The Ingenuity of Neutrosophic Topology via N-Topology

Neutrosophic Sets and Systems, Vol. 19, 2018

91

University of New Mexico

The Ingenuity of Neutrosophic Topology via N-Topology 1 M.

Lellis Thivagar

2

S. Jafari

3 V.

Antonysamy

4 V.

Sutha Devi

Abstract: In this paper we desire to extend the neutrosophic topological spaces into N -neutrosophic topological spaces. Also we show that this theory can be deduced to N -intuitionistic and N -fuzzy topological spaces etc. Further we develop not only the concept of classical generalized closed sets into N -neutrosophic topological spaces but also obtain its basic properties. Finally we investigate its continuous function and generalized continuous function. 2010 MSC: 54A05, 54A40, 03E72. Keywords: Nk -topology, Nk int(A), Nk cl(A), Nk -generalized Nk -continuous function, Nk -generalized continuous function.

1

closed

set,

Introduction

Set theory is the fundamental concept in mathematics developed by a Russian Mathematician George Cantor in 1877. He showed that the points on two dimensional square has a one to one correspondent with points on different line segment leading to the development of dimensional theory. Frechet and Hausdorff along with others studied general topology. Hausdorff, the German mathematician, following the footsteps of Cantor developed set theory. Set theory enabled us to study various precise concepts in mathematics. But in real life situation we do come across many imprecise concepts or uncertain situation. If a class has fifty students say, to distinguish the taller/stronger students we are left with some short of uncertainty or vagueness. We can overcome the vagueness by fixing the percentage of membership namely the percentage of membership enables us to find out the level of inexactness. This theory is known as fuzzy theory. The concepts of fuzzy set was established by Zadeh.A [12]. This is an essential tool to analyse imprecise mathematical information. Since 1965, this theory has been greatly acknowledged by the community of mathematicians, scientists, engineers and social scientists [4,9,10,11]. The idea of fuzzy topological space was introduced by Chang.C.L [3]. Atanassov.K introduced the seed of intuitionistic fuzzy set [1] and his colleagues [2] developed it further. Smarandache extended it to a neutrosophic set[7,8]. The notion of neutrosophic crisp sets and topological spaces were the contribution of Salama.A.A and Alblowi.S.A [6]. The geometric existence of N -topology was given by Lellis Thivagar et al. [5] which is a

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nonempty set equipped with N -arbitrary topologies. In this paper, we explore the possibility of expanding the classical neutrosophic topological spaces into N -neutrosophic topological spaces and also try to deduce N -intuitionistic and N -fuzzy topological spaces etc. Further we develop the concept of classical generalized closed sets into N -neutrosophic topological spaces and verify its properties. Finally, we investigate the related continuous function and generalized continuous function.

2

Preliminaries

In this section, we discuss some basic definitions and properties of N -topological spaces as well as fuzzy, intuitionistic and neutrosophic topological spaces which are useful in sequel. Definition 2.1 [5] Let X be a non empty set, then τ1 , τ∪ -arbitrary topologies 2 , ... , τN be ∩N N N defined on X and the collection N τ = {S ⊆ X : S = ( i=1 Ai ) ∪ ( i=1 Bi ), Ai , Bi ∈ τi } is called a N -topology on X if the following axioms are satisfied: (i) X, ∅ ∈ N τ . ∪ ∞ (ii) ∞ i=1 Si ∈ N τ for all {Si }i=1 ∈ N τ . ∩ (iii) ni=1 Si ∈ N τ for all {Si }ni=1 ∈ N τ . Then (X, N τ ) is called a N -topological space on X. The elements of N τ are known as N τ -open sets on X and its complement is called as N τ -closed on X. Definition 2.2 [12] Let X be a non empty set. A fuzzy set A is an object having the form A = {(x, µA (x)) : x ∈ X}, where 0 ≤ µA (x) ≤ 1 represents the degree of membership of each x ∈ X to the set A. Definition 2.3 [1,2] Let X be a non empty set. An intuitionistic set A is of the form A = {(x, µA (x), γA (x)) : x ∈ X}, where µA (x) and γA (x) represent the degree of membership and non membership function respectively of each x ∈ X to the set A and 0 ≤ µA (x) + γA (x) ≤ 1 for all x ∈ X. Definition 2.4 [7] Let X be a non empty set. A neutrosophic set A having the form A = {(x, µA (x), σA (x), γA (x)) : x ∈ X}, where µA (x), σA (x) and γA (x) represent the degree of membership function (namely µA (x)), the degree of indeterminacy (namely σA (x)) and the degree of non membership (namely γA (x)) respectively of each x ∈ X to the set A. Also − 0 ≤ µA (x) + σA (x) + γA (x) ≤ 3+ for all x ∈ X. Remark 2.5 The following definitions can be deduced into fuzzy if the percentages of indeterminacy and non membership are not taken into consideration so also for intuitionistic case the percentage of indeterminacy is not considered.

