EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6339 ISSN 1307-5543 – ejpam.com Published by New York Business Global An Approach to Closure and Kernel Defined Through N-Points José Sanabria1,∗, Ennis Rosas2, Carlos Granados3 1 Departamento de Matemáticas, Facultad de Educación y Ciencias, Universidad de Sucre, Sincelejo, Colombia 2 Departamento de Ciencias Naturales y Exactas, Universidad de la Costa, Barranquilla, Colombia 3 Escuela de Ciencias de la Educación, Universidad Nacional Abierta y a Distancia, Barranquilla, Colombia Abstract. In this paper, we introduce novel definitions of closure and kernel of a neutrosophic set (abbreviated as N -set) based on the newly formulated concept of a N -point. Building upon these foundational ideas, we develop and analyze the key properties of these new notions, which naturally lead to the construction of two previously unstudied N -topologies. These contributions offer fresh insights into the topological behavior of neutrosophic structures. We believe that the proposed framework not only deepens the theoretical understanding of N -sets but also opens new avenues for their application in various fields involving uncertainty, imprecision, and indeterminacy. 2020 Mathematics Subject Classifications: 03E72, 54A40. Key Words and Phrases: Neutrosophic set, neutrosophic point, neutrosophic topological space 1. Introduction Neutrosophic sets (briefly N -sets) were introduced by F. Smarandache in 2010 [1] as a generalization of classical and intuitionistic fuzzy sets. Unlike their predecessors, N - sets are characterized by the presence of three independent membership functions: truth (T ), indeterminacy (I), and falsity (F ), each of which varies independently within the unit interval [0, 1]. This allows N -sets to effectively model uncertainty, inconsistency, and incompleteness in a way that classical and intuitionistic frameworks cannot, making them particularly suitable for complex real-world applications in decision-making, artificial intelligence, and information systems. Today, researchers have contributed significantly to neutrosophic theory, driving the development of new concepts such as neutrosophic ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6339 Email addresses: jesanabri@gmail.com (J. Sanabria), ennisrafael@gmail.com (E. Rosas), carlosgranadosortiz@outlook.es (C. Granados) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 2 of 15 convex structures [2] and Pythagorean neutrosophic closure [3], among others. For the above, neutrosophic set theory represents a field of research in constant and rapid growth. Following the introduction of N -sets, researchers began exploring their topological aspects. In particular, Salama and Alblowi [4] introduced the concept of N -topological spaces, laying the groundwork for a new branch of neutrosophic topology. This devel- opment has stimulated a growing body of research aiming to extend classical topological concepts such as open and closed sets, continuity, and convergence into the neutrosophic setting. Subsequent contributions by Ray [5], Subasree and Basari [6, 7], among others, have enriched this field by proposing new forms of neutrosophic continuity, compactness, separation axioms, and bases for neutrosophic topologies (briefly N -topologies). Recently, in 2024, Açikgöz and Esenbel [8] introduced the concept of neutrosophic preclosure and neutrosophic strong semiclosure to study new classes of open sets and explore the prop- erties of modifications