic set A Ă— B on X Ă— X is of the form A Ă— B = (x, y), ÎźAĂ—B , ĎƒAĂ—B , ÎłAĂ—B where DeďŹ nition 2.17. An ordered neutrosophic bitopological space ÎźAĂ—B ((x, y)) = ÎźA (x) ∧ ÎźB (y), ĎƒAĂ—B ((x, y)) = ĎƒA (x) ∧ (X, Ď„1 , Ď„2 , ≤) is said to be a pairwise neutrosophic Gδ -Îą-locally ĎƒB (y) and ÎłAĂ—B ((x, y)) = ÎłA (x) ∨ ÎłB (y), for every (x, y) ∈ T2 -ordered space if for a, b ∈ X with a b, there exist a X Ă— X neutrosophic Gδ -Îą-locally open sets A = x, ÎźA , ĎƒA , ÎłA and Proposition 2.3. For an ordered neutrosophic bitopological B = x, ÎźB , ĎƒB , ÎłB such that A is an increasing Ď„i neutrosophic space (X, Ď„1 , Ď„2 , ≤) the following are equivalent Gδ -Îą-locally neighbourhood of a, B is a decreasing Ď„j neutro(i) X is a pairwise neutrosophic Gδ -Îą-locally T2 -ordered sophic Gδ -Îą-locally neighbourhood of b (i, j = 1, 2 and i = j) space. and A ∊ B = 0N . DeďŹ nition 2.18. Let (X, ≤) be a partially ordered set. Let G = {(x, y) ∈ X Ă— X | x ≤ y, y = f (x)} . Then G is called a graph of the partially ordered ≤.
(ii) For each pair a, b ∈ X such that a b, there exist a Ď„i neutrosophic Gδ -Îą-locally open set A = x, ÎźA , ĎƒA , ÎłA and Ď„j neutrosophic Gδ -Îą-locally open set B = x, ÎźB , ĎƒB , ÎłB such that A(a) > 0, B(b) > 0 and A(x) > 0, B(y) > 0 together imply that x y.
DeďŹ nition 2.19. Let X be any nonempty set. Let A ⊆ X. Then we deďŹ ne a neutrosophic set Ď‡âˆ—A is of the form (iii) The neutrosophic set Ď‡âˆ—G , where G is the graph of the par x, χA (x), χA (x), 1 − χA (x) . tial order of X is a Ď„ ∗ -neutrosophic Gδ -Îą-locally closed set, where Ď„ ∗ is either Ď„1 Ă— Ď„2 or Ď„2 Ă— Ď„1 in X Ă— X. DeďŹ nition 2.20. Let A = x, ÎźA , ĎƒA , ÎłA be a neutrosophic set in an ordered neutrosophic bitopological space (X, Ď„1 , Ď„2 , ≤). Proof: Then for i = 1(or)2, we deďŹ ne (i)⇒(ii) Let X be a pairwise neutrosophic Gδ -Îą-locally T2 IĎ„i -Gδ -Îą-li(A) = increasing Ď„i neutrosophic Gδ -Îą-locally in- ordered space. terior of A Assume that suppose A(x) > 0, B(y) > 0 and x ≤ y. Since A = the greatest increasing Ď„i neutrosophic Gδ -Îą- is an increasing Ď„ neutrosophic G -Îą-locally open set and B is a i δ locally open decreasing Ď„j neutrosophic Gδ -Îą-locally open set, A(x) ≤ A(y) set contained in A and B(y) ≤ B(x). Therefore 0 < A(x) ∊ B(y) ≤ A(y) ∊ B(x), DĎ„i -Gδ -Îą-li(A) = decreasing Ď„i neutrosophic Gδ -Îą-locally which is a contradiction to the fact that A ∊ B = 0 . Therefore N interior of A x y. = the greatest decreasing Ď„i neutrosophic Gδ -Îą- (ii)⇒(i) Let a, b ∈ X