Neutrosophic πgγ* closed sets in Neutrosophic Topological Spaces

Neutrosophic Sets and Systems, Vol. 98, 2026 University of New Mexico Sakthivel K and Gomathi A, Neutrosophic πgγ* closed sets in Neutrosophic Topological Spaces Neutrosophic πgγ* closed sets in Neutrosophic Topological Spaces Sakthivel K 1 and Gomathi A 2, * 1 Assistant professor, Department of Mathematics, Government Arts College, Udumalpet, Tamil Nadu, India; [email protected] * Assistant professor, Department of Science and Humanities, Hindusthan College of Engineering and Technology, Coimbatore, Tamil Nadu, India; [email protected] Abstract: This paper introduces and investigate the notion of Neutrosophic πgγ* closed sets within the framework of Neutrosophic topological spaces. The study begins defining Neutrosophic πgγ* closed sets and proceeds to investigate their fundamental properties and characterizations. Special attention is given to examining how these sets interrelate with and extend existing classes of Neutrosophic closed sets. By establishing inclusion relations and comparative hierarchies, the paper highlights the significance of Neutrosophic πgγ* closed sets in broadening the structural understanding of Neutrosophic topologies. Furthermore, several illustrative examples are provided to demonstrate the distinctive features of these sets and to clarify their role in the generalization process. The paper also derives various theorems that reveal the interplay between Neutrosophic πgγ* closed sets and other Neutrosophic closed families, thereby enriching the theoretical landscape of Neutrosophic topology. These results contribute to ongoing developments in generalized closed set theory, offering new insights and paths for further research in Neutrosophic mathematics. Keywords: Neutrosophic topological spaces, Neutrosophic open sets, Neutrosophic closed sets, Neutrosophic πgγ* closed sets. 1. Introduction The concept of fuzzy sets, introduced by Zadeh [13] in 1965, allow each element to have a degree of membership. This concept was expanded by K. Atanassov [1] in 1986 with the introduction of Intuitionistic Fuzzy sets, which assign both a degree of membership and a degree of non-membership to each element. Sakthivel K and Manikandan M [10] had studied the concept of πgγ* closed Sets in Intuitionistic Fuzzy Topological Spaces. Florentin Smarandache [2] introduced Neutrosophic Sets as a further generalization, which adds more flexibility. Later, Salama A A and Alblowi S A [11] extended the idea by developing Neutrosophic Topological Spaces. In this article we define Neutrosophic πgγ* closed sets in Neutrosophic topological spaces and investigate their properties. 