M. Lellis Thivagar, S. Jafari, V. Antonysamy, V. Sutha Devi: The Ingenuity of Neutrosophic Topology via N-Topology

Neutrosophic Sets and Systems, Vol. 19, 2018

Definition 2.6 [7] Let X be a non empty neutrosophic set. if A = {(x, µA (x), σA (x), γA (x)) : x ∈ X} and B = {(x, µB (x), σB (x), γB (x)) : x ∈ X} are two neutrosophic sets in X, then the following statements hold: (i) A ⊆ B if and only if µA (x) ≤ µB (x), σA (x) ≤ σB (x) and γA (x) ≥ γB (x) for all x ∈ X. (ii) A = B if and only if A ⊆ B and B ⊆ A. (iii) Ac = {(x, γA (x), σA (x), µA (x)) : x ∈ X} [Complement of A]. (iv) A ∩ B = {(x, min{µA (x), µB (x)}, min{σA (x), σB (x)}, max{γA (x), γB (x)}) : x ∈ X}. (v) A ∪ B = {(x,max{µA (x), µB (x)},max{σA (x), σB (x)}, min{γA (x), γB (x)}) : x ∈ X}. Remark 2.7 Let X be a non empty neutrosophic set. We consider the neutrosophic empty set 0 as 0 = {(x, 0, 0, 1) : x ∈ X} and the neutrosophic whole set 1 as 1 = {(x, 1, 1, 0) : x ∈ X}. Remark 2.8 By the notion k-set we mean any one of the following sets: fuzzy set, intuitionistic set, neutrosophic set. Definition 2.9 [6,7] Let X be a non empty set. A k-topology on X is a family k τ of k-sets in X satisfying the following axioms: (i) the sets 1 and 0 belong to the family k τ . (ii) an arbitrary union of sets of the family k τ belong to k τ . (iii) the finite intersection of sets of the family k τ belong to k τ . Then the ordered pair (X, k τ ) (simply X) is called k-topological space on X. The elements of k τ are known as k-open sets on X and its complement is called as k-closed on X. Definition 2.10 [6] The interior and closure of a k-set A of a k-topological space (X, k τ ) are respectively defined as (i) k int(A) = ∪{G : G ⊆ A and G is k-open in X}. (ii) k cl(A) = ∩{F : A ⊆ F and F is k-closed in X}. Corollary 2.11 [7] If A, B, C and D are k-sets in X, then the followings are true: (i) A ⊆ B and C ⊆ D ⇒ A ∩ C ⊆ B ∩ D and A ∪ C ⊆ B ∪ D. (ii) If A ⊆ B and A ⊆ C, then A ⊆ B ∩ C. If A ⊆ C and B ⊆ C, then A ∪ B ⊆ C. (iii) If A ⊆ B and B ⊆ C ⇒ A ⊆ C. (iv) (A ∩ B)c = Ac ∪ B c , (A ∪ B)c = Ac ∩ B c and (Ac )c = A. If A ⊆ B ⇒ B c ⊆ Ac .

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(v) 1c = 0 and 0c = 1. Now, we introduce the notions of image and pre-image of neutrosophic sets. Let us consider X and Y as two non empty sets and f : X → Y be a function. Definition 2.12 [6] Let X and Y be two non empty sets, A = {(x, µA (x), σA (x), γA (x)) : x ∈ X} be a neutrosophic set in X and B = {(y, µB (y), σB (y), γB (y)) : y ∈ Y } be a neutrosophic set in Y . Then (i) the pre-image of B under f , denoted by f −1 (B), is the neutrosophic set in X defined by f −1 (B) = {(x, f −1 (µB )(x), f −1 (σB )(x), f −1 (γB )(x)) : x ∈ X}. (ii) the image of A under f , denoted by f (A), is the neutrosophic set in Y defined by f (A) = {(y, f (µA )(y), f (σA )(y), (1 − f (1 − γA ))(y)) : y ∈ Y }, where { f (µA )(y) = { f (σA )(y) =