with a b, there exists a neutrosophic locally open sets A and B satisfying the properties in (ii). Since Ii â—Ś (A) is an set contained in A increasing Ď„i neutrosophic Gδ -Îą-locally open set and Dj â—Ś (B)is IĎ„i -Gδ -Îą-lc(A) = increasing Ď„i neutrosophic Gδ -Îą-locally decreasing Ď„ neutrosophic G -Îą-locally open set, we have j δ closure of A Ii â—Ś (A) ∊ Dj â—Ś (B)=0N . Suppose z ∈ X is such that Ii â—Ś (A)(z) = the smallest increasing Ď„i neutrosophic Gδ -Îą- ∊ D â—Ś (B)(z) >0. Then I â—Ś (A) > 0 and D â—Ś (B)(z) > 0. If j i j locally closed x ≤ z ≤ y, then x ≤ z implies that Dj â—Ś (B)(x) ≼ Dj â—Ś (B)(z) set containing in A >0 and z ≤ y implies that Ii â—Ś (A)(y) ≼ Ii â—Ś (A)(z) >0 then DĎ„i -Gδ -Îą-lc(A) = decreasing Ď„i neutrosophic Gδ -Îą-locally D â—Ś (B)(x) >0 and I â—Ś (A)(y) >0. Hence by (ii), x y but j i closure of A then x ≤ y. This is a contradiction. This implies that X is pair= the smallest decreasing Ď„i neutrosophic Gδ -Îą- wise neutrosophic G -Îą-locally T -ordered space. δ 2 locally closed (i)⇒ (iii) We want to show that Ď‡âˆ—G is a Ď„ ∗ neutrosophic Gδ set containing in A. Îą-locally closed set. That is to show that Ď‡âˆ— is Ď„ ∗ neutrosophic G
Notation 2.2. (i) The complement of a neutrosophic set χG ∗ , Gδ -Îą-locally open set. It is sufďŹ cient to prove that Ď‡âˆ—G is a neuwhere G is the graph of the partial order of X is denoted by trosophic Gδ -Îą-locally neighbourhood of a point (x, y) ∈ X Ă—X ∗ such that Ď‡âˆ—G (x, y) > 0. Suppose (x, y) ∈ X Ă— X is such . χG that Ď‡âˆ—G (x, y) > 0. We have Ď‡âˆ—G (x, y) < 1. This means (ii) IĎ„i -Gδ -Îą-lc(A) is denoted by Ii (A) and DĎ„j -Gδ -Îą-lc(A) Ď‡âˆ—G (x, y) = 0. Thus (x, y) ∈ G and hence x y. Therefore is denoted by Dj (A), where A = x, ÎźA , ĎƒA , ÎłA is a neuby assumption (i), there exist neutrosophic Gδ -Îą-locally open trosophic set in an ordered neutrosophic bitopological space sets A and B such that A is an increasing Ď„i neutrosophic Gδ -Îą(X, Ď„1 , Ď„2 , ≤), for i, j = 1, 2 and i = j. locally neighbourhood of a , B is an decreasing Ď„j neutrosophic (iii) IĎ„i -Gδ -Îą-li(A) is denoted by Ii â—Ś (A) and DĎ„j -Gδ -Îą-li(A) Gδ -Îą-locally neighbourhood of b (i, j = 1, 2 and i = j) and is denoted by Dj â—Ś (A), where A = x, ÎźA , ĎƒA , ÎłA is a neu- A ∊ B = 0N . Clearly A Ă— B is an IF Ď„ ∗ Gδ -Îą-locally neightrosophic set in an ordered neutrosophic bitopological space bourhood of (x, y). It is easy to verify that A Ă— B ⊆ χG . Thus (X, Ď„1 , Ď„2 , ≤), for i, j = 1, 2 and i = j. we ďŹ nd that χG is an Ď„ ∗ N Gδ -Îą-locally open set. Hence (iii) is