2. Preliminaries Neutrosophic Sets and Systems, Vol. 98, 2026 58 Sakthivel K and Gomathi A, Neutrosophic πgγ* closed sets in Neutrosophic Topological Spaces This section reviews essential definitions, operations, and key results related to Neutrosophic sets. Definition 2.1 [6] Let 𝑋 be a non-empty fixed set. A Neutrosophic Set (NS) A is an object having the form A={〈𝑥,𝜇A(𝑥),𝜎A(𝑥),𝑣A(𝑥) 〉:𝑥 ∈ 𝑋} where 𝜇A(𝑥),𝜎A(𝑥)and 𝑣A(𝑥) represent the degree of membership, degree of indeterminacy and the degree of non-membership respectively of each element x ∈ X the set A. Definition 2.2 [6] Let X be a non-empty set and let A be a Neutrosophic Set A = {〈𝑥,𝜇A(𝑥),𝜎A(𝑥),𝑣A(𝑥) 〉:𝑥 ∈ 𝑋}.Then the complement of A is AC = {〈𝑥,𝑣A(𝑥),1 − 𝜎A(𝑥), 𝜇A(𝑥)〉 ∶ 𝑥 ∈ 𝑋 } Definition 2.3 [6] Let A and B be two Neutrosophic Sets, ∀ x ∈ X A = {〈𝑥,𝜇A(𝑥),𝜎A(𝑥),𝑣A(𝑥) 〉:𝑥 ∈ 𝑋} B = {〈𝑥,𝜇B(𝑥),𝜎B(𝑥),𝑣B(𝑥) 〉:𝑥 ∈ 𝑋}. Then A ⊆ B ⇔ {⟨𝑥,𝜇A(𝑥) ≤ 𝜇𝐵(𝑥), 𝜎𝐴(𝑥)≤ 𝜎𝐵(𝑥) ,𝑣A(𝑥)≥ 𝑣B(𝑥)⟩ ∶ 𝑥 ∈ 𝑋} Definition 2.4 [6] Let X be a non-empty set and let A and B be two Neutrosophic Sets are A={〈𝑥,𝜇A(𝑥),𝜎A(𝑥), 𝑣A(𝑥) 〉:𝑥 ∈ 𝑋}, B = {〈𝑥,𝜇B(𝑥),𝜎B(𝑥),𝑣B(𝑥) 〉:𝑥 ∈ 𝑋}. Then 1. A ∩ B = {〈𝑥,𝜇A(𝑥)∧ 𝜇𝐵(𝑥),𝜎𝐴(𝑥)∧ 𝜎𝐵(𝑥),𝑣A(𝑥)∨ 𝑣B(𝑥)〉 ∶ 𝑥 ∈ 𝑋} 2. A ∪ B = {〈𝑥,𝜇A(𝑥)∨ 𝜇𝐵(𝑥),𝜎𝐴(𝑥)∨ 𝜎𝐵(𝑋),𝑣A(𝑥)∧ 𝑣𝐵(𝑥)〉 ∶ 𝑥 ∈ 𝑋} Definition 2.5 [6] Let X be a non-empty set and τN be the collection of Neutrosophic subsets of X satisfying the following properties: 1. 0N, 1N ∈ τN 2. T1 ∩ T2 ∈ τN for any T1, T2 ∈ τN 3. ∪ Ti ∈ τN for every {Ti: i ∈ j} ⊆ τN Then the space (X, τN) is called a Neutrosophic Topological Space (NTS). The elements of τN are called Neutrosophic Open Set (NOS) and its complement is Neutrosophic Closed Set (NCS). Definition 2.6 [6] Let (X, 𝜏N) be a NTS and A={〈𝑥,𝜇A(𝑥),𝜎A(𝑥),𝑣A(𝑥) 〉:𝑥 ∈ 𝑋} be a NS in 𝑋. Then Neutrosophic closure of A is N_Cl(A) = ∩ { M : M is a NCS in X and A ⊆ 𝑀} Neutrosophic interior of A is N_Int(A) = ∪ { H : H is a NOS in 𝑋 and H ⊆A} Definition 2.7 Let (X,𝜏N) be a NTS and A={〈𝑥,𝜇A(𝑥),𝜎A(𝑥),𝑣A(𝑥) 〉:𝑥 ∈ 𝑋} be a NS in 𝑋. Then A is said to be • Neutrosophic