Neutrosophic set and Neutrosophic Topological spaces has been introduced by Salama. Neutrosophic Closed set and Neutrosophic Continuous Functions were introduced by Salama et. al.. In this paper, we introduce the concept of Nω- closed sets and their properties in Neutrosophic topological spaces

Santhi R.

Udhayarani N.

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Neutrosophic Sets and Systems, Vol. 12, 2016 Santhi R. and Udhayarani N., Nω –CLOSED SETS IN NEUTROSOPHIC TOPOLOGICAL SPACES University of New Mexico Nω –CLOSED SETS IN NEUTROSOPHIC TOPOLOGICAL SPACES Santhi R.1 and Udhayarani N.2 1N.G.M. college, Pollachi, Tamil nadu- 642001, India.E-mail:santhifuzzy@gmail.com2N.G.M. college, Pollachi, Tamil nadu- 642001, India.E-mail: udhayaranin@gmail.com Abstract. Neutrosophic set and Neutrosophic Topological spaces has been introduced by Salama[5]. Neutrosophic Closed set and Neutrosophic Continuous Functions were introduced by Keywords: Intuitionistic Fuzzy set, Neutrosophic set, Neutrosophic Topology, Ns-open set, Ns-closed set, Nω- closed set, Nω- open set and Nω-closure. 1. Introduction Many theories like, Theory of Fuzzy sets[10], Theory of Intuitionistic fuzzy sets[1], Theory of Neutrosophic sets[8] and The Theory of Interval Neutrosophic sets[4] can be considered as tools for dealing with uncertainities. However, all of these theories have their own difficulties which are pointed out in[8]. In 1965, Zadeh[10] introduced fuzzy set theory as a mathematical tool for dealing with uncertainities where each element had a degree of membership. The Intuitionistic fuzzy set was introduced by Atanassov[1] in 1983 as a generalization of fuzzy set, where besides the degree of membership and the degree of nonmembership of each element. The neutrosophic set was introduced by Smarandache[7] and explained, neutrosophic set is a generalization of intuitionistic fuzzy set. In 2012, Salama, Alblowi[5] introduced the concept of  Neutrosophic topological spaces. They introduced neutrosophic topological space as a generalization of intuitionistic fuzzy topological space and a neutrosophic set besides the degree of membership, the degree of indeterminacy and the degree of nonmembership of each element. In 2014 Salama, Smarandache and Valeri [6]  were introduced the concept of neutrosophic closed sets and neutrosophic continuous functions. In this paper, we introduce the concept of Nω- closed sets and their properties in  neutrosophic topological spaces.  2. PreliminariesIn this paper, X denote a topological space (X, τN) on which no separation axioms are assumed unless otherwise explicitly  mentioned. We recall the following definitions, which will be used throughout  this paper. For a subset A of X, Ncl(A), Nint(A) and Ac denote the neutrosophic closure, neutrosophic interior, and the complement of neutrosophic set A respectively. Definition 2.1.[3] Let X be a non-empty fixed set. A neutrosophic set(NS for short) A is an object having the form A = {<x, µA(x), σA(x), υA(x)>: for all x ∈ X}. Where µA(x), σA(x), υA(x) which represent the degree of membership, the degree of indeterminacy and the degree of nonmembership of each element x ∈ X to the set A. Definition 2.2.