supx∈f −1 (y) µA (x) if f −1 (y) ̸=∅ 0 otherwise

supx∈f −1 (y) σA (x) if f −1 (y) ̸=∅ 0 otherwise {

(1 − f (1 − γA ))(y) =

infx∈f −1 (y) γA (x) if f −1 (y) ̸=∅ 1 otherwise

For the sake of simplicity, let us use the symbol f− (γA ) for (1 − f (1 − γA )). Corollary 2.13 [6] Let Ai∈J , Bi∈J be k-sets in X and Y respectively and f : X → Y a function. Then (a) A1 ⊆ A2 ⇒ f (A1 ) ⊆ f (A2 ). (b) B1 ⊆ B2 ⇒ f −1 (B1 ) ⊆ f −1 (B2 ). (c) Ai∈J ⊆ f −1 (f (Ai∈J )) { If f is injective, then Ai∈J = f −1 (f (Ai∈J ))}. (d) f (f −1 (Bi∈J )) ⊆ Bi∈J { If f is surjective, then f (f −1 (Bi∈J )) = Bi∈J }. (e) f −1 (∪Bi ) = ∪f −1 (Bi ). (f) f −1 (∩Bi ) = ∩f −1 (Bi ). (g) f (∪Ai ) = ∪f (Ai ). (h) f (∩Ai ) ⊆ f (Ai ) { If f is injective, then f (∩Ai ) = ∩f (Ai )}.

M. Lellis Thivagar, S. Jafari, V. Antonysamy, V. Sutha Devi: The Ingenuity of Neutrosophic Topology via N-Topology

Neutrosophic Sets and Systems, Vol. 19, 2018

(i) f −1 (1) = 1. (j) f −1 (0) = 0. (k) f (1) = 1, if f is surjective. (l) f (0) = 0. (m) (f (Ai∈J ))c ⊆ f (Aci∈J ), if f is surjective. c (n) (f −1 (Bi∈J ))c = f −1 (Bi∈J ).

3

Nk -Topological Spaces

In this section, we introduce N -fuzzy, N -intuitionistic and N -neutrosophic topological spaces and discuss their properties. Henceforth in this paper by the notion Nk τ we mean N -fuzzy topology (if k = f ), N -intuitionistic topology (if k = i) and N -neutrosophic topology ( if k = n). Definition 3.1 Let X be a non empty set, then k τ1 , k τ2 ,∪... , k τN be ∩ N -arbitrary k topologies N defined on X and the collection Nk τ = {G ⊆ X : G = ( N A ) ∪ ( i=1 i i=1 Bi ), Ai , Bi ∈ k τi } is called Nk -topology on X if the following axioms are satisfied: (i) 1, 0 ∈ Nk τ . ∪ ∞ (ii) ∞ i=1 Gi ∈ Nk τ for all {Gi }i=1 ∈ Nk τ . ∩ (iii) ni=1 Gi ∈ Nk τ for all {Gi }ni=1 ∈ Nk τ . Then (X, Nk τ ) is called Nk -topological space on X. The elements of Nk τ are known as Nk -open sets on X and its complement is called Nk -closed sets on X. Example 3.2 Let N = 3, X = {a, b, c}. Define the neutrosophic sets A = {(x, ( a1 , 1b , 1c ), a b c a b c ( a0 , 0b , 0c ), ( 0.7 , 0.7 , 0.7 ))} and B = {(x, ( 0.6 , 0.6 , 0.6 ), ( a0 , 0b , 0c ), ( a0 , 0b , 0c ))} in X. Then a b c a b c a b c a b c a b c a b c A ∪ B = {(x, ( 1 , 1 , 1 ), ( 0 , 0 , 0 ), ( 0 , 0 , 0 ))}, A ∩ B = {(x, ( 0.6 , 0.6 , 0.6 ), ( 0 , 0 , 0 ), ( 0.7 , 0.7 , 0.7 ))}. Considering n τ1 O(X) = {0, 1, A}, n τ2 O(X) = {0, 1, B} and n τ3 O(X) = {0, 1}, we get 3n τ O(X) = {0, 1, A, B, A ∪ B, A ∩ B} which is a tri-neutrosophic topology on X. The pair (X, 3n τ ) is called a tri-neutrosophic topological space on X. Remark 3.3 Considering N = 2 in definition 3.1 we get the required definition of bineutrosophic topology on X. The pair (X, 2n τ ) is called a bi-neutrosophic topological space on X. b c a , 0.3 , 0.6 ), Example 3.4 Let N = 2, X = {a, b, c}. Define the neutrosophic set A = {(x, ( 0.4 a b c a b c ( 0.4 , 0.5 , 0.6 ), ( 0.3 , 0.2 , 0.3 ))} in X. If n τ1 O(X) = {0, 1, A} and n τ2 O(X) = {0, 1} are two neutrosophic topologies then we get 2n τ O(X) = {0, 1, A} which is a bi-neutrosophic topology on X.

