Advancing Neutrosophic Topology through Gamma Generalized Alpha Closed Sets

Neutrosophic Sets and Systems, Vol. 98, 2026 University of New Mexico Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.kesavan Advancing Neutrosophic Topology through Gamma Generalized Alpha Closed Sets B.Kalaiselvi 1, K.Sivakumar 2*, S.Chandrasekar 3,P.Kalarani4 and A.Kesavan5 1 Department of Mathematics , Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai, India, 602105 ,e-mail: [email protected] 2 Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai, India, 602105 , e-mail: sivakumarkaliappan.[email protected] 3 Department of Mathematics, Arignar Anna Government Arts College, Namakkal (DT), Tamil Nadu, India e-mail: [email protected] 4 Department of Mathematics,Tagore Engineering College, Chennai, India, 602105, e-mail: [email protected] 5 Department of Mathematics, Tagore College of Arts and Science,Tamil Nadu, India e-mail: [email protected] * Correbondence: [email protected] Abstract: This paper introduces a novel class of Neutrosophic closed sets called Neutrosophic 𝛾 - generalized 𝞪 -closed sets (Ne.(𝛾G 𝞪 )CS), along with their corresponding open sets (Ne.(𝛾G 𝞪 )OS), within the structure of Neutrosophic Topological Spaces (NTS). The motivation for this study arises from the limitations observed in existing Neutrosophic closed sets such as 𝞪 -closed, semi-closed, and 𝛾 -closed sets, which often lack the flexibility to model hybrid structures involving partial membership and indeterminacy. To address this gap, we define the Ne.(𝛾G 𝞪 )CS using 𝞫-closure operators and α-open supersets, offering a broader framework that unifies and extends several earlier concepts. The proposed sets are systematically analyzed through formal claims, and their behavior is demonstrated using counterexamples to confirm that reverse implications do not generally hold. Additionally, we explore their algebraic properties including union, intersection, and inclusion relationships. A comparative analysis illustrates how these sets generalize previously defined structures while preserving essential topological characteristics. The findings not only contribute to the advancement of Neutrosophic set theory but also offer a solid foundation for further research in uncertainty modeling, generalized topology, and decision-making systems. This work enhances the expressiveness of Neutrosophic topology and opens potential pathways for practical applications in fields requiring nuanced treatment of imprecision. 1. Introduction and Preliminaries In recent decades, the limitations of classical set theory in handling real-world uncertainty have driven the development of more generalized mathematical frameworks. Among these, Smarandache’s Neutrosophic Set theory stands out as a significant advancement . This enables the representation of uncertain, incomplete, inconsistent, and vague information with greater flexibility. Building on this foundation, Neutrosophic Topology emerged as a natural extension of classical topology into the domain of indeterminacy. This new branch was initiated by A.A. Salama [10], who developed the concept of Neutrosophic Topological Spaces (NTS). In these spaces, the classical notions of open and closed sets are redefined to accommodate the presence of indeterminate and inconsistent information, which is especially relevant in areas such as artificial intelligence, decision support systems, and data analysis. Since the introduction of NTS, many researchers have contributed to its advancement by proposing various types of Neutrosophic open and closed sets. These generalized forms have helped to build a more complete understanding of topological structures under uncertain conditions. For example, Arokiarani I. and colleagues [2] proposed the notion of Neutrosophic α-CS, which broadened the traditional concept of closedness in topological spaces by integrating indeterminacy and partial membership. Their work added depth to the exploration of closure operations in generalized topologies. Neutrosophic Sets and Systems, Vol. xx, 20xx 22 Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.Kesavan Similarly, Ishwarya P. et al. [7] studied Neutrosophic Semi-Open Sets, which represent a hybrid category between open and closed sets. This intermediate classification has provided new insights into how Neutrosophic sets behave with respect to topological boundaries, particularly when information is incomplete or partially defined. Despite these advancements, existing classifications may not fully capture the intricate relationships between different types of Neutrosophic sets. To address this gap, the present study introduces a novel class of closed sets, termed Neutrosophic 𝛾-Generalized α -Closed Sets (abbreviated as Ne.