Semi closed set [4] (NSCS) if N_Int (N_Cl (A)) ⊆A. • Neutrosophic Semi open set [4] (NSOS) if A ⊆ N_Cl (N_Int(A)). • Neutrosophic Pre closed set [12] (NPCS) if N_Cl(N_Int(A)) ⊆A. • Neutrosophic Pre open set [12] (NPOS) if A⊆ N_Int(N_Cl(A)). • Neutrosophic Regular closed set [7] (NRCS) if A= N_Cl(N_IntA)). • Neutrosophic Regular open set [7] (NROS if A = N_Int(N_Cl(A)). • Neutrosophic α closed set [5] (NαCS) if N_Cl (N_Int (N_Cl (A))) ⊆A. • Neutrosophic α open set [5] (NαOS) if A ⊆ N_Int (N_Cl (N_Int (A))). • Neutrosophic β closed set [8] (NβCS) if N_Int (N_Cl (N_Int (A))) ⊆A. • Neutrosophic β open set [8] (NβOS in short) if A⊆ N_Cl (N_Int (N_Cl (A))). • Neutrosophic b closed set [6] (NbCS) if N_Cl (N_Int(A)) Ⴖ N_Int (N_Cl (A)) ⊆ A Neutrosophic Sets and Systems, Vol. 98, 2026 59 Sakthivel K and Gomathi A, Neutrosophic πgγ* closed sets in Neutrosophic Topological Spaces • Neutrosophic 𝑏 open set [6] (NbOS) if A ⊆ N_Int (N_Cl (A)) ꓴ N_Cl (N_Int(A)) • Neutrosophic 𝜋 closed set [9] if A is the union of Neutrosophic Regular closed sets. • Neutrosophic 𝜋 open set [9] if A is the union of Neutrosophic Regular open sets. Definition 2.8 Let (X,𝜏N) be a NTS and A={〈𝑥,𝜇A(𝑥),𝜎A(𝑥),𝑣A(𝑥) 〉:𝑥 ∈ 𝑋} be a NS in 𝑋. Then A is said to be • Neutrosophic Generalized closed set [3] (NGCS) if N_Cl(A)⊆ Ʋ whenever A ⊆ Ʋ and Ʋ is a NOS in 𝑋. • Neutrosophic Generalized semi closed set [3] (NGSCS) if N_sCl(A) ⊆ Ʋ whenever A ⊆ Ʋ andƲ is a NOS in 𝑋. • Neutrosophic α Generalized closed set [5] (NαGCS) if N_αCl(A) ⊆ Ʋ whenever A ⊆ Ʋ and Ʋ is a NOS in 𝑋 Remark 2.9 For any Neutrosophic Set A, • N_Cl (A) C = (N_Int (A)) C • N_Int (A)C = (N_Cl (A)) C • N_sCl (A) C = (N_sInt (A)) C • N_sInt (A)C = (N_sCl (A)) C • N_sCl (A) = A ∪ N_Int (N_Cl (A)) • N_sInt (A) = A ∩ N_Cl (N_Int (A)) • N_αCl(A) = A ∪ N_Cl (N_Int (N_Cl A))) • N_𝛼Int(A) = A ∩ N_Int (N_Cl (N_Int (A))) Remark 2.10 1. Each NOS is NPOS in NTS. 2. Each NRCS is NCS in NTS. 3. Each N 𝜋OS is NOS in NTS. 3. Neutrosophic πgγ* closed sets in Neutrosophic Topological Spaces We have introduced Neutrosophic πgγ* closed sets and explored some of their properties. Definition 3.1. A Neutrosophic Set A in (X, τN) is said to be a Neutrosophic πgγ* closed sets (Nπgγ*CS in short) if N_Cl (N_Int(A)) ∩ N_Int (N_Cl (A)) ⊆ Ʋ whenever A⊆ Ʋ and Ʋ is NπOS in (X, τN). Example 3.2. Let X = {p, q} with τN = {0N, B, 