[5] Let A and B be NSs of the form  A = {<x, µA(x), σA(x), υA(x)>: for all x ∈ X} and B = {<x, µB(x), σB(x), υB(x)>: for all x ∈ X} . Then  i. A ⊆ B if and only if µA(x) ≤ µB(x), σA(x) ≥ σB(x)and υA(x) ≥ υB(x) for all x ∈ X,ii. A = B if and only if A ⊆ B and B ⊆ A,iii. Ac = {<x, υA(x), 1− σA(x), µA(x)>: for all x ∈ X },iv. A ∪ B = {<x, µA(x) ∨ µB(x), σA(x) ∧ σB(x),υA(x) ∧ υB(x): for all x ∈ X >},v. A ∩ B = {<x, µA(x) ∧µB(x), σA(x) ∨ σB(x),υA(x) ∨ υB(x): for all x ∈ X >}.Definition 2.3.[5] A neutrosophic topology(NT for short) on a non empty set X is a family τ of neutrosophic subsets in X satisfying the following axioms: i) 0N, 1N ∈ τ,ii) G1 ∩ G2 ∈ τ, for any G1, G2 ∈ τ,iii) ∪Gi ∈ τ, for all  Gi: i ∈ J ⊆ τIn this pair (X, τ) is called a neutrosophic topological space (NTS for short) for neutrosophic set (NOS for short) τ in X.  The elements of τ are called open neutrosophic sets. A neutrosophic set F is called closed if and only if the complement of F(Fc for short) is neutrosophic open. Definition 2.4.[5] Let (X, τ) be a neutrosophic topological space. A neutrosophic set A in (X, τ) is said to be neutrosophic closed(N-closed for short) if Ncl(A) ⊆ G whenever A ⊆ G and G is neutrosophic open. Salama et. al.. In this paper, we introduce the concept of Nω- closed sets and their properties in Neutrosophic topological spaces.  114Neutrosophic Sets and Systems, Vol. 12, 2016  Santhi R. and Udhayarani N., Nω –CLOSED SETS IN NEUTROSOPHIC TOPOLOGICAL SPACESDefinition 2.5.[5] The complement of N-closed set is N-open set. Proposition 2.6.[6] In a neutrosophic topological space (X, T), T = ℑ (the family of all neutrosophic closed sets) iff every neutrosophic subset of (X, T) is a neutrosophic closed set. 3. Nω -closed setsIn this section, we introduce the concept of Nω-closed set and some of their properties. Throughout this paper (X, τN) represent a neutrosophic topological spaces. Definition 3.1. Let (X, τN) be a neutrosophic topological space. Then A is called neutrosophic semi open set(Ns-open set for short) if A ⊂ Ncl(Nint(A)). Definition 3.2. Let (X, τN) be a neutrosophic topological space. Then A is called neutrosophic semi closed set(Ns-closed set for short) if Nint(Ncl(A)) ⊂ A. Definition 3.3. Let A be a neutrosophic set of  a neutrosophic topological space   (X, τN).  Then,  i. The neutrosophic semi closure of A is defined asNscl(A) = ∩ {K: K is a Ns-closed in X and A ⊆K}ii. The neutrosophic semi interior of A is defined asNsint(A) = ∪ {G: G is a Ns-open in X and G ⊆ A}Definition 3.4. Let (X, τN) be a neutrosophic topological space. Then A is called Neutrosophic ω closed set(Nω-closed set for short) if Ncl(A) ⊆ G whenever A ⊆ G and G is Ns-open set. Theorem 3.5. Every neutrosophic closed set is Nω-closed set, but the converse may not be true. Proof: If A is any neutrosophic set in X and G is any Ns-open set containing A, then Ncl (A)  ⊆ G. Hence A is Nω-closed set. The converse of the above theorem need not be true as seen from the following example. Example 3.6. Let X = {a,b,c} and τN = {0N,G1, 1N} is a neutrosophic topology and (X, τN) is a neutrosophic topological spaces. Take G1 = <x, (0.5, 0.6, 0.4), (0.4, 0.5, 0.2), (0.7, 0.6, 0.9)>, A = <x, (0.2, 0.2, 0.1), (0, 1, 0.2), (0.8, 0.6,0.9)>.  