(𝛾G 𝞪 )CS, along with their corresponding open sets, known as Neutrosophic γ\gamma-Generalized α\alpha-Open Sets (Ne.(𝛾G 𝞪 )CS). These newly defined set classes are proposed to further refine and generalize the concepts of closure and openness in Neutrosophic Topological Spaces. The key idea is that a set Λ1 in a Neutrosophic topological space (𝜒𝔫𝔢.,Ne.τ) is said to be a Ne.(𝛾G 𝞪 )CS if 𝑁𝑒.bcl(Λ1)  Ω whenever Λ1Ω and Ω is a Neutrosophic 𝞪 - alpha-Open Set in the same space. This framework incorporates both the 𝛾-closure and α\alpha-openness concepts, leading to a more layered and flexible understanding of set boundaries. By doing so, it bridges the gap between multiple earlier notions and offers a unified structure to study more complex topological behaviors under uncertainty. The objective of this research is threefold: • To formally define and introduce the new classes of Ne.(𝛾G 𝞪 )CS closed and open sets; • To analyze and prove their fundamental properties, including behavior under standard set operations like union, intersection, and complement; • To explore their relationships with existing types of Neutrosophic sets, such as N(α\alpha)CS, N(G)CS, and N(GS)CS. By addressing these goals, this paper aims to contribute both theoretical and structural value to the growing domain of Neutrosophic topology. These developments have the potential to enhance future investigations in topology, logic, and their interdisciplinary applications. Moreover, this study lays the groundwork for further exploration into continuity, compactness, and separation axioms using the newly defined set types. It also opens the possibility for practical applications where vague, incomplete, or inconsistent information must be systematically analyzed. In conclusion, the introduction of Neutrosophic 𝛾 −Generalized 𝞪 -Closed and Open Sets represents a significant step forward in the evolution of Neutrosophic topology. It offers refined tools for topological analysis in the presence of indeterminacy and strengthens the theoretical foundation for further research in uncertainty modeling and applied mathematics. 1.1 Motivation for the Study Many real-life problems involve situations where things are not fully true or false, and we face uncertainty or incomplete information. Traditional set theories like classical sets or fuzzy sets cannot properly deal with this kind of uncertainty. To solve this, Neutrosophic Set Theory was introduced, which allows us to separately consider truth, falsity, and indeterminacy. Building on this idea, Neutrosophic topology was developed to study open and closed sets in uncertain environments. Several types of Neutrosophic closed sets already exist, like 𝞪 , 𝛾 and semiCS. But these sets often work separately and don’t give a complete picture when openness and closeness overlap. They are not flexible enough to handle all types of uncertain or mixed cases. This creates a need for a new, more general type of set that can combine and extend the features of the Neutrosophic Sets and Systems, Vol. xx, 20xx 23 Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.Kesavan existing ones. That’s why this paper introduces a new kind of set called the Neutrosophic 𝛾-generalized 𝞪 - closed set, which is designed to be broader and more useful in dealing with complex uncertain situations. 