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established. (iii)⇒(i) Suppose x y. Then (x, y) ∈ G,where G is a graph of the partial order. Given that Ď‡âˆ—G is Ď„ ∗ neutrosophic Gδ -Îąlocally closed set. That is Ď‡âˆ—G is an Ď„ ∗ neutrosophic Gδ -Îą-locally open set. Now (x, y) ∈ G implies that Ď‡âˆ—G (x, y) > 0. Therefore Ď‡âˆ—G is an Ď„ ∗ neutrosophic Gδ -Îą-locally neighbourhood of (x, y) ∈ X Ă— X. Hence we can ďŹ nd that Ď„ ∗ neutrosophic Gδ Îą-locally open set A Ă— B such that A Ă— B ⊆ Ď‡âˆ—G and A is Ď„i neutrosophic Gδ -Îą-locally open set such that A(x) > 0 and B is an Ď„j neutrosophic Gδ -Îą-locally open set such that B(y) > 0. We now claim that Ii â—Ś (A) ∊ Dj â—Ś (B)=0N . For if z ∈ X is such that (Ii â—Ś (A) ∊ Dj â—Ś (B))(z)> 0, then Ii â—Ś (A)(z) ∊ Dj â—Ś (B)(z) > 0. This means Ii â—Ś (A)(z)> 0 and Dj â—Ś (B)(z)> 0. And if a ≤ z ≤ b, then z ≤ b implies that Ii â—Ś (A)(b)≼ Ii â—Ś (A)(z)> 0 and a ≤ z implies that Dj â—Ś (B)(a)≼ Dj â—Ś (B)(z)> 0. Then Dj â—Ś (B)(a)> 0 and Ii â—Ś (A)(b)> 0 implies that a b but then a ≤ b. This is a contradiction. Hence (i) is established. DeďŹ nition 2.22. An ordered neutrosophic bitopological space (X, Ď„1 , Ď„2 , ≤) is said to be a weakly pairwise neutrosophic T2 ordered space if given b < a (that is b ≤ a and b = a), there exist an Ď„i neutrosophic open set A = x, ÎźA , ĎƒA , ÎłA such that A(a) > 0 and Ď„j neutrosophic open set B = x, ÎźB , ĎƒB , ÎłB such that B(b) > 0 (i, j = 1, 2 and i = j) such that if x, y ∈ X, A(x) > 0, B(y) > 0 together imply that y < x. DeďŹ nition 2.23. An ordered neutrosophic bitopological space (X, Ď„1 , Ď„2 , ≤) is said to be a weakly pairwise neutrosophic Gδ Îą-locally T2 -ordered space if given b < a (that is b ≤ a and b = a), there exist an Ď„i neutrosophic Gδ -Îą-locally open set A = x, ÎźA , ĎƒA , ÎłA such that A(a) > 0 and Ď„j neutrosophic Gδ -Îą-locally open set B = x, ÎźB , ĎƒB , ÎłB such that B(b) > 0 (i, j = 1, 2 and i = j) such that if x, y ∈ X, A(x) > 0, B(y) > 0 together imply that y < x. DeďŹ nition 2.24. The symbol x y means that x ≤ y and y ≤ x. DeďŹ nition 2.25. An ordered neutrosophic bitopological space (X, Ď„1 , Ď„2 , ≤) is said to be an almost pairwise neutrosophic T2 ordered space if given a b, there exist a Ď„i neutrosophic open set A = x, ÎźA , ĎƒA , ÎłA such that A(a) > 0 and Ď„j neutrosophic open set B = x, ÎźB , ĎƒB , ÎłB such that B(b) > 0 (i,j=1,2 and i = j) such that if x, y ∈ X, A(x) > 0 and B(y) > 0 together imply that x y.