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Definition 3.5 Let (X, Nk τ ) be a Nk -topological space on X and A be a k-set on X then the Nk int(A) and Nk cl(A) are respectively defined as (i) Nk int(A) = ∪{G : G ⊆ A and G is a Nk -open set in X}. (ii) Nk cl(A) = ∩{F : A ⊆ F and F is a Nk -closed set in X}. Proposition 3.6 Let (X, Nk τ ) be any Nk -topological space. If A and B are any two k-sets in (X, Nk τ ), then the Nk -closure operator satisfy the following properties: (i) A ⊆ Nk cl(A). (ii) Nk int(A) ⊆ A. (iii) A ⊆ B ⇒ Nk cl(A) ⊆ Nk cl(B). (iv) A ⊆ B ⇒ Nk int(A) ⊆ Nk int(B). (v) Nk cl(A ∪ B) = Nk cl(A) ∪ Nk cl(B). (vi) Nk int(A ∩ B) = Nk int(A) ∩ Nk int(B). (vii) (Nk cl(A))c = Nk int(A)c . (viii) (Nk int(A))c = Nk cl(A)c . Proof (i) Nk cl(A) = ∩{G : G is a Nk -closed set in X and A ⊆ G}. Thus, A ⊆ Nk cl(A). (ii) Nk int(A) = ∪{G : G is a Nk -open set in X and G ⊆ A}. Thus, Nk int(A) ⊆ A. (iii) Nk cl(B) = ∩{G : G is a Nk -closed set in X and B ⊆ G} ⊇ ∩{G : G is a Nk -closed set in X and A ⊆ G} ⊇ Nk cl(A). Thus, Nk cl(A) ⊆ Nk cl(B). (iv) Nk int(B) = ∪{G : G is a Nk -open set in X and B ⊇ G} ⊇ ∪{G : G is a Nk -open set in X and A ⊇ G} ⊇ Nk int(A). Thus, Nk int(A) ⊆ Nk int(B). (v) Nk cl(A ∪ B) = ∩{G : G is a Nk -closed set in X and A ∪ B ⊆ G} = (∩{G : G is a Nk -closed set in X and A ⊆ G}) ∪ (∩{G : G is a Nk -closed set in X and B ⊆ G}) = Nk cl(A) ∪ Nk cl(B). Thus, Nk cl(A ∪ B) = Nk cl(A) ∪ Nk cl(B). (vi) Nk int(A ∩ B) = ∪{G : G is a Nk -open set in X and A ∩ B ⊇ G} = (∪{G : G is a Nk -open set in X and A ⊇ G}) ∩ (∪{G : G is a Nk -open set in X and B ⊇ G}) = Nk int(A) ∩ Nk int(B). Thus, Nk int(A ∩ B) = Nk int(A) ∩ Nk int(B). (vii) Nk cl(A) = ∩{G : G is a Nk -closed set in X and A ⊆ G}, (Nk cl(A))c = ∪{Gc : Gc is a Nk -open set in X and Ac ⊇ Gc } = Nk int(A)c . Thus, (Nk cl(A))c = Nk int(A)c . (viii) Nk int(A) = ∪{G : G is a Nk -open set in X and A ⊇ G}, (Nk int(A))c = ∩{Gc : Gc is a Nk -closed set in X and Ac ⊇ Gc } = Nk cl(A)c . Thus, (Nk int(A))c = Nk cl(A)c .