1.2 Research Gap Although several classes of Neutrosophic closed sets have been introduced in recent years—such as Neutrosophic 𝞪 -closed sets, semi-closed sets, pre-closed sets, and 𝛾-closed sets—these concepts are limited in scope. Most of them address specific types of closure behavior and do not offer a unified structure that combines multiple closure and openness properties. As a result, they fall short in representing more complex topological structures that may arise in uncertain systems. Another issue is that the relationships between these different types of Neutrosophic closed sets are not fully explored in the literature. There is a lack of generalized set definitions that can include these existing types as special cases while offering new insights into how they interact or differ. Additionally, many existing models do not account for how sets behave under different closure operations, such as semi-closure or 𝞫-closure, in a combined or comparative manner. Therefore, there is a clear gap in developing a broader class of Neutrosophic closed sets that can generalize and unify various existing structures under a single theoretical framework. This limitation inspires the introduction and investigation of Neutrosophic 𝛾-generalized -generalized 𝞪 -CS in the present study. 1.3 Objective of this study The main aim of this research is to introduce and explore a novel category of closed sets in Neutrosophic topology, referred to as Neutrosophic γ -generalized 𝞪 -CS (Ne.(γG𝞪)CS). This class is introduced to generalize and unify several existing Neutrosophic closed set types, such as α-closed, semi-closed, pre-closed, and 𝛾 -closed sets, under a broader and more inclusive framework. The study aims to establish the foundational properties of Ne.(𝛾G 𝞪 )CS, examine their algebraic behavior, and explore their interactions with other wellknown closed sets. In addition, the paper provides formal proofs and counterexamples to demonstrate that while Ne.(𝛾G 𝞪 )CS include many existing classes as special cases, the reverse inclusions do not hold. Another key objective is to introduce the corresponding open sets, namely Neutrosophic 𝛾 -generalized 𝞪 -open sets, and investigate their characteristics. Through this work, the paper seeks to enrich the structure of Neutrosophic topological spaces and support further theoretical development and practical application in fields that require refined treatment of uncertainty and imprecision. 1.4 Discussion of Existing Problems and Core Contributions The study addresses a key limitation in Neutrosophic topology—namely, the lack of a unified structure that can generalize and relate various existing Neutrosophic closed sets such as 𝞪 -closed, semi-closed, preclosed, and 𝛾-CS. These earlier set types are defined in isolated contexts and are often insufficient for representing the complex interplay between openness and closedness in uncertain systems. They do not capture all types of boundary behaviors, nor do they offer a general framework that allows comparison or inclusion among multiple closure concepts. In response to this limitation, the paper introduces a new and more inclusive class called Neutrosophic 𝛾generalized 𝞪 -closed sets (Ne.(𝛾G 𝞪 )CS), which incorporates 𝞫-closure operations with α-open supersets. This framework not only generalizes several known classes of Neutrosophic closed sets but also establishes their interrelationships through a series of logical claims. The paper rigorously proves that (Ne.(𝛾G 𝞪 )CS) Neutrosophic Sets and Systems, Vol. xx, 20xx 24 Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.Kesavan includes all of these earlier classes as special cases and presents counterexamples to show that the converse is not generally true. This distinction is crucial for deepening the theoretical structure of Neutrosophic topology. The core contributions of the paper are as follows: • Formal definition and development of the new class (Ne.(𝛾G 𝞪 )CS)and its corresponding open set (Ne.(𝛾G 𝞪 )OS). • Establishment of inclusion relationships between (Ne.(𝛾G 𝞪 )CS)and existing Neutrosophic closed sets ( 𝞪 -closed, semi-closed, 𝛾-closed, etc.). • Presentation of multiple claims supported by proofs and counterexamples to clarify boundary conditions. • Analysis of set operations (such as union and intersection) on (Ne.(𝛾G 𝞪 )CS)and their closure properties. • Introduction of generalization theorems showing how (Ne.(𝛾G 𝞪 )CS) can serve as a broader framework for future topological investigations. By resolving the fragmented nature of existing closed set definitions and offering a unified approach, this work significantly enhances the expressive power of Neutrosophic topological structures and provides a solid foundation for further applications and theoretical extensions. 