32
Gδ -Îą-locally T2 -ordered and almost pairwise neutrosophic Gδ Îą-locally T2 -ordered space. Proof: Let (X, Ď„1 , Ď„2 , ≤) be a pairwise neutrosophic Gδ -Îą-locally T2 ordered space. Then by Proposition 3.3 and DeďŹ nition 3.20, it is a weakly pairwise neutrosophic Gδ -Îą-locally T2 -ordered space. Let a b . Then a b and b a.Since a b and X is a pairwise neutrosophic Gδ -Îą-locally T2 -ordered space. We have Ď„i neutrosophic Gδ -Îą-locally open set A = x, ÎźA , ĎƒA , ÎłA and Ď„j neutrosophic Gδ -Îą-locally open set B = x, ÎźB , ĎƒB , ÎłB such that A(a) > 0, B(b) > 0 and A(x) > 0 , B(y) > 0 together imply that x y. Also since b a,there exist Ď„i neutrosophic Gδ -Îą-locally open set A∗ = x, ÎźA∗ , ÎłA∗ and Ď„j neutrosophic Gδ -Îą-locally open set B ∗ = x, ÎźB ∗ , ÎłB ∗ such that A∗ (a) > 0, B ∗ (b) > 0 and A∗ (x) > 0 , B ∗ (y) > 0 together imply that y x. Thus Ii â—Ś (A∊A∗ ) is an Ď„i neutrosophic Gδ -Îą-locally open set such that Ii â—Ś (A ∊ A∗ )(a) > 0 and Ij â—Ś (B ∊ B ∗ ) is a Ď„j neutrosophic Gδ -Îą-locally open set such that Ij â—Ś (B ∊ B ∗ )(b) > 0.Also Ii â—Ś (A ∊ A∗ )(x) > 0 and Ij â—Ś (B ∊ B ∗ )(y) > 0 togetherimply that x y. Hence X is an almost pairwise neutrosophic Gδ -Îą-locally T2 -ordered space. Conservely, let X be a weakly pairwise neutrosophic Gδ -Îąlocally T2 -ordered and almost pairwise neutrosophic Gδ -Îąlocally T2 -ordered space. We want to show that X is a pairwise neutrosophic Gδ -Îą-locally T2 -ordered space. Let a b. Then either b < a (or) b a. If b < a then X being weakly pairwise neutrosophic Gδ -Îą-locally T2 -ordered space, there exist Ď„i neutrosophic Gδ -Îą-locally open set A and Ď„j neutrosophic Gδ Îą-locally open set B such that A(a) > 0,B(b) > 0 and such that A(x) > 0,B(y) > 0 together imply that y < x. Thus x y. If b a, then a b and the result follows easily since X is an almost pairwise neutrosophic Gδ -Îą-locally T2 -ordered space. Hence X is a pairwise neutrosophic Gδ -Îą-locally T2 ordered space. DeďŹ nition 2.27. Let A = x, ÎźA , ĎƒA , ÎłA and B = x, ÎźB , ĎƒB , ÎłB be neutrosophic sets in an ordered neutrosophic bitopological space (X, Ď„1 , Ď„2 , ≤). Then A is said to be a Ď„i neutrosophic neighbourhood of B if B ⊆ A and there exists Ď„i neutrosophic open set C= x, ÎźC , ĎƒC , ÎłC such that B ⊆ C ⊆ A,(i = 1(or)2). DeďŹ nition 2.28. Let A = x, ÎźA , ĎƒA , ÎłA and B = x, ÎźB , ĎƒB , ÎłB be neutrosophic sets in an ordered neutrosophic bitopological space (X, Ď„1 , Ď„2 , ≤). Then A is said to be a Ď„i neutrosophic Gδ -Îą-locally neighbourhood of B if B ⊆ A and there exists Ď„i neutrosophic Gδ -Îą-locally open set C = x, ÎźC , ĎƒC , ÎłC such that B ⊆ C ⊆ A, (i = 1(or)2).