M. Lellis Thivagar, S. Jafari, V. Antonysamy, V. Sutha Devi: The Ingenuity of Neutrosophic Topology via N-Topology

Neutrosophic Sets and Systems, Vol. 19, 2018

4

Generalized Closed Sets in Nk -topology

We introduce here the generalized closed sets in Nk -topological spaces and investigate their properties. Definition 4.1 Let (X, Nk τ ) be a Nk -topological space. A k-set A in (X, Nk τ ) is said to be a Nk -generalized closed set if Nk cl(A) ⊆ G, whenever A ⊆ G and G is a Nk -open set. The complement of a Nk -generalized closed set is called a Nk -generalized open set. Definition 4.2 Let (X, Nk τ ) be a Nk -topological space and A be a k-set in X. Then the Nk -generalized closure and Nk -generalized interior of A are defined as: (i) Nk Gcl(A) = ∩{G : G is a Nk -generalized closed set in X and A ⊆ G} (ii) Nk Gint(A) = ∪{G : G is a Nk -generalized open set in X and G ⊆ A}. Proposition 4.3 Let (X, Nk τ ) be any Nk -topological space. If A and B are any two k-sets in (X, Nk τ ), then the Nk -generalized closure operator satisfies the following properties: (i) A ⊆ Nk Gcl(A). (ii) Nk Gint(A) ⊆ A. (iii) A ⊆ B ⇒ Nk Gcl(A) ⊆ Nk Gcl(B). (iv) A ⊆ B ⇒ Nk Gint(A) ⊆ Nk Gint(B). (v) Nk Gcl(A ∪ B) = Nk Gcl(A) ∪ Nk Gcl(B). (vi) Nk Gint(A ∩ B) = Nk Gint(A) ∩ Nk Gint(B). (vii) (Nk Gcl(A))c = Nk Gint(A)c . (viii) (Nk Gint(A))c = Nk Gcl(A)c . Proof The proof is analogous to Proposition 3.6. Proposition 4.4 Let (X, Nk τ ) be a Nk -topological space. If B is a Nk -generalized closed set and B ⊆ A ⊆ Nk cl(B), then A is a Nk - generalized closed set. Proof. Let G be a Nk -open set in (X, Nk τ ) such that A ⊆ G. Since B ⊆ A, B ⊆ G. Now, B is a Nk -generalized closed set and Nk cl(B) ⊆ G. But Nk cl(A) ⊆ Nk cl(B). Since Nk cl(A) ⊆ Nk cl(B) ⊆ G, Nk cl(A) ⊆ G. Hence, A is a Nk -generalized closed set. Proposition 4.5 Let (X, Nk τ ) be a Nk -topological space. Then A is a Nk -generalized open set if and only if B ⊆ Nk int(A), whenever B is an Nk -closed set and B ⊆ A.

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Proof. Let A be a Nk -generalized open set and B a Nk -closed set such that B ⊆ A. Now, B ⊆ A ⇒ Ac ⊆ B c and since Ac is a Nk -generalized closed set, then Nk cl(Ac ) ⊆ B c . This means that B = (B c )c ⊆ (Nk cl(Ac ))c . But (Nk cl(Ac ))c = Nk int(A). Hence, B ⊆ Nk int(A). Conversely, suppose that A is a k-set such that B ⊆ Nk int(A), whenever B is a Nk -closed set and B ⊆ A. Now, Ac ⊆ B ⇒ B c ⊆ A. Hence by assumption, B c ⊆ Nk int(A). That is, (Nk int(A))c ⊆ B. But (Nk int(A))c = Nk cl(A)c . Hence, Nk cl(A)c ⊆ B. This means that A is a Nk -generalized closed set. Therefore, A is a Nk -generalized open set. Proposition 4.6 If Nk int(A) ⊆ B ⊆ A and A is a Nk -generalized open set, then B is also a Nk -generalized open set. Proof. Now, Ac ⊆ B c ⊆ (Nk int(A))c = Nk cl(A)c . Since A is a Nk -generalized open set, then Ac is a Nk -generalized closed set. By Proposition 3.6, B c is a Nk -generalized closed set. That is, B is a Nk -generalized open set.