Then the set A is Nω-closed set but A is not a neutrosophic closed. Theorem 3.7. Every Nω-closed set is N-closed set but not conversely. Proof:  Let A be any Nω-closed set in X and G be any neutrosophic open set such that A ⊆ G. Then G is Ns-open, A ⊆ G and Ncl (A) ⊆ G. Thus A is N-closed. The converse of the above theorem proved by the following example. Example 3.8. Let X = {a,b,c} and τN = {0N,G1,1N} is a neutrosophic topology and (X, τN) is a neutrosophic topological spaces. Let G1 = <x, (0.5, 0.6, 0.4), (0.4, 0.5, 0.2), (0.7, 0.6, 0.9)> and A = <x, (0.55, 0.45, 0.6), (0.11, 0.3, 0.1), (0.11, 0.25, 0.2)>. Then the set A is N-closed but A is not a Nω-closed set. Remark 3.9. The concepts of Nω-closed sets and Ns-closed sets are independent. Example 3.10. Let X = {a,b,c} and τN = {0N,G1, 1N} is a neutrosophic topology and (X, τN) is a neutrosophic topological spaces. Take G1 = <x, (0.5, 0.6, 0.4), (0.4, 0.5, 0.2), (0.7, 0.6, 0.9)>, A = <x, (0.2, 0.2, 0.1), (0, 1, 0.2), (0.8, 0.6,0.9)>.  Then the set A is Nω-closed set but A is not a Ns-closed set.  Example 3.11. Let X = {a,b} and τN = {0N,G1, G2, 1N} is a neutrosophic topology and (X, τN) is a neutrosophic topological spaces. Take G1 = <x, (0.6, 0.7), (0.3, 0.2), (0.2, 0.1)> and A = <x, (0.3, 0.4), (0.6, 0.7), (0.9,0.9)>.  Then the set A is Ns-closed set but A is not a Nω-closed.   Theorem 3.12. If A and B are Nω-closed sets, then A ∪ B is Nω-closed set. Proof: If   A ∪ B ⊆ G and G is Ns-open set, then A ⊆ G and B ⊆ G. Since A and B are Nω-closed sets, Ncl(A) ⊆ G and Ncl(B) ⊆ G and hence Ncl(A) ∪ Ncl(B) ⊆  G. This implies Ncl(A ∪ B) ⊆  G.  Thus A ∪ B is Nω-closed set in X. Theorem 3.13. A neutrosophic set A is Nω-closed set then Ncl(A) – A does not contain any nonempty neutrosophic closed sets. Proof: Suppose that A is  Nω-closed set. Let F be a neutrosophic closed subset of  Ncl(A) – A. Then A ⊆ Fc. But A is Nω-closed set. Therefore Ncl(A) ⊆ Fc.Consequently F ⊆ (Ncl(A))c. We have F ⊆ Ncl(A). Thus F ⊆ Ncl(A)∩(Ncl(A))c = ϕ. Hence F is empty. The converse of the above theorem need not be true as seen from the following example. Example 3.14. Let X = {a,b,c} and τN = {0N,G1, 1N} is a neutrosophic topology and (X, τN) is a neutrosophic topological spaces. Take G1 = <x, (0.5, 0.6, 0.4), (0.4, 0.5, 0.2), (0.7, 0.6, 0.9)> and A = <x, (0.2, 0.2, 0.1), (0.6, 0.6, 0.6), (0.8, 0.9,0.9)>.  Then the set A is not a Nω-closed set and Ncl(A) – A = <x, (0.2, 0.2, 0.1), (0.6, 0.6, 0.6), (0.8, 0.9,0.9)> does not contain non-empty neutrosophic closed sets.  Theorem 3.15. A neutrosophic set A is Nω-closed set if and only if Ncl(A) – A contains no non-empty Ns-closed set. 115Neutrosophic Sets and Systems, Vol. 12, 2016  Santhi R. and Udhayarani N., Nω –CLOSED SETS IN NEUTROSOPHIC TOPOLOGICAL SPACESProof:  Suppose that A is Nω-closed set. Let S be a Ns-closed subset of Ncl(A) – A. Then A ⊆ Sc. Since A is Nω-closed set, we have Ncl(A)  ⊆ Sc. Consequently S ⊆ (Ncl(A))c. Hence S ⊆ Ncl(A)∩(Ncl(A))c = ϕ. Therefore S is empty. Conversely, suppose that Ncl(A) – A contains no non-empty Ns-closed set. Let A ⊆ G and that G be Ns-open. If Ncl(A) ⊈ G, then Ncl(A) ∩ Gc is a non-empty Ns-closedsubset of Ncl(A) – A. Hence A is Nω-closed set. Corollary 3.16. A Nω-closed set A is Ns-closed if and only if Nscl(A) – A is Ns-closed. Proof: Let A be any Nω-closed set. If A is Ns-closed set, then Nscl(A) – A = ϕ. Therefore Nscl(A) – A is  Ns-closed set. Conversely, suppose that Ncl(A) – A is Ns-closed set. But A is Nω-closed set and Ncl(A) – A contains