DeďŹ nition 2.26. An ordered neutrosophic bitopological space (X, Ď„1 , Ď„2 , ≤) is said to be an almost pairwise neutrosophic Gδ Îą-locally T2 -ordered space if given a b, there exist a Ď„i neutrosophic Gδ -Îą-locally open set A = x, ÎźA , ĎƒA , ÎłA such that A(a) > 0 and Ď„j neutrosophic Gδ -Îą-locally open set B = x, ÎźB , ĎƒB , ÎłB such that B(b) > 0 (i, j = 1, 2 and i = j) such that if x, y ∈ X, A(x) > 0 and B(y) > 0 together imply that DeďŹ nition 2.29. An ordered neutrosophic bitopological space (X, Ď„1 , Ď„2 , ≤) is said to be a strongly pairwise neutrosophic x y. Gδ -Îą-locally normally ordered space if for every pair A = Proposition 2.4. An ordered neutrosophic bitopological space x, ÎźA , ĎƒA , ÎłA is a decreasing Ď„i neutrosophic Gδ -Îą-locally (X, Ď„1 , Ď„2 , ≤) is a pairwise neutrosophic Gδ -Îą-locally T2 - closed set and B = x, ÎźB , ĎƒB , ÎłB is an decreasing Ď„j ordered space if and only if it is a weakly pairwise neutrosophic neutrosophic Gδ -Îą-locally open set such that A ⊆ B then
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there exist decreasing Ď„j neutrosophic Gδ -Îą-locally open set A1 = x, ÎźA1 , ÎłA1 such that A ⊆ A1 ⊆ Di (A1 ) ⊆ B,(i, j = 1, 2 and i = j). Proposition 2.5. An ordered neutrosophic bitopological space (X, Ď„1 , Ď„2 , ≤) the following are equivalent (i) (X, Ď„1 , Ď„2 , ≤) is a strongly pairwise neutrosophic Gδ -Îąlocally normally ordered space. (ii) For each increasing Ď„i neutrosophic Gδ -Îą-locally open set A= x, ÎźA , ĎƒA , ÎłA and decreasing Ď„j neutrosophic Gδ -Îąlocally open set B= x, ÎźB , ĎƒB , ÎłB with A ⊆ B there exists an decreasing Ď„j neutrosophic Gδ -Îą-locally open set A1 such that A ⊆ A1 ⊆ N Gδ -Îą-lclĎ„i (A1 ) ⊆ B,(i, j = 1, 2 and i = j). Proof:
The Proof is simple.
for all x ∈ X and t ∈ R. Then f is a Ď„i lower∗ (resp.Ď„j upper∗ ) neutrosophic Gδ -Îą-locally continuous function if and only if A is an increasing (or) a decreasing Ď„i (resp. Ď„j ) neutrosophic Gδ Îą-locally open (resp. closed) set (i, j = 1, 2 and i = j). Proof: ⎧ âˆź if t < 0 ⎨ 1 if 0 ≤ t ≤ 1 Aâˆź f −1 (RI t ) = ⎊ âˆź if t > 1 0 implies that f is Ď„i lower∗ neutrosophic Gδ -Îą-locally continuous function if and only if A is an increasing (or) a decreasing Ď„i neutrosophic Gδ -Îą-locally ⎧ âˆź open set in X. if t < 0 ⎨ 1 if 0 ≤ t ≤ 1 Aâˆź f −1 (LI t ) = ⎊ âˆź if t > 1 0 implies that f is Ď„j upper∗ neutrosophic Gδ -Îą-locally continuous function if and only if A is an increasing (or) a decreasing Ď„j neutrosophic Gδ -Îą-locally closed set in X (i, j = 1, 2 and i = j).
Notation 2.3. (i) The collection of all neutrosophic set in nonempty set X is denoted by Îś X . Uryshon’s lemma (ii) Let X be any nonempty set and A ∈ Îś X . Then for x ∈ X, ÎźA (x), ĎƒA (x), ÎłA (x) is denoted by Aâˆź . DeďŹ nition 2.30. A neutrosophic real line RI (I) is the set of all monotone decreasing neutrosophic A ∈ Îś R satisfying âˆŞ{A(t) : t ∈ R} = 1âˆź and ∊{A(t) : t ∈ R} = 0âˆź after the identiďŹ cation of neutrosophic sets A, B ∈ RI (I) if and only if A(t−) = B(t−) and A(t+) = B(t+) for all t ∈ R where A(t−) = ∊{A(s) : s < t} and A(t+) = âˆŞ{A(s) : s > t}. The neutrosophic unit interval II (I) is a subset of RI (I) such that [A] ∈ II (I) if the membership , indeterminancy and nonmembership of A are deďŹ ned by
ÎźA (t) =
1, 0,
t<0; Ďƒ (t) = t>1. A
1, 0,
t<0; t>1.