1.5 Proposed Methodology This study adopts a theoretical methodology to define and explore a new class of closed sets in Neutrosophic Topological Spaces (NTS), called Neutrosophic 𝛾-generalized 𝞪 -closed sets (Ne.(𝛾G 𝞪 )CS). The method begins with a review of existing closed set types—such as 𝞪 -closed, semi-closed, pre-closed, and 𝛾closed sets—to highlight the need for a unifying structure. The new class is defined using β-closure and 𝞪 - open sets: a set Λ1 is (Ne.(𝛾G 𝞪 )CS) if its 𝞫 -closure is contained in every 𝞪 -open superset that includes it. Several claims are then established to show that well-known Neutrosophic closed sets are special cases of (Ne.(𝛾G 𝞪 )CS), with counterexamples demonstrating that the converse is not generally true. Illustrative examples clarify the behavior of these sets, including their response to set operations like union and intersection. The study also introduces the corresponding open set class, Neutrosophic 𝛾 -generalized 𝞪 -open sets (Ne.(𝛾G 𝞪 )OS), and explores their properties. Overall, this methodology provides a step-by-step generalization framework that strengthens and extends the theory of Neutrosophic topology. The rationale for selecting a theoretical and axiomatic approach in this study stems from the need to generalize and unify multiple existing classes of Neutrosophic closed sets within a single framework. Traditional Neutrosophic closed sets—such as 𝞪 -closed, semi-closed, pre-closed, and 𝛾-closed sets—are defined independently and lack a shared structure that allows for direct comparison or integration. By employing 𝞫 -losure and α-open set operations, the proposed Neutrosophic 𝛾-generalized 𝞪 - closed sets (Ne.(γG 𝞪 )CS) offer a flexible yet rigorous extension that includes these existing sets as particular cases. This formal method ensures mathematical clarity, enables the derivation of inclusion relations, and allows the formulation of claims with both proofs and counterexamples. The goal was to address the structural gaps in current Neutrosophic topology and to enrich the theoretical landscape for future developments. The selected methodology thus Neutrosophic Sets and Systems, Vol. xx, 20xx 25 Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.Kesavan provides a solid foundation for extending closure-based reasoning under uncertainty and lays the groundwork for potential applications in decision theory, data analysis, and soft computing. 2. Basic Definitions and Preliminaries Definition 2.1 [5,6] Consider a fixed non-empty set NX. A Neutrosophic set V1 ∗ defined on NX can be expressed as V1 ∗= {〈x,μV1 ∗(x),σV1 ∗(x),νV1 ∗(x)〉|x ∈ N x },where μV1 ∗(x):The membership degree is denoted by NX→, and the function νV1 ∗(x):NX→[0,1] specifies the non-membership degree for the Neutrosophic set V1 ∗, whereas σV1 ∗(x), represents the indeterminacy degree. Definition 2.2 [10] A Neutrosophic topology (abbreviated as NT) on the set Nx defined as a collection Nτof Neutrosophic sets within Nx that satisfies the following conditions: 1. The null Neutrosophic set 0N and the universal Neutrosophic set 1N are elements of Nτ 2. The intersection J1∩J2 belongs to Nτfor any two sets J1,J2∈ Nτ 3. For any collection {Ji|i ∈ j}⊆ Nτ. In this situation, the couple (NX,Nτ)(NX,Nτ) is denoted to as a NTS. A NOS is any subclass of NX that fits to Nτ. The counterpart V1 ∗c of a NOS V1 ∗ in the NTS (NX,Nτ) is recognized as a NCS in NX.. Claim 2. 3 [10]. For any NS V1 ∗ in (NX,Nτ), we have 1. Nint(0N)= 0N and Ncl(0N)= 0N 2. (Nint(V1 ∗))c= Ncl(V1 ∗c) 3. (Ncl(V1 ∗))c= Nint(V1 ∗c) 4. Nint(1N)= 1N and Ncl(1N)= 1N Definition 2.4 A NS V1 ∗of a NTS (NX,Nτ) is a 1. A Neutrosophic semi preclosed set (denoted as (N(γ)CS) is well-defined as a set V1 ∗ for which ∃an N(P) closed set V2 ∗ in N(P) Closed set and there exists a Neutrosophic preclosed set V2 ∗ in which contains the neutrosophic interior of V2 ∗ contains V1 ∗. 