Ns-closed. By theorem 3.15, Nscl(A) – A = ϕ. Therefore  Nscl(A) = A. Hence A is Ns-closed set. Theorem 3.17. Suppose that B ⊆ A ⊆ X, B is a Nω-closed set relative to A and  that A is Nω-closed set in X. Then B is Nω-closed set in X. Proof:  Let B ⊆ G, where G is Ns-open in X. We have B ⊆ A∩G and A∩G is Ns-open in A. But B is a Nω-closed set relative to A. Hence NclA(B) ⊆ A∩G. Since NclA(B) = A ∩ Ncl(B). We have A∩Ncl(B) ⊆ A∩G. It implies A ⊆ G∪(Ncl(B))c and G∪(Ncl(B))c is a Ns-open set in X. SinceA is Nω-closed in X, we have Ncl(A) ⊆ G∪(Ncl(B))c.Hence Ncl(B) ⊆ G∪(Ncl(B))c and Ncl(B) ⊆ G. Therefore B is Nω-closed set relative to X. Theorem 3.18. If A is Nω-closed and A ⊆ B ⊆ Ncl(A), then B is Nω-closed. Proof:  Since B ⊆ Ncl(A), we have Ncl(B) ⊆ Ncl(A) and Ncl(B) – B ⊆ Ncl(A) – A. But A is Nω-closed. Hence Ncl(A) – A has no non-empty Ns-closed subsets, neither does Ncl(B) – B. By theorem 3.15, B is Nω-closed. Theorem 3.19. Let A ⊆ Y ⊆ X and suppose that A is Nω-closed in X. Then A is Nω-closed relative to Y. Proof:  Let A ⊆ Y∩G where G is Ns-open in X. Then A ⊆ G and hence Ncl(A) ⊆ G. This implies, Y ∩ Ncl(A) ⊆ Y∩G. Thus A is Nω-closed relative to Y. Theorem 3.20. If A is Ns-open and Nω-closed, then A is neutrosophic closed set. Proof:  Since A is Ns-open and Nω-closed, then Ncl(A) ⊆ A. Therefore Ncl(A) =A. Hence A is neutrosophic closed. 4. Nω-open setsIn this section, we introduce and study about Nω-open sets and some of their properties. Definition 4.1. A Neutrosophic set A in X is called Nω-open in X if Ac is Nω-closed in X.Theorem 4.2. Let (X, τN) be a neutrosophic topological space. Then (i) Every neutrosophic open set is Nω-open but not conversely. (ii) Every Nω-open set is N-open but not conversely. The converse part of the above statements are proved by the following example. Example 4.3. Let X = {a,b,c} and τN = {0N,G1, 1N} is a neutrosophic topology and (X, τN) is a neutrosophic topological space. Take G1 = <x, (0.7, 0.6, 0.9), (0.6, 0.5, 0.8), (0.5, 0.6, 0.4)> and A = <x, (0.8, 0.6, 0.9), (1, 0, 0.8), (0.2, 0.2, 0.1)>.  Then the set A is Nω-open set but not  a neutrosophic open and B = <x, (0.11, 0.25, 0.2), (0.89, 0.7, 0.9), (0.55, 0.45, 0.6)>  is N-open but not a Nω-open set. Theorem 4.4. A neutrosophic set A is Nω-open if and only if F ⊆  Nint(A) where F is Ns-closed and F ⊆ A. Proof: Suppose that F ⊆ Nint(A) where F is Ns-closed and F ⊆ A. Let Ac ⊆ G where G is Ns-open. Then Gc ⊆ A andGc is Ns-closed. Therefore Gc ⊆ Nint(A). Since Gc ⊆Nint(A), we have (Nint(A))c ⊆ G. This implies Ncl((A)c) ⊆ G. Thus Ac is Nω-closed. Hence A is Nω-open.Conversely, suppose that A is Nω-open, F ⊆ A and F is Ns-closed. Then Fc is Ns-open and Ac ⊆ Fc. ThereforeNcl((A)c) ⊆ Fc. But Ncl((A)c) = (Nint(A))c. Hence F ⊆ Nint(A). Theorem 4.5. A neutrosophic set A is Nω-open in X if and only if G = X whenever G is Ns-open and (Nint(A)∪Ac) ⊆ G. Proof: Let A be a Nω-open, G be Ns-open and (Nint(A)∪Ac) ⊆ G. This implies Gc ⊆ (Nint(A))c ∩((A)c)c = (Nint(A))c – Ac = Ncl((A)c) – Ac. Since Ac is Nω-closedand Gc is Ns-closed, by Theorem 3.15, it follows that Gc =ϕ. Therefore X = G. Conversely, suppose that F is Ns-closed and F ⊆ A. Then Nint(A) ∪ Ac ⊆ Nint(A) ∪ Fc. This implies Nint(A) ∪ Fc = X and hence F ⊆ Nint(A). Therefore A is Nω-open. Theorem 4.6. If Nint(A)  ⊆ B ⊆ A and if A is