and ÎłA (t) =
0, 1,
respectively. The natural neutrosophic topology on RI (I) is generated from the subbasis {LI t , RI t : t ∈ R} where LIt , RtI : RI (I) → II (I) are given by LIt [A] = A(t−) and RtI [A] = A(t+), respectively. DeďŹ nition 2.31. Let (X, Ď„1 , Ď„2 , ≤) be an ordered neutrosophic bitopological space. A function f : X → RI (I) is said to be a Ď„i lower∗ (resp.upper∗ ) neutrosophic Gδ -Îą-locally continuous function if f −1 (RI t ) (resp.f −1 (LI t )) is an increasing (or)an decreasing Ď„i (resp.Ď„j ) neutrosophic Gδ -Îą-locally open set, for each t ∈ R (i, j = 1, 2 and i = j). Proposition 2.6. Let (X, Ď„1 , Ď„2 , ≤) be an ordered neutrosophic bitopological space. Let A = x, ÎźA , ĎƒA , ÎłA be a neutrosophic set in X and let f : X → RI (I) be such that ⎧ âˆź if t < 0 ⎨ 1 if 0 ≤ t ≤ 1 Aâˆź f (x)(t) = ⎊ âˆź 0 if t > 1
Proposition 2.7. An ordered neutrosophic bitopological space (X, Ď„1 , Ď„2 , ≤) is a strongly pairwise neutrosophic Gδ -Îą-locally normally ordered space if and only if for every A = x, ÎźA , ĎƒA , ÎłA is decreasing Ď„i neutrosophic closed set and B = x, ÎźB , ĎƒB , ÎłB is an increasing Ď„j neutrosophic closed set with A ⊆ B, there exists increasing neutrosophic function f : X → II (I) such that A ⊆ f −1 (L1 ) ⊆ f −1 (R0 ) ⊆ B and f is a Ď„i upper∗ neutrosophic Gδ -Îą-locally continuous function and Ď„j lower∗ neutrosophic Gδ -Îą-locally continuous function (i, j = 1, 2 and i = j). Proof: Suppose that there exists a function f satisfying the given conditions. Let C = x, ÎźC , ĎƒC , ÎłC =t<0; f −1 (LI t ) and D = x, ÎźD , ĎƒD , ÎłD =f −1 (RI t ) for some 0 ≤ t t>1. ≤ 1. Then C ∈ Ď„i and D ∈ Ď„j and such that A ⊆ C ⊆ D ⊆ B. It is easy to verify that D is a decreasing Ď„j neutrosophic Gδ Îą-locally open set and C is an increasing Ď„i neutrosophic Gδ -Îąlocally closed set. Then there exists decreasing Ď„j neutrosophic Gδ -Îą-locally open set C1 such that C ⊆ C1 ⊆ Di (C1 ) ⊆ D, (i, j = 1, 2 and i = j). This proves that X is a strongly pairwise neutrosophic Gδ -Îą-locally normally ordered space. Conversely, let X be a strongly pairwise neutrosophic Gδ -Îąlocally normally ordered space. Let A be a decreasing Ď„i neutrosophic Gδ -Îą-locally closed set and B be an increasing Ď„j neutrosophic Gδ -Îą-locally closed set. By the Proposition 3.6, we can construct a collection {Ct | t ∈ I} ⊆ Ď„j , where C = x, ÎźCt , ÎłCt , t ∈ I such that A ⊆ Ct ⊆ B, N Gδ -Îą-lclĎ„i (Cs ) ⊆ Ct whenever s < t, A ⊆ C0 C1 = B and Ct = 0N for t < 0,Ct = 1N for t > 1. We deďŹ ne a function f : X → II (I) by f (x)(t) = C1−t (x). Clearly f is well deďŹ ned. Since A ⊆ −1 −1 I I C1−t ⊆ B, for t ∈ I. We have A ⊆ f (L 1 ) ⊆ f (R 0 ) ⊆ B. −1 I Furthermore f (R t ) = s<1−t Cs is a Ď„j neutrosophic Gδ -Îą locally open set and f −1 (LI t ) = s>1−t Cs = s>1−t N Gδ -ÎąlclĎ„i (Cs ) is an Ď„i neutrosophic Gδ -Îą-locally closed set. Thus f is
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a Ď„j lower∗ neutrosophic Gδ -Îą-locally continuous function and whenever s < t. We have, Ď„i upper∗ neutrosophic Gδ -Îą-locally continuous function and is Vt = Cr an increasing neutrosophic function. t∈R r<t
t∈R
⊇
Tietze extension theorem
Br
t∈R r<t
= Proposition 2.8. Let (X, τ1 , τ2 , ≤) be an ordered neutrosophic bitopological space the following statements are equivalent. (i) (X, τ1 , τ2 , ≤) is a strongly pairwise neutrosophic Gδ -ιlocally normally ordered space.