2. In [15] (N(γ)OS ∃ N(P)OS V2 ∗ such that V1 ∗⊆(V1 ∗)⊆ Ncl(V2 ∗)V1 ∗ Definition 2.5 Let V1 ∗be an NS of a NTS (NX,Nτ). Then 1. Nαcl(V1 ∗)=∩{I|I is a N(α)CS in NX and V1 ∗⊆ I} 2. Nαint(V1 ∗)=∪{I|I is a N(α)OS in NX and I ⊆ V1 ∗} Definition 2.6 Let V1 ∗ be a Neutrosophic set (NS) in the Neutrosophic Topological Space (NTS) (NX,Nτ). Then: V1 ∗ is called a Neutrosophic Generalized Closed Set (abbreviated as N(G)CS if Ncl(V1 ∗)⊆ Ψ whenever V1 ∗⊆ Ψ is a Neutrosophic Open Set (NOS) in NX.. 1. V1 ∗ is called a Neutrosophic Generalized Semi Closed Set (abbreviated as N(GS)CS if Ncl(V1 ∗)⊆ Ψ where Ψ is a NOS in NX. Neutrosophic Sets and Systems, Vol. xx, 20xx 26 Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.Kesavan 2. V1 ∗is called an Alpha-Neutrosophic Generalized Closed Set (abbreviated as (N(α)GCS if Ncl(V1 ∗)⊆ Ψ, and Ψ is a NOS in NX.. 3. V1 ∗ is called a Neutrosophic Generalized Alpha Closed Set (abbreviated as (N(α)GCS if Ncl(V1 ∗)⊆ Ψ, and Ψ is a Neutrosophic Alpha Open Set (NαOS) in NX Remark 2.7 Let V1 ∗ be a NS in (NX,Nτ). Then 1. NS−cl(V1 ∗)= V1 ∗∩Nint(Ncl(V1 ∗)) 2. NS−int(V1 ∗)= V1 ∗∪Ncl(Nint(V1 ∗)) If V1 ∗ is a NS of NX then NScl(V1 ∗c)= (NScl(V1 ∗))c 3. (𝐍𝐞.( 𝛄𝐆𝛂)𝐂𝐒)- Neutrosophic γ generalized α - CS Definition 3.1 A Neutrosophic set Λ1 in the Neutrosophic Topological Space (𝜒𝔫𝔢.,Ne.τ) is called a Neutrosophic (Ne.( γGα)CS) if 𝑁𝑒.bcl(Λ1)  Ω wheneverΛ1Ωand Ω is a Ne.(α)OS in (𝜒𝔫𝔢.,Ne.τ) in the space Ne.TS (𝜒𝔫𝔢.,Ne.τ). . Example 3.2: Let 𝜒𝔫𝔢. ={s1 ∗,s2 ∗}, K1 ∗=〈x,(5 10,5 10,5 10),(5 10,5 10,5 10)〉,and K2 ∗=〈x,(4 10,5 10,6 10),(3 10,5 10,7 10)〉. Then 𝜏𝔫𝔢. = {0N,K1 ∗,K2 ∗,1N} is a 𝑁𝑒.T on 𝜒𝔫𝔢.. Here Λ1 =〈x,(3 10,5 10,7 10),(2 10,5 10,8 10)〉 stands an 𝑁𝑒.𝑠 in (𝜒𝔫𝔢., 𝑁𝑒.τ). Claim 3.3: In the space (𝜒𝔫𝔢.,𝑁𝑒.τ) every 𝑁𝑒.CS is also a𝑁𝑒.( γGα)CS but the converse does not generally hold. Proof: Assume Λ1 is a 𝑁𝑒. Closed set in 𝜒𝔫𝔢. suppose Λ1 Ω where Ω is a 𝑁𝑒.(𝛼) openset in 𝜒𝔫𝔢.. As given that 𝑁𝑒.𝑏𝑐𝑙(Λ1) 𝑁𝑒.cl(Λ1)= Λ1 Ω is follows that Ne.bcl(Λ1)  Ω. Then Λ1 is in the space (𝜒𝔫𝔢.) with the neutrosophic topology 𝑁𝑒.τ and is a a 𝑁𝑒.(𝑏𝐺𝛼) Illustration 3.4: Let χ𝔫𝔢. ={s1 ∗,s2 ∗}, K1 ∗=〈x,(5 10,5 10,5 10),(5 10,5 10,5 10)〉 and K2 ∗=〈x,(4 10,5 10,6 10),(3 10,5 10,7 10)〉. Then τ𝔫𝔢. = {0N,K1 ∗,K2 ∗,1N} is a Ne.T on 𝜒𝔫𝔢.. Here Λ1 =〈x,(3 10,5 10,7 10),(2 10,5 10,8 10)〉 is a neutrosophic topology is in (𝜒𝔫𝔢. , Ne.τ), is a neutrosophic topology ( γGα) closed set nonetheless non Ne.closed in (𝜒𝔫𝔢. , Ne. τ) as Ne.cl(Λ1)= K1 ∗C ≠ Λ1. Claim 3.5: In the space (𝜒𝔫𝔢., 𝑁𝑒.τ) every 𝑁𝑒.(𝑆)𝐶𝑆 is also 𝑁𝑒.(𝑏𝐺𝛼)𝐶𝑆 but the converse does not generally hold. Proof: Neutrosophic Sets and Systems, Vol. xx, 20xx 27 Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.Kesavan Let Λ1 be a Ne.SCS in 𝜒𝔫𝔢. . Let Λ1 Ω and Ω is a Ne.(α)OS in 𝜒𝔫𝔢. . As Ne.(γ)cl(Λ1) Ne.(S)cl(Λ1)= Λ1 Ω by hypothesis, we have Ne.(γ)cl(Λ1) Ω. Then Λ1 is a Ne.( γGα)CS in (𝜒𝔫𝔢., Nτ). Illustration 3.6: Let χ𝔫𝔢. ={s1 ∗,s2 ∗}, K1 ∗=〈x,(5 10,5 10,5 10),(5 10,5 10,5 10)〉 and K2 ∗=〈x,(4 10,5 10,6 10),(3 10,5 10,7 10)〉. Then τ𝔫𝔢. = {0N,K1 ∗,K2 ∗,1N} is a Ne.T on 𝜒𝔫𝔢.. Here Λ1 =〈x,(3 10,5 10,7 10),(2 10,5 10,8 10)〉 is a Ne.s in (𝜒𝔫𝔢., Ne.τ), stands a neutrosophi ( γGα) closed set nevertheless non a Ne.(S) closed set in ( 𝜒𝔫𝔢. , Nτ) as 𝑁𝑒.𝑖𝑛𝑡(𝑁𝑒.𝑐𝑙(Λ1))= 𝑁𝑒.int(K1 ∗C)= K1 ∗ ⊈ Λ1. Claim 3.7 Every Neutrosophic Ne.(𝑃) Closed Set in the space(𝑋, 𝑁𝑒.τ) is also a Neutrosophic N( γGα) Closed Set, In general, however, the converse is not necessarily true. Proof: Let Λ1 is a Ne.(P)CS in 𝜒𝔫𝔢.. Let Λ1 Ω and Ω is a Ne.(α)OS in 𝜒𝔫𝔢.. As Ne.(γ)cl(Λ1) Ne.(P)cl(Λ1) =Λ1 Ω by hypothesis, we have Ne.(γ)cl(Λ1) Ω. Then Λ1 is a Ne.( γGα)CS in (𝜒𝔫𝔢., Nτ). Illustration 3.8: Let χ𝔫𝔢. ={s1 ∗,s2 ∗}, K1 ∗=〈x,(5 10,5 10,5 10),(6 10,5 10,4 10)〉 and K2 ∗=〈x,(4 10,5 10,6 10),(3 10,5 10,7 10)〉. Then τ𝔫𝔢. = {0N,K1 ∗,K2 ∗,1N} is a Ne.T on 𝜒𝔫𝔢.. Here Λ1 =〈x,(3 10,5 10,7 10),(2 10,5 10,8 10)〉 is a Ne.s in (𝜒𝔫𝔢., Ne.τ), stands a Ne.(𝑏𝐺𝛼)𝐶𝑆 nevertheless not anNe.(𝑃)𝐶𝑆 in (𝜒𝔫𝔢., Nτ) as 𝑁𝑒.𝑐𝑙(𝑁𝑒.𝑖𝑛𝑡(Λ1)) = 𝑁𝑒.𝑐𝑙(K2 ∗) = K1 ∗C ⊈ 𝛬1. Claim 3.9: Every Neutrosophic 𝛼 Closed Set in the space (𝜒𝔫𝔢., 𝑁𝑒.τ) is also a Neutrosophic 𝑁𝑒.(𝑏𝐺𝛼) Closed Set, but the converse is not true in general. Proof: Let Λ1 is a Ne.(α)CS in 𝜒𝔫𝔢.. Let Λ1Ω and Ω is a Ne.(α)OS in 𝜒𝔫𝔢.. As Ne.(γ)cl(Λ1)  Ne.(α)cl(Λ1) = Λ1Ω by hypothesis, we have Ne.bcl(Λ1)  Ω. Therefore, in (𝜒𝔫𝔢., Ne.τ)., Λ1 is a Ne.( γGα)CS. Illustration 3.10: Let χ𝔫𝔢. ={s1 ∗,s2 ∗}, K1 ∗=〈x,(5 10,5 10,5 10),(5 10,5 10,5 10)〉 and K2 ∗=〈x,(4 10,5 10,6 10),(3 10,5 10,7 10)〉. Then τ𝔫𝔢. = {0N,K1 ∗,K2 ∗,1N} is a Ne.T on 𝜒𝔫𝔢.. Here Λ1 =〈x,(3 10,5 10,7 10),(2 10,5 10,8 10)〉 is a Ne.s in (𝜒𝔫𝔢., Ne.τ), stands a Neutrosophic𝑁𝑒.(𝑏𝐺𝛼 closed set nevertheless non an 𝑁𝑒.(𝛼)𝐶𝑆 in (𝜒𝔫𝔢., 𝑁𝑒.τ) as 𝑁𝑒.𝑐𝑙(𝑁𝑒.𝑖𝑛𝑡(𝑁𝑒.𝑐𝑙(𝛬1))) = 𝑁𝑒.