of key topological notions such as connectivity and continuity in neutrosophic topological spaces (briefly N -topological spaces). In this year, Tyagi and Ku- mar Gupta [9] studied the neutrosophic λ-closed sets, generalizing closed and pre-closed neutrosophic sets within N -topological spaces, along with relevant concepts and its proper- ties. Despite this progress, certain fundamental constructs in topology, such as the closure and kernel operators, have not been thoroughly studied in the context of N -topological spaces. Closure and kernel play pivotal roles in classical and fuzzy topology by character- izing the boundaries and interior structures of sets, as well as influencing the properties of continuity, connectedness, and compactness. Their proper generalization to N -sets is therefore essential for a deeper understanding of neutrosophic topological behavior. In this paper, we introduce novel definitions of closure and kernel for N -sets, employing the concept of a N -point as the central building block. We then systematically explore the fundamental properties of these new notions and demonstrate how they lead to the construction of two previously unexamined N -topologies. In comparison with most of the studies carried out in N -topological spaces, where variations of the closure and kernel operators have been defined by means of global tools, our study incorporates the novelty of defining these operators by means of a local tool such as the notion of N -point. With this, we intend to fill the gap in the existing literature and bring new perspectives to the theory of N -topological spaces. Our results not only enrich the mathematical framework of N -sets, but also offer potential for future applications in areas where uncertainty and indeterminacy play a central role. 2. Neutrosophic sets Throughout this paper, let X be a nonempty set, called universe of discourse. Definition 1. [1] A neutrosophic set (briefly N -set) N on X is an object of the form N = {⟨x, µN (x), σN (x), γN (x)⟩ : x ∈ X} , where µN , σN , γN are functions from X to [0, 1]. We denote by N (X) the collection of all N -sets over X. J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 3 of 15 Definition 2. [10] For N,M ∈ N (X) we define the following: (1) (Inclusion) N is called a neutrosophic subset of M , denoted by N ⊑ M , if µN (x) ≤ µM (x), σN (x) ≥ σM (x) and γN (x) ≥ γM (x) for all x ∈ X. Also, we can say that M is a neutrosophic super set of N . (2) (Equality) N is called neutrosophic equal to M , denoted by N = M , if N ⊑ M and M ⊑ N . (3) (Universal set) N is called the neutrosophic universal set, denoted by X̃, if µN (x) = 1, σN (x) = 0 and γN (x) = 0 for all x ∈ X. (4) (Empty set) N is called neutrosophic empty set, denoted by ∅̃, if µN (x) = 0, σN (x) = 1 and γN (x) = 1 for all x ∈ X. (5) (Intersection) The neutrosophic intersection of N andM , denoted by N⊓M , is defined as N ⊓M = {(x, µN (x) ∧ µM (x), σN (x) ∨ σM (x), γN (x) ∨ γM (x)⟩ : x ∈ X} . (6) (Union) The neutrosophic union of N and M , denoted by N ⊔M , is defined as N ⊔M = {⟨x, µN (x) ∨ µM (x), σN (x) ∧ σM (x), γN (x) ∧ γM (x)⟩ : x ∈ X} . (7) (Complement) The neutrosophic complement of N , denoted by N c, is defined as N c = {⟨x, γN (x), 1− σN (x), µN (x)⟩ : x ∈ X} . Proposition 1. [10] If N,M ∈ N (X), then we have the following properties: (1) N ⊓N = N and N ⊔N = N . (2) N ⊓M =M ⊓N and N ⊔M =M ⊔N . (3) N ⊓ ∅̃ = ∅̃ and N ⊓ X̃ = N . (4) N ⊔ ∅̃ = N and N ⊔ X̃ = X̃. (5) N ⊓ (M ⊓O) = (N ⊓M) ⊓O and N ⊔ (M ⊔O) = (N ⊔M) ⊔O. (6) (N c)c = N . The union and intersection operations given in Definition 2 can be extended as follows. Definition 3. [4] For {Nj : j ∈ J} ⊆ N (X) we define the following operations: J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 4 of 15 (1) (Arbitrary intersection) The arbitrary neutrosophic intersection of the collection {Nj : j ∈ J}, denoted by l j∈J Nj , is defined as l j∈J Nj = {〈 x, inf j∈J µNj (x), sup j∈J σNj (x), sup j∈J γNj (x) 〉 : x ∈ X } . (2) (Arbitrary union) The arbitrary neutrosophic union of the collection {Nj : j ∈ J}, denoted by ⊔ j∈J Nj , is defined as ⊔ j∈J Nj = {〈 x, sup j∈J µNj (x), inf j∈J σNj (x), inf j∈J γNj (x) 〉 : x ∈ X } . Proposition 2. [10] If {Nj : j ∈ J} ⊆ N (X) and M ∈ N (X), then we have the following properties: (1) M ⊓ ⊔ j∈J Nj  = ⊔ j∈J (M ⊓Nj). (2) M ⊔ l j∈J Nj  = l j∈J (M ⊔Nj). (3) l j∈J Nj c = ⊔ j∈J N c j . (4) ⊔ j∈J Nj c = l j∈J N c j . Definition 4. [10] A neutrosophic topology (briefly N -topology) on a set X is a collection τ ⊆ N (X) which satisfies the following conditions: (1) ∅̃ and X̃ are in τ . (2) The intersection of two N -sets belonging to τ is in τ . (3) The union of any collection of N -sets belonging to τ is in τ . A set X for which a N -topology τ has been defined is called a N -topological space and is denoted as a pair (X, τ). If N ∈ τ , then N is called a N -open set and if N c ∈ τ , then N is called a N -closed set. We denote by τ c the collection of all N -closed sets in the N -topological space (X, τ). J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 5 of 15 Proposition 3. [10] Let (X, τ) be a N -topological space. Then, the following conditions hold: (1) ∅̃ and X̃ are in τ c. (2) The union of two N -sets belonging to τ c is in τ c. (3) The intersection of any collection of N -sets belonging to τ c is in τ c. Definition 5. [10] Let (X, τ) be a N -topological space and N ∈ N (X). The N -closure of N , denoted by Cl(N), is defined as Cl(N) = l {F ∈ N (X) : N ⊑ F and F ∈ τ c} . Proposition 4. [10] Let (X, τ) be a N -topological space and N,M ∈ N (X). Then, the following conditions hold: (1) N ⊑ Cl(N). (2) Cl(Cl(N)) = Cl(N). (3) Cl(N ⊔M) = Cl(N) ⊔ Cl(M). (4) Cl(∅̃) = ∅̃. (5) Cl(X̃) = X̃. (6) If N ⊑M , then Cl(N) ⊑ Cl(M). (7) Cl(N ⊓M) ⊑ Cl(N) ⊓ Cl(M). (8) N ∈ τ c if and only if N = Cl(N). Definition 6. [5] A N -set M = {⟨x, µM (x), σM (x), γM (x)⟩ : x ∈ X} is called a N -point if for any element y ∈ X, µM (y) = a, σM (y) = b, γM (y) = c for y = x and µM (y) = 0, σM (y) = 1, γM (y) = 1 for y ̸= x, where a ∈ (0, 1] and b, c ∈ [0, 1). In this case, the N -point M is denoted by Mx a,b,c or simply by xa,b,c. Also, x is called the support of the N -point xa,b,c. The N -point x1,0,0 is called a N -crisp point. Definition 7. [5] Let N ∈ N (X). A N -point xa,b,c is said to belong to N , denoted by xa,b,c ∈ N , if µN (x) ≥ a, σN (x) ≤ b and γN (x) ≤ c. Remark 1. It is important to note that ∅̃ is not the only N -set that does not have points belonging to it. For example, if X = {x, y}, then N = {⟨x, 0, 0.5, 1⟩, ⟨y, 0, 0.4, 1⟩} is a N -set over X for which there are not N -points belonging to it. Lemma 1. [5] Let N,M ∈ N (X). Then, we have: (1) N = ⊔ {xa,b,c : xa,b,c ∈ N}. J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 6 of 15 (2) If xa,b,c ∈ N and N ⊑M , then xa,b,c ∈M . Definition 8. [11] A N -ideal on a set X is a nonempty collection L ⊆ N (X), which satisfies the following conditions: (1) N ∈ L and M ⊑ N imply that M ∈ L. (Hereditary property) (2) N,M ∈ L imply that N ⊔M ∈ L. (Finite additivity property) Given a N -topological space (X, τ), a N -ideal L on X and N ∈ N (X), the N -local function [12] of N , denoted by N⋆(L, τ), is defined as N⋆(L, τ) = ⊔ {xa,b,c ∈ N (X) : U ⊓N /∈ L for every U ∈ τ(xa,b,c)}, where τ(xa,b,c) = {U ∈ τ : xa,b,c ∈ U}. We will denote N⋆(L, τ) by N⋆ or N⋆(L). Theorem 1. [12] Let (X, τ) be a N -topological space with two N -ideals L, L′ on X. If N,M ∈ N (X), then the following properties hold: (1) If N ⊑M , then N⋆ ⊑M⋆. (2) If L ⊆ L′, then N⋆(L′) ⊑ N⋆(L). (3) N⋆ = Cl(N⋆) ⊑ Cl(N) (N⋆ is a N -closed set). (4) (N⋆)⋆ ⊑ N⋆. (5) (N ⊔M)⋆ = N⋆ ⊔M⋆. (6) (N ⊓M)⋆ ⊑ N⋆ ⊓M⋆. (7) If M ∈ L, then (N ⊔M)⋆ = N⋆. 3. N-point-closure The concept of N -closure introduced by Karatas and Kuru [10] has inspired recent research in the neutrosophic environment, which has extended the related theory of neu- trosophic topological spaces to other contexts, some of which can be found in references [8] and [3]. Motivated by these recent advances, in this section, we introduce and investigate the concept of N -point-closure using N -points, which is independent of the concept of N -closure, as we show in two examples below. First, we establish new results related to the concepts of N -points and N -closure. Proposition 5. Let N,M ∈ N (X). If N ⊓M = ∅̃, then M ⊑ N c and N ⊑M c. Proof. We will only show the neutrosophic inclusion M ⊑ N c, because the other neutrosophic inclusion is shown in the same way. Assume that N ⊓M = ∅̃. Then, {⟨x, µN (x) ∧ µM (x), σN (x) ∨ σM (x), γN (x) ∨ γM (x)⟩ : x ∈ X} = ∅̃, J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 7 of 15 so the following equalities hold: (1) µN (x) ∧ µM (x) = 0, ∀x ∈ X. (2) σN (x) ∨ σM (x) = 1, ∀x ∈ X. (3) γN (x) ∨ γM (x) = 1, ∀x ∈ X. From equality (1), we have γN (x) = 1− µN (x) ≥ µM (x), ∀x ∈ X. From equality (2), let us observe that one of the quantities σN (x) or σM (x) is equal to 1 for an arbitrary x ∈ X. Since 0 ≤ σN (x) ≤ 1 and 0 ≤ σM (x) ≤ 1, ∀x ∈ X, it follows that σN (x) + σM (x) ≥ 1, ∀x ∈ X. Thus, σM (x) ≥ 1− σN (x), ∀x ∈ X. Using a similar reasoning from equality (3), we show that γM (x) ≥ 1− γN (x) = µN (x), ∀x ∈ X. Therefore, M = {⟨x, µM (x), σM (x), γM (x)⟩ : x ∈ X} ⊑ {⟨x, γN (x), 1− σN (x), µN (x)⟩ : x ∈ X} = N c. In the following example, we show that the converse of Proposition 5, in general, is not true. Example 1. Let X = {x, y} and O,U ∈ N (X) such that N = {⟨x, 0.4, 0.8, 0.6⟩, ⟨y, 0.6, 0.7, 0.4⟩}, M = {⟨x, 0.3, 0.3, 0.7⟩, ⟨y, 0.4, 0.3, 0.6⟩}. Then, M ⊑ N c = {⟨x, 0.6, 0.2, 0.4⟩, ⟨y, 0.4, 0.3, 0.6⟩}, but N ⊓M = {⟨x, 0.3, 0.8, 0.7⟩, ⟨y, 0.4, 0.7, 0.6⟩} ≠ ∅̃. Remark 2. In Example 1, we have N ⊔N c = {⟨x, 0.6, 0.2, 0.4⟩, ⟨y, 0.6, 0.3, 0.4⟩} ≠ X̃ and N ⊓ N c = {⟨x, 0.4, 0.8, 0.6⟩, ⟨y, 0.4, 0.7, 0.6⟩} ≠ ∅̃. Thus, we deduce that the equalities N ⊔N c = X̃ and N ⊓N c = ∅̃ are not satisfied, in general. Proposition 6. Let N,M ∈ N (X). Then, the following properties are equivalent: (1) N ⊑M . (2) xa,b,c ∈ N implies that xa,b,c ∈M . Proof. The proof follows directly from Lemma 1. Definition 9. An application Υ : N (X) → N (X) is called a N -closure operator if it satisfies the following conditions: (1) N ⊑ Υ(N), (expansivity) J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 8 of 15 (2) Υ(Υ(N)) = Υ(N), (idempotency) (3) Υ(N ⊔M) = Υ(N) ⊔Υ(M), (additivity) (4) Υ(∅̃) = ∅̃, (non-spontaneous creation) whenever M,N ∈ N (X). Proposition 7. If Υ : N (X) → N (X) is a N -closure operator, then the collection τ(Υ) = {N ∈ N (X) : Υ(N c) = N c} is a N -topology on X