∗
(ii) If g, h : X → RI (I),g is an Ď„i upper neutrosophic Gδ -Îąlocally continuous function, h is a Ď„j lower∗ neutrosophic Gδ -Îą-locally continuous function and g ⊆ h, then there exists f : X → II (I) such that g ⊆ f ⊆ h and f is a Ď„i upper∗ neutrosophic Gδ -Îą-locally continuous function and Ď„j lower∗ neutrosophic Gδ -Îą-locally continuous function (i, j = 1, 2 and i = j).
Proof: (ii)⇒(i) Let A= x, ÎźA , ĎƒA , ÎłA and B= x, ÎźB , ĎƒB , ÎłB be a neutrosophic Gδ -Îą-locally open sets such that A ⊆ B. DeďŹ ne g, h : X → RI (I) ⎧ byâˆź if t < 0 ⎨ 1 if 0 ≤ t ≤ 1 and h(x)(t) = Aâˆź g(x)(t) = ⎊ âˆź if t > 1 0 ⎧ âˆź if t < 0 ⎨ 1 if 0 ≤ t ≤ 1 Bâˆź ⎊ âˆź 0 if t > 1 for each x ∈ X. By Proposition 3.6, g is an Ď„i upper∗ neutrosophic Gδ -Îą-locally continuous function and h is an Ď„j lower∗ neutrosophic Gδ -Îą-locally continuous function. Clearly, g ⊆ h holds,so that there exists f : X → RI (I) such that g ⊆ f ⊆ h. Suppose t ∈ (0, 1). Then A = g −1 (RI t ) ⊆ f −1 (RI t ) ⊆ f −1 (LI t ) ⊆ h−1 (LI t ) = B. By Proposition 3.7, X is a strongly pairwise neutrosophic Gδ -Îą-locally normal ordered space. (i)⇒(ii) DeďŹ ne two mappings A, B : Q → I by A(r) = Ar = h−1 (RI r ) and B(r) = Br = g −1 (LI r ), for all r ∈ Q (Q is the set of all rationals). Clearly, A and B are monotone increasing families of a decreasing Ď„i neutrosophic Gδ -Îą-locally closed sets and decreasing Ď„j neutrosophic Gδ -Îą-locally open sets of X. Moreover Ar ⊂ Br if r < r . By Proposition 3.5, there exists an decreasing Ď„j neutrosophic Gδ -Îą-locally open set C = x, ÎźC , ĎƒC , ÎłC such that Ar ⊆ N Gδ -Îą-lintĎ„i (Cr ), N Gδ Îą-lclĎ„i (Cr ) ⊆ N Gδ -Îą-lintĎ„i (Cr ), N Gδ -Îą-lclĎ„i (C r ) ⊆ Br whenever r < r (r, r ∈ Q). Letting Vt = r<t Cr for t ∈ R, we deďŹ ne a monotone decreasing family {Vt | t ∈ R} ⊆ I. Moreover we have N Gδ -Îą-lclĎ„i (Vt ) ⊆ N Gδ -Îą-lintĎ„i (Vs )
g −1 (LI r )
t∈R r<t
=
g −1 (LI r )
t∈R
= g −1 (
LI t )
t∈R
= 1N Similarly, Now deďŹ ne a function f : t∈R Vt = 0N . (X, Ď„1 , Ď„2 , ≤) → RI (I) satisfying the required conditions. Let f (x)(t) = Vt (x), for all x ∈ X and t ∈ R. By the above discussion, it follows that f is well deďŹ ned. To prove f is a Ď„i upper∗ neutrosophic Gδ -Îą-locally continuous function and Ď„j lower∗ neutrosophic Gδ -Îą-locally continuous function (i,j=1,2 and i = j). Observe that V = N G δ -Îą-lintĎ„i (Vs ) s>t s s>t V = N G -Îą-lcl (V ). Then f −1 (Rt ) = and δ Ď„i s s>t s s>t Ď„ neutros>t Vs = s>t N Gδ -Îą-lintĎ„i (Vs ) is an increasing i −1 sophic G -Îą-locally open set. Now f (L ) = δ t s>t Vs = N G -Îą-lcl (V ) is a decreasing Ď„ neutrosophic Gδ -ιδ Ď„i s j s>t locally closed set. So that f is a Ď„i upper∗ neutrosophic Gδ -Îąlocally continuous function and Ď„j lower∗ neutrosophic Gδ -Îąlocally continuous function. To conclude the proof it remains to show that g ⊆ f ⊆ h. That is g −1 (LI t ) ⊆ f −1 (LI t ) ⊆ h−1 (LI t ) and g −1 (RI t ) ⊆ f −1 (RI t ) ⊆ h−1 (RI t ) for each t ∈ R. We have, g −1 (LI s ) g −1 (LI t ) = s<t