1N} be a NTS on X, where B = <x, (0.4,0.5,0.7), (0.4,0.4,0.6)>. Let us consider the NS, A = <x, (0.3,0.2,0.7), (0.3,0.2,0.8)>. Here NπOS is Ʋ = {1N, B}. Clearly A ⊆ Ʋ . Now N_Cl (N_Int(A)) ∩ N_Int (N_Cl (A)) = 0N ⊆ Ʋ . Therefore NS, A is a Nπgγ*CS in (X, τN). Theorem 3.3. Every NCS in (X, τN) is a N πgγ*CS (X, τN) but not conversely in general. Proof: Consider ‘A’ is a NCS in (X, τN). Assume that A⊆ Ʋ and Ʋ is a N𝜋OS in (X, τN). Since A is a NCS in (X, τN), N_Cl(A)= A . Now N_Cl (N_Int(A)) ∩ N_Int (N_Cl (A)) = N_Int(A) ∩ N_Int(A) = N_Int(A) ⊆ A ⊆ Ʋ. Thus N_Cl (N_Int(A)) ∩ N_Int(N_Cl (A)) ⊆ Ʋ , whenever A⊆ Ʋ and Ʋ is N𝜋OS in (X, τN) Therefore, A is a N πgγ*CS in (X, τN). However, the reverse implication is not true. Neutrosophic Sets and Systems, Vol. 98, 2026 60 Sakthivel K and Gomathi A, Neutrosophic πgγ* closed sets in Neutrosophic Topological Spaces Example 3.4. Let 𝑋 = {p, q} with 𝜏𝑁 = { 0𝑁 ,𝐵,1𝑁} be a NTS on 𝑋 , where B = <x, (0.2,0.4,0.6), (0.3,0.5,0.7)>. Let us consider the NS, A = <x, (0.1,0.3,0.8), (0.2,0.4,0.7)>. Here N𝜋OS is Ʋ = {1𝑁 ,𝐵}. Clearly A ⊆ Ʋ. Now N_Cl (N_Int(A)) ∩ N_Int (N_Cl (A)) = 0N ⊆ Ʋ . But N_Cl(A) = Bc ⊈ 𝐴. Therefore NS, A is a N πgγ*CS but not NCS in (X, τN). Theorem 3.5. Every NSCS in (X, τN) is a Nπgγ*CS (X, τN) but not conversely in general. Proof: Consider A is a NSCS in (X, τN). Suppose that A ⊆ Ʋ and Ʋ is a N𝜋OS in (X, τN). Since A is a NSCS in (𝑋,𝜏𝑁), N_Int (N_Cl (A)) ⊆ A. Now N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) ⊆ N_Cl (N_Int (A)) Ⴖ A ⊆ N_Cl (A) ∩ A = A ⊆ Ʋ. Thus N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) ⊆ Ʋ , whenever A⊆ Ʋ and Ʋ is N𝜋OS in (X, τN). Therefore, A is a Nπgγ*CS in (X, τN). However, the reverse implication is not true. Example 3.6. Let 𝑋= {p, q} with 𝜏𝑁 = { 0𝑁,B, 1𝑁} be a NTS on 𝑋 , where B = <x, (0.2,0.4,0.6), (0.3,0.5,0.7)>. Let us consider the NS, A = <x, (0.1,0.3,0.8), (0.2,0.4,0.7)> . Here N𝜋OS is Ʋ = {1𝑁 ,𝐵}. Clearly A ⊆ Ʋ . Now N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) = 0N ⊆ Ʋ . But N_Int (N_Cl (A)) = B ⊈ 𝐴. Therefore NS, A is a Nπgγ*CS but not NSCS in (X, τN). Theorem 3.7. Every NPCS in (X, τN) is a Nπgγ*CS (X, τN) but not conversely in general. Proof: Consider A is a NPCS in (X, τN). Suppose that A ⊆ Ʋ and Ʋ is a N𝜋OS in (X, τN). Since A is a NPCS in (X, τN), N_Cl (N_Int (A)) ⊆ A. Now