5

Continuous Functions in Nk -Topology

In this section, we generalize continuous functions in N -neutrosophic topological spaces and also establish its relationship with other existing continuous functions. Definition 5.1 Let (X, Nk τ )) and (Y, Nk σ) be any two Nk -topological spaces. A map f : (X, Nk τ ) → (Y, Nk σ) is said to be Nk -continuous if the inverse image of every Nk closed set in (Y, Nk σ) is a Nk -closed set in (X, Nk τ ). Equivalently if the inverse image of every Nk -open set in (Y, Nk σ) is a Nk -open set in (X, Nk τ ). Remark 5.2 By considering N = 2 in definition 5.1 we obtain bi-neutrosophic continuous function. The following properties can be extended to N -fuzzy and N -intuitionistic topological spaces too. Proposition 5.3 Let (X, Nk τ )) and (Y, Nk σ) be any two Nk -topological spaces. Let f : (X, Nk τ ) → (Y, Nk σ) be a Nk - continuous function. Then for every k-set A in X, f (Nk cl(A)) ⊆ Nk cl(f (A)). Proof. Let A be a k-set in (X, Nk τ ). Since Nk cl(f (A)) is a Nk -closed set and f is a Nk continuous function, f −1 (Nk cl(f (A)) is a Nk -closed set and f −1 (Nk cl(f (A))) ⊇ A. Now, Nk cl(A) ⊆ f −1 (Nk cl(f (A))). Therefore, f (Nk cl(A)) ⊆ Nk cl(f (A)). Proposition 5.4 Let (X, Nk τ ) and (Y, Nk σ) be any two Nk -topological spaces. Let f : (X, Nk τ ) → (Y, Nk σ) be a Nk -continuous function. Then for every Nk -set A in Y , Nk cl(f −1 (A)) ⊆ f −1 (Nk cl(A)). Proof. Let A be a Nk -set in (Y, Nk σ). Let B = f −1 (A). Then, f (B) = f (f −1 (A)) ⊆ A. By Proposition 5.3, f (Nk cl(f −1 (A))) ⊆ Nk cl(f (f −1 (A))). Thus, Nk cl(f −1 (A)) ⊆ f −1 (Nk cl(A)).

M. Lellis Thivagar, S. Jafari, V. Antonysamy, V. Sutha Devi: The Ingenuity of Neutrosophic Topology via N-Topology

Neutrosophic Sets and Systems, Vol. 19, 2018

Definition 5.5 Let (X, Nk τ )) and (Y, Nk σ) be any two Nk -topological spaces. A map f : (X, Nk τ ) → (Y, Nk σ) is said to be Nk -generalized continuous if the inverse image of every Nk -closed set in (Y, Nk σ) is a Nk -generalized closed set in (X, Nk τ ). Equivalently if the inverse image of every Nk -open set in (Y, Nk σ) is a Nk -generalized open set in (X, Nk τ ). Remark 5.6 For N = 2 in the above definition we aquire the needed definition of bigeneralized neutrosophic continuous function. Proposition 5.7 Let (X, Nk τ )) and (Y, Nk σ) be any two Nk -topological spaces. Let f : (X, Nk τ ) → (Y, Nk σ) be a Nk -generalized continuous function. Then for every Nk -set A in X, f (Nk Gcl(A)) ⊆ Nk cl(f (A)). Proof. Let A be a Nk -set in (X, Nk τ ). Since Nk cl(f (A)) is a Nk -closed set and f is a Nk continuous function, f −1 (Nk cl(f (A)) is a Nk -generalized closed set and f −1 (Nk cl(f (A))) ⊇ A. Now, Nk Gcl(A) ⊆ f −1 (Nk cl(f (A))). Therefore, f (Nk Gcl(A)) ⊆ Nk cl(f (A)). Proposition 5.8 Let (X, Nk τ ) and (Y, Nk σ) be any two Nk -topological spaces. Let f : (X, Nk τ ) → (Y, Nk σ) be a Nk -generalized continuous function. Then for every Nk -set A in Y , Nk Gcl(f −1 (A)) ⊆ f −1 (Nk cl(A)). Proof. Let A be a Nk -set in (Y, Nk σ). Let B = f −1 (A). Then, f (B) = f (f −1 (A)) ⊆ A. By Proposition 5.7, f (Nk Gcl(f −1 (A))) ⊆ Nk cl(f (f −1 (A))). Thus, Nk Gcl(f −1 (A)) ⊆ f −1 (Nk cl(A)). Proposition 5.9 Let (X, Nk τ ) and (Y, Nk σ) be any two Nk -topological spaces. If f : (X, Nk τ ) → (Y, Nk σ) is a Nk -continuous function, then it is a Nk -generalized continuous function. Proof. Let A be a Nk -open set in (Y, Nk σ). Since f is a Nk -continuous function, f −1 (A) is a Nk -open set in (X, Nk τ ). Every Nk -open set is a Nk -generalized open set. Now, f −1 (A) is a Nk -generalized open set in (X, Nk τ ). Hence, f is a Nk -generalized continuous function. The converse of Proposition 5.9 need not be true as it is shown in the following example. Example 5.10 Let N = 2, X = {a, b, c} and Y = {p, q, r}. Define the neutrosophic sets p q a b c a b c a b c r A = {(x, ( 0.4 , 0.4 , 0.5 ), ( 0.4 , 0.4 , 0.5 ), ( 0.2 , 0.4 , 0.3 ))} in X and B = {(y, ( 0.4 , 0.5 , 0.6 ), p q p q r r ( 0.4 , 0.5 , 0.6 ), ( 0.3 , 0.2 , 0.3 ))} in Y . Considering n τ1 O(X) = {0, 1, A} and n τ2 O(X) = {0, 1} we get 2n τ O(X) = {0, 1, A}. Also by considering n σ1 O(Y ) = {0, 1} and n σ2 O(Y ) = {0, 1, B} we get 2n σO(Y ) = {0, 1, B}. Thus, (X, 2n τ ) and (Y, 2n σ) are bi-neutrosophic topological space on X and Y , respectively. Define f : X → Y as f (a) = q, f (b) = p, f (c) = r. Then f is bi-generalized neutrosophic continuous but not bi-neutrosophic continuous. Conclusion Neutrosophic topology is well equipped to deal with imprecise data. By employing neutrosophic set in spacial data models, we can express the vagueness of the object as expected. This paper has gone a step forward in extending the theory to N -neutrosophic topology that can be used to determine the uncertain situation effectively. Further we also extended the same to N -Fuzzy and N -Intuitionistic topologies and discussed not only the relations but also its properties.