𝑐𝑙(𝑁𝑒.𝑖𝑛𝑡(K1 ∗C)) = 𝑁𝑒.cl(K1 ∗) = K1 ∗C ⊈ 𝛬1. Claim 3.11: Neutrosophic Sets and Systems, Vol. xx, 20xx 28 Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.Kesavan Every Neutrosophic γ− Closed Set in the space (𝜒𝔫𝔢., Ne.τ) is also a Neutrosophic Ne.( γGα) Closed Set, but the converse does not hold in general Proof: Let Λ1 Neutrosophic γ− Closed Set in the space (𝜒𝔫𝔢.). Let Λ1Ω and Ω is aNeutrosophic− α open set in 𝜒𝔫𝔢.. while Ne.(γ)cl(Λ1) Ne. (γ)cl(Λ1) = Λ1  Ω by hypothesis, here consume Ne.(γ)cl(Λ1)  Ω. Therefore, in (𝜒𝔫𝔢., Ne.τ)., Λ1 is a Neutrosophic ( γGα) closed set. Illustration 3.12: Let χ𝔫𝔢. ={s1 ∗,s2 ∗}, K1 ∗=〈x,(5 10,5 10,5 10),(3 10,5 10,7 10)〉 and K2 ∗=〈x,(4 10,5 10,6 10),(3 10,5 10,7 10)〉. Then τ𝔫𝔢. = {0N,K1 ∗,K2 ∗,1N} is a Ne.T on 𝜒𝔫𝔢.. Here Λ1 =〈x,(4 10,5 10,4 10),(6 10,5 10,4 10)〉 is a Ne.s in (𝜒𝔫𝔢., Ne.τ), stands a Neutrosophic Ne.( γGα) closed set but non an Ne.(b) closed set in (𝜒𝔫𝔢. , Ne.τ) as Ne.int(Ne.cl(Λ1))∩ Ne.cl(Ne.int(Λ1)) =K1 ∗∩K1 ∗C = K1 ∗⊈ Λ1 . Claim 3.13: Every Neutrosophic 𝑁𝑒.(𝑅) Closed Set in the space in (𝜒𝔫𝔢., 𝑁𝑒.τ) is also a Neutrosophic 𝑁𝑒.(𝑏𝐺𝛼) Closed Set, but the converse is not generally true. Proof: Let Λ1 is a Ne.(R)CS in 𝜒𝔫𝔢.. Since every Ne.(R)CS is a Ne.CS., Λ1 is a Ne.CS. Therefore by claim 2.3, Λ1 is a Ne.( γGα)CS in (𝜒𝔫𝔢., Nτ). Illustration 3.14: Let χ𝔫𝔢. ={s1 ∗,s2 ∗}, K1 ∗=〈x,(5 10,5 10,5 10),(6 10,5 10,4 10)〉 and K2 ∗=〈x,(4 10,5 10,6 10),(3 10,5 10,7 10)〉. Then τ𝔫𝔢. = {0N,K1 ∗,K2 ∗,1N} is a Ne. T on 𝜒𝔫𝔢. .Here Λ1 =〈x,(4 10,5 10,6 10),(3 10,5 10,7 10)〉 is a Ne. ( γGα)CS but not an Ne.(R)CS in (𝜒𝔫𝔢., Ne.τ) as Ne.cl(Ne.int(Λ1)) = Ne.cl(K2 ∗) = K1 ∗C ≠ Λ1. Claim 3.15: Every Neutrosophic 𝑁𝑒.(γ)Closed Set in the space (𝜒𝔫𝔢., 𝑁𝑒.τ) is also a Neutrosophic 𝑁𝑒.(𝑏𝐺𝛼) Closed Set; however, the converse does not necessarily hold. Proof: Let Λ1 be a Ne.(γ)CS in 𝜒𝔫𝔢.. Let Λ1  Ω and Ω is a Ne.(α)OS in 𝜒𝔫𝔢.. As Ne.bcl(Λ1)  Ne.(γ)cl(Λ1) = Λ1  Ω by hypothesis, we have Ne.(γ)cl(Λ1)  Ω. Therefore, in (𝜒𝔫𝔢., Ne.τ)., Λ1 is a Ne.( γGα)CS. Illustration 3.16: Neutrosophic Sets and Systems, Vol. xx, 20xx 29 Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.Kesavan Let χ𝔫𝔢. ={s1 ∗,s2 ∗}, K1 ∗=〈x,(5 10,5 10,5 10),(3 10,5 10,7 10)〉 and K2 ∗=〈x,(4 10,5 10,6 10),(3 10,5 10,7 10)〉. Then τ𝔫𝔢. = {0N,K1 ∗,K2 ∗,1N} is a Ne. T on 𝜒𝔫𝔢. . Here Λ1 =〈x,(4 10,5 10,4 10),(6 10,5 10,4 10)〉 is a Ne.( γGα)CS but not a Ne.(γ)CS in (𝜒𝔫𝔢., Ne.τ), as we could not find any Ne.(P)CS Λ2 such that Ne.int(Λ2)  Λ1  Λ2 in 𝜒𝔫𝔢.. Claim 3.17: Every Neutrosophic 𝑁𝑒.(𝑏) Closed Set in the space (𝜒𝔫𝔢., 𝑁𝑒.τ) is also a Neutrosophic 𝑁𝑒.(𝑏𝐺𝛼) Closed Set, but the converse is not true in general. Proof: Let Λ1 is a 𝑁𝑒.(𝑏)𝐶𝑆 in 𝜒𝔫𝔢.. Let Λ1  Ω and Ω is a 𝑁𝑒.(𝛼)𝑂𝑆 in 𝜒𝔫𝔢.. Now 𝑁𝑒.(𝛾)cl(Λ1)=Λ1Ω, by hypothesis. Therefore we have 𝑁𝑒. (𝑏)𝑐𝑙(Λ1)Ω. Hence Λ1 is a 𝑁𝑒.(𝑏𝐺𝛼)𝐶𝑆 in (𝜒𝔫𝔢., 𝑁𝑒.τ). Illustration 3.18: Let 𝜒𝔫𝔢. ={s1 ∗,s2 ∗}, K1 ∗=〈x,(5 10,5 10,3 10),(5 10,5 10,7 10)〉, and K2 ∗=〈x,(4 10,5 10,6 10),(3 10,5 10,7 10)〉. Then 𝜏𝔫𝔢. = {0N,K1 ∗,K2 ∗,1N} is a 𝑁𝑒. T on 𝜒𝔫𝔢. . Here Λ1 =〈x,(4 10,5 10,4 10),(6 10,5 10,4 10)〉 is a 𝑁𝑒.(𝑏𝐺𝛼)𝐶𝑆 but not an 𝑁𝑒.(𝑏)𝐶𝑆 in (𝜒𝔫𝔢., 𝑁𝑒.τ) as 𝑁𝑒.𝑖𝑛𝑡(𝑁𝑒.𝑐𝑙(𝑁𝑒.𝑖𝑛𝑡(Λ1))) = 𝑁𝑒.int(𝑁𝑒.𝑐𝑙(K2 ∗)) = 𝑁𝑒.int(K1 ∗C ) = K1 ∗⊈ Λ1 Remark 3.19: In general, the union of two Neutrosophic 𝑁𝑒.(𝑏𝐺𝛼) Closed Sets in the space (𝜒𝔫𝔢., 𝑁𝑒.τ) is not necessarily a Neutrosophic 𝑁𝑒.(𝑏𝐺𝛼)𝐶𝑆 Closed Set, as demonstrated in the following example. Illustration 3.20: Let 𝜒𝔫𝔢. ={s1 ∗,s2 ∗}, K1 ∗=〈x,(5 10,5 10,5 10),(6 10,5 10,4 10)〉,K2 ∗=〈x,(2 10,5 10,8 10),(3 10,5 10,7 10)〉and K3 ∗= 〈x,(6 10,5 10,4 10),(7 10,5 10,3 10)〉. Then 𝜏𝔫𝔢. ={0N,K1 ∗,K2 ∗,K3 ∗,1N} is a 𝑁𝑒. T on 𝜒𝔫𝔢. . Here Λ1 = 〈x,(1 10,5 10,9 10),(5 10,5 10,5 10)〉, Λ2 =〈x,(5 10,5 10,5 10),(2 10,5 10,8 10)〉, are 𝑁𝑒.(𝑏𝐺𝛼)𝐶𝑆𝑠 in ( 𝜒𝔫𝔢. , 𝑁𝑒. τ). But Λ1∪ Λ2 is not an 𝑁𝑒.(𝑏𝐺𝛼)𝐶𝑆 as Λ1∪ Λ2 = 〈x,(5 10,5 10,5 10),(5 10,5 10,5 10)〉⊆ K1 ∗ but 𝑁𝑒.(𝑏𝑐𝑙(Λ1∪Λ2) =〈x,(6 10,5 10,4 10),(7 10,5 10,3 10)〉 ⊈ K1 ∗. Remark 3.21: The intersection of any two 𝑁𝑒.(𝑏𝐺𝛼)𝐶𝑆𝑠 is not an 𝑁𝑒.(𝑏𝐺𝛼)𝐶𝑆 in general as seen in the following example. Illustration 3.22: Let 𝜒𝔫𝔢. ={s1 ∗,s2 ∗}, K1 ∗=〈x,(5 10,5 10,5 10),(6 10,5 10,4 10)〉,K2 ∗=〈x,(2 10,5 10,8 10),(3 10,5 10,7 10)〉and K3 ∗= 〈x,(6 10,5 10,4 10),(7 10,5 10,3 10)〉. Then 𝜏𝔫𝔢. ={0N,K1 ∗,K2 ∗,K3 ∗,1N} is a 𝑁𝑒. T on 𝜒𝔫𝔢. .Here Λ1 = Neutrosophic Sets and Systems, Vol. xx, 20xx 36 Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.Kesavan Proof: (𝑖) → (𝑖𝑖) Let Λ1 is a 𝑁𝑒.