Nω-open, then B is Nω-open. Proof:  Suppose that Nint(A) ⊆ B ⊆ A and A is Nω-open. Then Ac ⊆Bc ⊆ Ncl(Ac) and since Ac is Nω-closed. Wehave by Theorem 3.15, Bc is Nω-closed. Hence B is Nω-open. Theorem 4.7. A neutrosophic set A is Nω-closed, if and only if Ncl(A) – A is Nω-open. Proof: Suppose that A is Nω-closed. Let F  ⊆ Ncl(A) – A Where F is  Ns-closed. By Theorem 3.15, F = ϕ. Therefore F ⊆ Nint((Ncl(A) – A) and by Theorem 4.4, we have  Ncl(A) – A is Nω-open.  116Neutrosophic Sets and Systems, Vol. 12, 2016  Santhi R. and Udhayarani N., Nω –CLOSED SETS IN NEUTROSOPHIC TOPOLOGICAL SPACESConversely, let A ⊆ G where G is a Ns-open set. Then Ncl(A) ∩ Gc ⊆ Ncl(A) ∩ Ac = Ncl(A) – A. Since Ncl(A) ⊆ Gc is Ns-closed and Ncl(A) – A is Nω-open. By Theorem4.4, we have Ncl(A) ∩ Gc ⊆ Nint(Ncl(A) – A) = ϕ. Hence A is Nω-closed. Theorem 4.8. For a subset A⊆X the following are equivalent: (i) A is Nω-closed. (ii) Ncl(A) – A contains no non-empty Ns-closed set. (iii) Ncl(A) – A is Nω-open set. Proof: Follows from Theorem 3.15 and Theorem 4.7. 5. Nω-closure and Properties of Nω-closureIn this section, we introduce the concept of Nω-closure and some of their properties. Definition 5.1. The  Nω-closure (briefly Nωcl(A)) of a subset A of a neutrosophic topological space (X, τN) is defined as follows:  Nωcl(A) = ∩ { F ⊆ X / A ⊆ F and F is Nω-closed in (X, τN)}. Theorem 5.2. Let A be any subset of (X, τN). If A is Nω-closed in (X, τN) then A = Nωcl(A). Proof: By definition, Nωcl(A) = ∩ { F ⊆ X / A ⊆ F and F is a Nω-closed in (X, τN)} and we know that A ⊆ A. Hence A = Nωcl(A). Remark 5.3. For a subset A of (X, τN), A ⊆ Nωcl(A) ⊆ Ncl(A). Theorem 5.4. Let A and B be subsets of  (X, τN). Then the following statements are true: i. Nωcl(A) = ϕ and Nωcl(A) = X.ii. If A ⊆ B, then Nωcl(A) ⊆ Nωcl(B)iii. Nωcl(A) ∪ Nωcl (B) ⊂ Nωcl (A ∪ B)iv. Nωcl (A ∩ B) ⊂ Nωcl(A)  ∩ Nωcl (B)References [1]   K.T. Atanassov, Intuitionistic fuzzy sets,  Fuzzy sets and Systems, 20(1986), 87-96. [2] C.L. Chang, Fuzzy topological spaces, Journal of Mathematical   Analysis and Applications, 24(1968),182-190. [3]  F.G. Lupianez, On Neutrosophic sets and Topology, Kybernetes, 37(2008), 797-800. [4]  F.G. Lupianez, Interval Neutrosophic sets and Topology, Proceedings of 13th WSEAS. International Conference on Applied Mathematics(MATH’08) Kybernetes,  38(2009),  621-624. [5]  A.A. Salama and  S.A. Alblowi S.A., Neutrosophic set and Neutrosophic Topological Spaces, IOSR-JM., 3(2012), 31-35. [6]  A.A. Salama, F. Samarandache and K. Valeri, Neutrosophic closed set and Neutrosophic continuous functions, Neutrosophic sets and Systems, 4(2014), 4-8. [7]  R. Santhi and D. Jayanthi, Intuitionistic fuzzy semi-pre closed sets, Tripura Math. Soc., 12(2010), 10-20. [8]  F. Smarandache, Neutrosophic set:- A generalization of the Intuitionistic Fuzzy set, Journal of Defense Resources Management. 1(2010),107-116. [9]  P. Sundaram, and M. Sheik John, On ω-closed sets in topology, Acta Ciencia Indica, 4(2000) ,389-392.  [10]  L.A. Zadeh, Fuzzy set,  Inform. and Control, 8(1965), 338-353.    117Received: May 30, 2016.   Accepted: July 06, 2016. 

Nω –CLOSED SETS IN NEUTROSOPHIC TOPOLOGICAL SPACES

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Nω –CLOSED SETS IN NEUTROSOPHIC TOPOLOGICAL SPACES

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