N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) ⊆ A ∩ N_Int (N_Cl (A)) ⊆ A ∩ N_Cl (A) = A ⊆ Ʋ. Thus N_Cl (N_Int(A)) ∩ N_Int (N_Cl (A)) ⊆ Ʋ , whenever A ⊆ Ʋ and Ʋ is N𝜋OS in (X, τN). Therefore, A is a Nπgγ*CS in (X, τN). However, the reverse implication is not true. Example 3.8. Let 𝑋= { p, q } with 𝜏𝑁 = { 0𝑁, B1, B2, 1𝑁} be a NTS on 𝑋 , where B1 = <x, (0.5,0.5,0.5), (0.6,0.5,0.4)> and B2 = <x, (0.4,0.5,0.6), (0.3,0.5,0.7)>. Let us consider the NS, A = <x, (0.4,0.5,0.6), (0.4,0.5,0.6)> . Here N𝜋OS is Ʋ = {1𝑁,𝐵1,𝐵2}. Now N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) = B2 ⊆ Ʋ . But N_Cl (N_Int (A) = B1C ⊈ 𝐴. Therefore NS, A is a Nπgγ*CS but not NPCS in (X, τN). Theorem 3.9. Every NRCS in (X, τN) is a Nπgγ*CS in (X, τN) but not conversely in general. Proof: Let A be a NRCS in (X, τN). Since every NRCS is a NCS, by Theorem 3.3 A is a Nπgγ*CS in (X, τN). However, the reverse implication is not true. Example 3.10. Let 𝑋= { p, q } with 𝜏𝑁 ={ 0𝑁,B, 1𝑁} be a NTS on 𝑋 , where B= <x, (0.5,0.5,0.5), (0.4,0.5,0.6)>. Let us consider the NS, A=<x, (0.5,0.5,0.5), (0.5,0.5,0.4)>. Here N𝜋OS is Ʋ = {1𝑁 ,𝐵}. Clearly A ⊆ Ʋ. Now N_Cl (N_Int (A))∩N_Int (N_Cl (A)) = B ⊆ Ʋ . But N_Cl (N_Int (A) = 𝐵𝐶≠ 𝐴 .Therefore NS, A is a N𝜋𝑔𝛾∗CS but not NRCS in (X, τN). Theorem 3.11. Every NROS in (𝑋,𝜏𝑁) is a Nπgγ*CS in (𝑋,𝜏𝑁) but not conversely in general. Proof: Let A be a NROS in (X, τN). A is a Nπgγ*CS in (X, τN). Then N_Int(N_Cl(A)) = A and N_Int(A) = A. Suppose that A ⊆ Ʋ and Ʋ is a N𝜋OS in (X, τN). Now N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) ⊆ N_Cl (A) ∩ A = A ⊆ Ʋ . Thus N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A))⊆Ʋ , whenever A⊆ Ʋ and Ʋ is N𝜋OS in (X, τN). Therefore, A is a Nπgγ*CS in (X, τN). However, the reverse implication is not true. Example 3.12. Let 𝑋= { p, q } with 𝜏𝑁 = { 0𝑁,B, 1𝑁} be a NTS on 𝑋 , where B = <x, (0.5,0.5,0.5), (0.4,0.5,0.6)>. Let us consider the NS, A = <x, (0.4,0.5,0.6), (0.4,0.5,0.6)>. Here Neutrosophic Sets and Systems, Vol. 98, 2026 61 Sakthivel K and Gomathi A, Neutrosophic πgγ* closed sets in Neutrosophic Topological Spaces N𝜋OS is Ʋ = {1𝑁 ,𝐵}. Clearly A ⊆ Ʋ. Now N_Cl (N_Int(A)) ∩ N_Int(N_Cl (A)) = 0N ⊆ Ʋ . But N_Int (N_Cl (A)) = B ≠ A. Therefore NS, A is a Nπgγ*CS but not NROS in (X, τN). Theorem 3.13. Every NαCS in (X, τN) is a Nπgγ*CS (X, τN) but not conversely in general. Proof: Consider A