( γGα)OS in 𝜒𝔫𝔢.. Then since 𝜒𝔫𝔢. is a 𝑁𝑒.𝑏𝑔𝛼𝑏𝑇1/2bace, Λ1 is a 𝑁𝑒.(γ)OS in 𝜒𝔫𝔢.. Therefore Λ1𝑁𝑒.cl(𝑁𝑒.int(𝑁𝑒.cl(Λ1))). (𝑖𝑖) → (𝑖𝑖𝑖) Let Λ1𝑁𝑒. cl( 𝑁𝑒. int( 𝑁𝑒. cl( Λ1))). Then 𝑁𝑒. cl( Λ1)𝑁𝑒. cl( 𝑁𝑒. cl( 𝑁𝑒. int( 𝑁𝑒. cl( Λ1)))) = 𝑁𝑒.cl(𝑁𝑒.int(𝑁𝑒.cl(Λ1)))𝑁𝑒.cl(Λ1).Therefore 𝑁𝑒.cl(Λ1) = 𝑁𝑒.cl(𝑁𝑒.int(𝑁𝑒.cl(Λ1))). Hence 𝑁𝑒.cl(Λ1) 𝑁𝑒.RC(𝜒𝔫𝔢.). (𝑖𝑖𝑖) → (𝑖) Since cl(Λ1) is a 𝑁𝑒.(R)CS in 𝜒𝔫𝔢., 𝑁𝑒.cl(Λ1)=𝑁𝑒.cl(𝑁𝑒.int(𝑁𝑒.cl(Λ1))) and since Λ1  𝑁𝑒.cl(Λ1), Λ1  𝑁𝑒.cl(𝑁𝑒.int(𝑁𝑒.cl(Λ1))). Therefore Λ1 is a N(γ)OS. Hence Λ1 is a 𝑁𝑒.( γGα)OS in 𝜒𝔫𝔢. Claim 5.13: Let (𝜒𝔫𝔢., 𝑁𝑒.τ) is a 𝑁𝑒. γGαbT1/2 space, then the following conditions are equivalent: (i) Λ1 is a 𝑁𝑒.( γGα)CS in 𝜒𝔫𝔢., (ii) 𝑁𝑒.int(𝑁𝑒.cl(𝑁𝑒.int(Λ1)))  Λ1, (iii) 𝑁𝑒.int(Λ1) ∈ 𝑁𝑒.RO(𝜒𝔫𝔢.). Proof: This claim can be easily proved by taking complement in claim 4.16 6. Limitations of the Study While the study successfully introduces and generalizes the concept of Neutrosophic 𝜸 -generalized α - closed sets, it is not without limitations. Firstly, the research is entirely theoretical and lacks practical applications or real-world data validation. The examples used are limited to small, finite Neutrosophic spaces, which may not reflect the behavior of these sets in large or complex topological systems. Secondly, no algorithmic or computational methods are developed to detect or implement these sets in applied settings. Thirdly, the study does not address the dynamic behavior of these sets under changes in the underlying topological space. Lastly, potential applications in decision-making, data analysis, or artificial intelligence are not explored, leaving the practical relevance of the proposed sets for future investigation. 7. Future Work The proposed class of Neutrosophic 𝜸 -generalized α-closed sets (Ne.(γG 𝞪 )CS opens multiple avenues for future investigation. One notable avenue is the creation of computational algorithms to detect and analyze (Ne.(γG 𝞪 )CS in large Neutrosophic topological spaces, making the concept applicable to practical decisionmaking and uncertainty modeling.. Future work may also investigate dynamic Neutrosophic systems where the topology evolves over time, requiring adaptive closure properties. In addition, exploring the application of (Ne.(γG 𝞪 )CS) in fields such as digital topology, image processing, data clustering, and granular computing could provide real-world relevance. Another direction involves studying dual concepts like Neutrosophic 𝜸 - generalized α-interior sets and their topological implications. Overall, the foundational structure developed in this study paves the way for further theoretical expansion and interdisciplinary applications in systems that involve incomplete, imprecise, or inconsistent information. Neutrosophic Sets and Systems, Vol. xx, 20xx 37 Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.Kesavan The comparative analysis table.1 evaluates the proposed Neutrosophic 𝛾 -generalized 𝞪 -closed sets (Ne.( 𝛾 G 𝞪 )CS) alongside traditional Neutrosophic closed set types—namely 𝞪 -closed, semi-closed, pre-closed, and 𝛾 -closed sets. Each class is compared based on criteria such as openness foundation, closure operator used, scope of generalization, and inclusion relationships. Traditional set types depend on specific types of open sets ( 𝞪 , semi, pre, 𝛾) and corresponding closures, often with narrow generalization and limited structural relationships. In contrast, (Ne.(γG 𝞪 )CS) utilizes 𝞫 -closure and α-open sets, providing a unified and more flexible framework. The table confirms that (Ne.(γG 𝞪 )CS) includes all traditional types as special cases, while none of the others offer similar inclusiveness. Reverse implications do not generally hold for (Ne.(γG 𝞪 )CS), which is supported through counterexamples in the paper. The proposed class also demonstrates improved behavior under operations like union and intersection, which is often not preserved in other types. Furthermore, it better captures uncertainty and hybrid behavior due to its broader formulation. This enhanced expressiveness makes (Ne.