=
g −1 (LI r )
s<t r<s
=
Br
s<t r<s
⊆
Cr
s<t r<s
=
Vs
s<t
= f −1 (LI t )
R. Narmada Devi, R. Dhavaseelan, S. Jafari, On Separation Axioms in an Ordered Neutrosophic Bitopological Space
Neutrosophic Sets and Systems, 18/2017
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DeďŹ ne g : X → II (I) by
and f −1 (LI t ) =
g(x) = f (x) if x ∈ A /A = [A0 ] if x ∈
Vs
s<t
=
Cr
and also deďŹ ne h : X → II (I) by
s<t r<s
⊆
Ar
h(x) = f (x) if x ∈ A /A = [A1 ] if x ∈
s<t r<s
=
h−1 (RI r )
s<t r<s
=
h
s<t −1
=h
−1
(LI
s)
A0 (t) = 1âˆź = 0âˆź
(Lt )
Similarly, we obtain g
−1
I
(R t ) =
where [A0 ] is the equivalence class determined by A0 : RI (I) → II (I) such that
g
−1
I
(R s )
A1 (t) = 1âˆź = 0âˆź
g −1 (LI r )
s>t r>s
=
Cr
s>t r>s
=
if if
t<1 t>1
g is a Ď„i upper∗ neutrosophic Gδ -Îą-locally continuous function and h is a Ď„j lower∗ neutrosophic Gδ -Îą-locally continuous function and g ⊆ h. Hence by Proposition 3.8, there exists a function F : X → II (I) such that F is a Ď„i upper∗ neutrosophic Gδ Îą-locally continuous function and Ď„j lower∗ neutrosophic Gδ Îą-locally continuous function and g(x) ⊆ F (x) ⊆ h(x) for all x ∈ X. Hence for all x ∈ A, f (x) ⊆ F (x) ⊆ f (x). So that F is a required extension of f over X.
Br
s>t r>s
⊆
t<0 t>0
and [A1 ] is the equivalence class determined by A1 : RI (I) → II (I) such that
s>t
=
if if
Vs
s>t
= f −1 (RI t ) and f −1 (RI t ) =
References
Vs
s>t
=
[1] I. Arokiarani, R. Dhavaseelan, S. Jafari, M. Parimala. On some new notions and functions in neutrosophic topological spaces(submitted).
Cr
s>t r>s
⊆
Ar
s>t r>s
=
h
−1
(RI r )
s>t r>s
=
h−1 (RI s )
s>t
= h−1 (RI t )
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Hence the proof. Proposition 2.9. Let (X, Ď„1 , Ď„2 , ≤) be a strongly pairwise neutrosophic Gδ -Îą-locally normally ordered space. Let A ∈ Ď„1 and A ∈ Ď„2 be crisp and let f : (A, Ď„1 /A, Ď„2 /A) → II (I) be a Ď„i upper∗ neutrosophic Gδ -Îą-locally continuous function and Ď„j lower∗ neutrosophic Gδ -Îą-locally continuous function (i,j=1,2 and i = j). Then f has a neutrosophic extension over (X, Ď„1 , Ď„2 , ≤) (that is, F : (X, Ď„1 , Ď„2 , ≤) → II (I). Proof:
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Neutrosophic Sets and Systems, 18/2017
36
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R. Narmada Devi, R. Dhavaseelan, S. Jafari, On Separation Axioms in an Ordered Neutrosophic Bitopological Space