is a NαCS in (X, τN). Suppose that A⊆ Ʋ and Ʋ is a N𝜋OS in (X, τN). Since A is a NαCS in (X, τN), N_Cl (N_Int (N_Cl (A))) ⊆ A. Now N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) ⊆ N_Cl (N_Int (N_Cl (A))) ∩ N_Cl (N_Int (N_Cl (A))) ⊆ A ∩ A = A ⊆ Ʋ . Thus N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) ⊆ Ʋ , whenever A ⊆ Ʋ and Ʋ is N𝜋OS in (X, τN). Therefore, A is a Nπgγ*CS in (X, τN). However, the reverse implication is not true. Example 3.14. Let 𝑋= { p, q } with 𝜏𝑁 ={ 0𝑁, B, 1𝑁} be a NTS on 𝑋 , where B = <x, (0.5,0.5,0.5), (0.4,0.5,0.6)>. Let us consider the NS, A = <x, (0.4,0.5,0.4), (0.5,0.5,0.6)>. Here N𝜋OS is Ʋ = {1𝑁 ,𝐵}. Clearly A ⊆ Ʋ. Now N_Cl (N_Int(A)) ∩ N_Int (N_Cl(A)) = 0N ⊆ Ʋ . But N_Cl (N_Int (N_Cl (A))) = 𝐵𝐶⊈ 𝐴. Therefore NS, A is a Nπgγ*CS but not NαCS in (X, τN). Theorem 3.15. Every NbCS in (X, τN) is a Nπgγ*CS (X, τN), but not conversely in general. Proof: Consider A is a NbCS in (X, τN). Suppose that A ⊆ Ʋ and Ʋ is a N𝜋OS in (X, τN). Since A is a NbCS in (X, τN), N_Cl (N_Int(A)) ∩ N_Int (N_Cl (A)) ⊆ A. Now N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) ⊆ A ⊆ Ʋ. Thus N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) ⊆ Ʋ, whenever A ⊆ Ʋ and Ʋ is N𝜋OS in (X, τN). Therefore, A is a Nπgγ*CS in (X, τN). However, the reverse implication is not true. Example 3.16. Let 𝑋={ p , q } with 𝜏𝑁 ={ 0𝑁, B1, B2, 1𝑁} be a NTS on 𝑋 , where B1 = <x, (0.5,0.5,0.5), (0.6,0.5,0.4)> and B2 = <x, (0.4,0.5,0.6), (0.3,0.5,0.7)>. Let us consider the NS, A =<x, (0.4,0.5,0.6), (0.6,0.5,0.4)> . Here N𝜋OS is Ʋ = {1𝑁,𝐵1,𝐵2}. Now N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) ⊆ Ʋ whenever A ⊆ Ʋ . But N_Cl (N_Int(A)) ∩ N_Int (N_Cl(A)) = B1C ⊈ 𝐴. Therefore NS, A is a Nπgγ*CS but not NbCS in (X, τN). Theorem 3.17. Every NGCS in (X, τN) is a Nπgγ*CS in (X, τN) but not conversely. Proof: Consider A is a NGCS in (X, τN). Assume that A⊆ Ʋ and Ʋ is a N𝜋OS in (X, τN). Since A is a NGCS in (X, τN), N_Cl(A)⊆ Ʋ . Now N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) ⊆ N_Cl (A) ⊆ Ʋ .Thus N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) ⊆ Ʋ , whenever A ⊆ Ʋ and Ʋ is N𝜋OS in (X, τN). Therefore, A is a Nπgγ*CS in (X, τN). However, the reverse implication is not true. Example 3.18. Let 𝑋={ p, q } with 𝜏𝑁 ={ 0𝑁, B, 1𝑁} be a NTS on 𝑋 , where B = <x, (0.3,0.5,0.4), (0.2,0.5,0.3)>. Consider NS, A = <x, (0.2,0.4,0.6), (0.2,0.4,0.4)>. Here N𝜋OS Ʋ = {1𝑁 ,𝐵}. Clearly A ⊆ Ʋ. Now N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) = 0N ⊆ Ʋ. But N_Cl(A) = 𝐵𝐶 ⊈ 𝐵, whenever A ⊆ 𝐵 . Therefore NS, A is a Nπgγ*CS but not NGCS in (X, τN). Theorem 3.19. Every NGSCS in (X, τN) is a Nπgγ*CS in (X, τN) but not conversely. Proof: Consider ‘A’ is a NGSCS in (X, τN). Assume that A⊆ Ʋ and Ʋ is a N𝜋OS in (X, τN). Since A is a NGSCS in (X, τN), N_sCl(A) = A ∪ N_Int(N_Cl(A)) ⊆ Ʋ .This implies N_Int (N_Cl(A) ⊆ Ʋ. Now N_Cl (N_Int(A)) ∩ N_Int (N_Cl(A)) ⊆ N_Cl (N_Int(A)) ∩ Ʋ ⊆ Ʋ .Thus N_Cl (N_Int(A)) ∩ N_Int (N_Cl(A)) ⊆ Ʋ , whenever A⊆ Ʋ and Ʋ is N𝜋OS in (X, τN) Therefore, A is a Nπgγ*CS in (X, τN). However, the reverse implication is not true. Example 3.20. Let 𝑋 = { p , q } with τN ={ 0𝑁, B1, B2, 1𝑁} be a NTS on 𝑋 , where B1 = <x, (0.5,0.5,0.5), (0.3,0.5,0.7)> and B2 = <x, (0.4,0.5,0.6), (0.3,0.5,0.7)>. Let us consider the NS, A = <x, (0.3,0.5,0.4), (0.2,0.5,0.8)>. Here N𝜋OS is Ʋ = {1𝑁, 𝐵1, 𝐵2}. Now N_Cl (N_Int(A)) ∩ N_Int Neutrosophic Sets and Systems, Vol. 98, 2026 62 Sakthivel K and Gomathi A, Neutrosophic πgγ* closed sets in Neutrosophic Topological Spaces (N_Cl(A)) = 0N ⊆ Ʋ whenever A ⊆ Ʋ . But N_sCl (A) = A ꓴ N_Int (N_Cl (A)) = A ꓴ B1 = B1 ⊈ B and A ⊆ B2. Therefore NS, A is a Nπgγ*CS but not NGSCS in (X, τN). Theorem 3.21. Every NαGCS in (X, τN) is a Nπgγ*CS (X, τN) but not conversely in general. Proof: Consider A is a NαGCS in (X, τN). Suppose that A ⊆ Ʋ and Ʋ is a N𝜋OS in (X, τN). Since A is a NαGCS in (𝑋,𝜏𝑁), N_αCl(A)⊆ Ʋ. Therefore A ∪ N_Cl (N_Int (N_Cl (A)))⊆ Ʋ , so N_Cl (N_Int (N_Cl A)))⊆ Ʋ and N_Int (N_Cl A)) ⊆ Ʋ. Now N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A))⊆ N_Cl (N_Int (A)) ∩ Ʋ ⊆ Ʋ. Thus N_Cl (N_Int (A)) ∩ N_Int (N_Cl (A)) ⊆ Ʋ, whenever A ⊆ Ʋ and Ʋ is N𝜋OS in (X, τN) . Therefore, A is a Nπgγ*CS in (X, τN). However, the reverse implication is not true. Example 3.22. In Example 3.20, the NS A =<x, (0.3,0.5,0.7), (0.2,0.5,0.8)> is Nπgγ*CS in (X, τN) but not a NαGCS in (X, τN) as N_αCl(A) = A ∪ N_Cl (N_Int (N_Cl (A)))= A ꓴ B1C = B1C ⊈ 𝐵1 ,𝐵2 and A ⊆ 𝐵1 ,𝐵2 . In the following figure (a) we have provided relation between Nπgγ*CS and other closed sets in Neutrosophic topological space. Remark 3.23. The union of two Nπgγ*CSs need not be a Nπgγ*CS in (X, τN) in general. Example 3.24. Let 𝑋 = {p , q} with