(γG 𝞪 )CS) more applicable to advanced modeling in uncertain topological environments. The comparison validates the generality, strength, and necessity of the proposed class within Neutrosophic topology. Table 1: Comparison between the proposed Neutrosophic γ-generalized α-closed sets and traditional Neutrosophic closed set types Feature 𝞪 - Closed Sets SemiClosed Sets Pre-Closed Sets 𝜸 - Closed Sets Proposed Ne.( 𝜸 G 𝞪 )CS Openness Basis 𝞪 -open sets Semi-open sets Pre-open sets 𝛾 -open sets 𝞪 -open sets Closure Type Used 𝞪 -closure or identity Semiclosure Pre-closure 𝛾 -closure 𝞫 -closure (broader) Defined via Inclusion via 𝞪 -open set Superset’s semi-open relation Pre-open neighborhood inclusion 𝛾 -open neighborhood containment 𝞫 -closure inclusion inside α-open sets Scope of Generalization Narrow Moderate Moderate Broader than 𝞪 Broadest – generalizes all Inclusion of Other Sets Does not include others Does not include others Does not include others Partial inclusion of 𝞪 and semi Includes α, semi, pre, and γ as special cases Reverse Implication May hold in special cases Not always true Often fails Rarely holds Proven false via counterexamples Support for Hybrid Behavior Limited Limited Limited Partial High – designed for uncertain overlap Behavior under Union/Intersection Not always closed Not preserved Not preserved Sometimes preserved Analyzed in claims; flexible Neutrosophic Sets and Systems, Vol. xx, 20xx 38 Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.Kesavan Expressiveness under Uncertainty Low Moderate Moderate Moderate High – handles mixed/indeterminate membership Application Readiness Theoretical Theoretical Theoretical Theoretical Theoretical; open for future applications 8. Conclusion In this articles, introduced and examined a new class of sets in Neutrosophic topology, namely Neutrosophic 𝛾 -generalized α-closed sets (𝛾 GS-closed sets) and their counterparts, Neutrosophic 𝛾 - generalized 𝞪 -open sets (𝛾 GS-open sets). These sets represent a meaningful generalization of existing Neutrosophic closed and open set concepts, enriching the structural framework of Neutrosophic topological spaces. We have discussed several foundational properties of these sets and explored their relationships with previously established classes of Neutrosophic sets, highlighting their uniqueness and broader applicability. The results obtained in this work not only contribute to the theoretical development of Neutrosophic topology but also pave the way for further generalizations and refinements. Future research could focus on extending these sets under different topological operators, examining their behavior in product spaces, and exploring their role in Neutrosophic continuity, compactness, and separation axioms. Additionally, potential applications in fields dealing with uncertainty, such as decision-making, data analysis, and artificial intelligence, can be explored by leveraging the flexible nature of 𝛾 GS-closed and 𝛾 GS-open sets. This work thus lays a solid foundation for advancing both theoretical investigations and practical applications within the broader domain of Neutrosophic mathematics. Funding No financial or external support from individuals or organizations was received for this study. Acknowledgments Sincere gratitude is extended to all who offered support, motivation, and valuable insights throughout the progression of this research. Appreciation is also due to the readers for their interest and to the authors of the referenced works, whose contributions laid the groundwork for this study. Thanks are given to the individuals and organizations that provided the necessary facilities and resources for the successful execution and dissemination of this paper. Lastly, acknowledgment is given to everyone who contributed to this project in any capacity. Data Availability The present work is wholly theoretical, with no inclusion of data gathering or analytical evaluation. Prospective researchers are invited to conduct empirical research to further investigate and substantiate the ideas outlined in this paper. Neutrosophic Sets and Systems, Vol. xx, 20xx 39 Neutrosophic 𝜸 generalized α closed sets and its Properties.B.Kalaiselvi ,K.Sivakumar,P.Kalarani, S.Chandrasekarand A.Kesavan Ethical Approval Ethical approval is not required, as this purely theoretical research involves no animal subjects or human participants. 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