M. Lellis Thivagar, S. Jafari, V. Antonysamy, V. Sutha Devi: The Ingenuity of Neutrosophic Topology via N-Topology

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Neutrosophic Sets and Systems, Vol. 19, 2018

References [1] Atanassov. K, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 1986, 87-96. [2] Atanassov. K. and Stoeva. S, Intuitionistic fuzzy sets, in Polish Syrup. on Interval & Fuzzy Mathematics, Poznan, 1983, 23-26. [3] Chang. C.L, Fuzzy topological spaces, J. Math. Anal. Appl., 24, 1968, 182-190. [4] Dhavaseelan. R, Roja. E and Uma. M. K, Generalized intuitionistic fuzzy closed sets, Advances in Fuzzy Mathematics, 5, 2010, 152-172. [5] Lellis Thivagar. M, Ramesh.V, Arockia Dasan. M, On new structure of N -topology, Cogent Mathematics (Taylor and Francis),3, 2016:1204104. [6] Salama. A. A. and Albowi. S. A, Neutrosophic Set and Neutrosophic Topological Spaces, IOSR Journal of Mathematics, 3(4), 2012, 31-35. [7] Smarandache. F, Neutrosophy and Neutrosophic Logic, First International Conference on Neutrosophy, Neutrosophic Logic , Set, Probability, and Statistics University of New Mexico, Gallup, NM 87301, USA(2002). [8] Smarandache. F, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability, American Research Press, Rehoboth, NM, 1999. [9] Smets. P, The degree of belief in a fuzzy event, Information sciences 25, 1981, 1-19. [10] Sugeno. M, An Introductory survey of fuzzy control, Information sciences, 36, 1985, 59-83. [11] Thakur. S. S and Chaturvedi. R, Generalized continuity in intuitionistic fuzzy topological spaces, NIFS, 12(1), 2006, 38-44. [12] Zadeh. L. A, Fuzzy sets, Inform. and Control, 8, 1965, 338-353. 1,3,4

School of Mathematics Madurai Kamaraj University Madurai - 625 021, Tamil Nadu, INDIA 2 Department of Mathematics College of Vestsjaelland South Herrestraede 11, 4200 Slagelse, Denmark. 1

E-mail: mlthivagar@yahoo.co.in E-mail: jafaripersia@gmail.com 3 E-mail: tonysamsj@yahoo.com 4 E-mail: vsdsutha@yahoo.co.in

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Received : February 19, 2018. Accepted : March 12, 2018. M. Lellis Thivagar, S. Jafari, V. Antonysamy, V. Sutha Devi: The Ingenuity of Neutrosophic Topology via N-Topology