τN = { 0𝑁 ,B1, B2, 1𝑁} be a NTS on 𝑋, where B1 = <x, (0.4,0.5,0.6), (0.2,0.5,0.8) > and B2 = <x, (0.4,0.5,0.5), (0.4,0.5,0.5)>. Here A = <x, (0.4,0.5,0.5), (0.5,0.5,0.4) > and B = <x, (0.5,0.5,0.4), (0.2,0.5,0.6) > are Nπgγ*CS in (X, τN), but A ꓴ B = <x, (0.5,0.5,0.4), (0.5,0.5,0.4)> is not a Nπgγ*CS in (X, τN). Theorem 3.25. Let (X, τN) be a NTS. Then for every A ∈ Nπgγ*CS in (X, τN) and for every B ∈ NS in (X, τN), A ⊆ B ⊆ N_Cl (N_Int(A)) which implies B ∈ Nπgγ*CS in (X, τN). Proof: Let B ⊆ Ʋ and Ʋ be an NπOS in X. Since A ⊆ B, A ⊆ Ʋ. Also, B ⊆ N_Cl (N_Int(A)) which implies N_Cl (N_Int(B)) ⊆ N_Cl (N_Int(A)). Now N_Int (N_Cl (B)) ⊆ N_Int (N_Cl (A)). Therefore N_Cl (N_Int(B)) Ⴖ N_Int (N_Cl (B)) ⊆ N_Cl (N_Int(A)) Ⴖ N_Int (N_Cl (A)) ⊆ Ʋ, by hypothesis. Hence B ∈ Nπgγ*CS in (X, τN). Theorem 3.26. If A is both NπOS and Nπgγ*CS in (X, τN) then A is a NbCS in (X, τN). Neutrosophic Sets and Systems, Vol. 98, 2026 63 Sakthivel K and Gomathi A, Neutrosophic πgγ* closed sets in Neutrosophic Topological Spaces Proof: Let A be a NπOS and a Nπgγ*CS in (X, τN). Then N_Cl (N_Int(A)) Ⴖ N_Int (N_Cl (A)) ⊆A, as A ⊆ A, by hypothesis. Therefore N_Cl (N_Int(A)) Ⴖ N_Int (N_Cl (A)) ⊆ A. Hence A is a NbCS in (X, τN). Theorem 3.27. If A is both a NπOS and a Nπgγ*CS in (X, τN) then A is a NβCS in (X, τN). Proof: Let A be a NπOS and a Nπgγ*CS in (X, τN). Then N_Cl (N_Int(A)) Ⴖ N_Int (N_Cl (A)) ⊆ A, as A ⊆ A, by hypothesis. Now N_Int (N_Cl (N_Int(A))) = N_Int (N_Cl (N_Int(A))) Ⴖ N_Cl (N_Int(A)) ⊆ N_Int (N_Cl (A)) Ⴖ N_Cl (N_Int(A)) = N_Cl (N_Int(A)) Ⴖ N_Int (N_Cl (A)) ⊆ A. Therefore N_Int (N_Cl (N_Int(A))) ⊆ A . Hence A is a NβCS in (X, τN). Theorem 3.28. If A is both a NπOS and a Nπgγ*CS in (X, τN) then A is a NSCS in (X, τN). Proof: Let A be a NπOS and a Nπgγ*CS in (X, τN). That is A is a NOS in (X, τN). Then N_Cl (N_Int(A)) Ⴖ N_Int (N_Cl (A)) ⊆ A, as A ⊆ A, by hypothesis. Clearly N_Int (N_Cl (A)) = N_Cl (A) Ⴖ N_Int (N_Cl (A)) = N_Cl (N_Int(A)) Ⴖ N_Int (N_Cl (A)) ⊆ A. Therefore N_Int (N_Cl (A)) ⊆ A. Hence A is a NSCS in (X, τN). Theorem 3.29. If A is both a NπOS and a Nπgγ*CS in (X, τN), then A is a NROS in (X, τN). Proof: Let A be a NπOS and a Nπgγ*CS in (X, τN). That is A is a NOS in (X, τN). Then N_Int (N_Cl (A)) = N_Int (N_Cl (A)) Ⴖ N_Cl (A) = N_Int (N_Cl (A)) Ⴖ N_Cl (N_Int(A)) ⊆ A. 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