and Υ is the N -closure in the N -topological space (X, τ(Υ)). Proof. We verify that τ(Υ) satisfies the conditions of a N -topology on X. Indeed: (1) X̃ ∈ τ(Υ) because by the non-spontaneous creation of Υ, we have Υ(X̃c) = Υ(∅̃) = ∅̃ = X̃c. Also, ∅̃ ∈ τ(Υ) because X̃ ⊑ Υ(X̃) ⊑ X̃ by expansivity of Υ and the fact that Υ(X̃) ∈ N (X). (2) Let {Mj : j ∈ J} ⊆ N (X). Then, Υ(M c j ) =M c j for each j ∈ J . By expansivity Υ, we have ⊔ j∈J Mj c ⊑ Υ ⊔ j∈J Mj c. For other neutrosophic inclusion, let us note that⊔ j∈J Mj c ⊑M c j , for each j ∈ J . By idempotency of Υ, it follows that Υ ⊔ j∈J Mi c ⊑ Υ(M c j ) =M c j for each j ∈ J . Thus, Υ ⊔ j∈J Mj c ⊑ l j∈J M c j = ⊔ j∈J Mj c and hence, Υ ⊔ j∈J Mj c =⊔ j∈J Mj c . This shows that ⊔ j∈J Mj belongs to τ(Υ). (3) Suppose that N,M ∈ N (X). Then, Υ(N c) = N c and Υ(M c) = M c and so, by additivity of Υ, we get that Υ((N ⊓M)c) = Υ(N c ⊔M c) = Υ(N c)⊔Υ(M c) = N c ⊔M c = (N ⊓M)c, which proves that N ⊓M belongs to τ(Υ). From (1)-(3), we conclude that τ(Υ) is a N -topology on X. Now, we will prove the remainder of the statement. First, Υ(N) ∈ τ c(Υ), because Υ(N) = Υ(Υ(N)) by (2). Since N ⊑ Υ(N) and Cl(N) is the smallest N -τ(Υ)-closed set containing N , it follows that Cl(N) ⊑ Υ(N). Secondly, N ⊑ Cl(N) and Cl(N) ⊑ Υ(N) imply that Cl(N) ⊑ Υ(Cl(N)) = Cl(N). Therefore, Υ(N) = Cl(N) whatever is N ∈ N (X). Let Np(X) = {N ∈ N (X) : there exists a N -point xa,b,c ∈ N} and let N ′(X) = {∅̃} ∪ Np(X). In the remainder of this paper, we will use the definitions and results described in the previous section, restricted to the collection N ′(X). J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 9 of 15 Definition 10. Let (X, τ) be a N -topological space and N ∈ N ′(X). The N -point-closure of N , denoted by Clp(N), is defined as Clp(N) = ⊔ {xa,b,c ∈ N ′(X) : U ⊓N ̸= ∅̃ for every U ∈ τ(xa,b,c)}. Remark 3. In general, it is not true that Cl(N) = Clp(N) for eachN ∈ N ′(X). Moreover, none of the neutrosophic inclusions Cl(N) ⊑ Clp(N) and Clp(N) ⊑ Cl(N) is true in general, as we can see in the following two examples. Example 2. Let X = {x, y} and O,U ∈ N ′(X) such that O = {⟨x, 0.5, 0.5, 0.5⟩, ⟨y, 0.4, 0.4, 0.6⟩}, U = {⟨x, 0.6, 0.6, 0.4⟩, ⟨y, 0.3, 0.3, 0.7⟩}. Observe that τ = {∅̃, X̃, O, U,O ⊓ U,O ⊔ U} is a N -topology on X. Then, the collection of all N -closed sets on X is τ c = { X̃, ∅̃, Oc, U c, (O ⊓ U)c, (O ⊔ U)c } , where Oc = {⟨x, 0.5, 0.5, 0.5⟩, ⟨y, 0.6, 0.6, 0.4⟩}, U c = {⟨x, 0.4, 0.4, 0.6⟩, ⟨y, 0.7, 0.7, 0.3⟩}, (O ⊓ U)c = {⟨x, 0.5, 0.4, 0.5⟩, ⟨y, 0.7, 0.6, 0.3⟩}, (O ⊔ U)c = {⟨x, 0.4, 0.5, 0.6⟩, ⟨y, 0.6, 0.7, 0.4⟩}. Consider the N -set N = {⟨x, 0.1, 0.8, 0.9⟩, ⟨y, 0.4, 0.9, 0.6⟩} and the N -point x0.4,0.3,0.6. Then, Cl(N) = (O ⊔U)c and X̃ is the only N -open set to which x0.4,0.3,0.6 belongs. Since X̃ ⊓N ̸= ∅̃, it follows that x0.4,0.3,0.6 belongs to Clp(N), but x0.4,0.3,0.6 does not belong to Cl(N) = {⟨x, 0.4, 0.5, 0.6⟩, ⟨y, 0.6, 0.7, 0.4⟩}. Example 3. Let X = {x, y} and O,U ∈ N ′(X) such that O = {⟨x, 0.2, 0.4, 0.8⟩, ⟨y, 0.4, 0.6, 0.6⟩}, U = {⟨x, 0, 0.3, 1⟩, ⟨y, 0.1, 1, 0.9⟩}. Consider the N -topology on X given by τ = {∅̃, X̃, O, U,O ⊓ U,O ⊔ U}. The collection of all N -closed sets on X is τ c = { X̃, ∅̃, Oc, U c, (O ⊓ U)c, (O ⊔ U)c } , J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 10 of 15 where Oc = {⟨x, 0.8, 0.6, 0.2⟩, ⟨y, 0.6, 0.4, 0.4⟩}, U c = {⟨x, 1, 0.7, 0⟩, ⟨y, 0.9, 0, 0.1⟩}, (O ⊓ U)c = {⟨x, 1, 0.6, 0⟩, ⟨y, 0.9, 0, 0.1⟩}, (O ⊔ U)c = {⟨x, 0.8, 0.7, 0.2⟩, ⟨y, 0.6, 0.4, 0.4⟩}. Consider the N -set N = {⟨x, 0.1, 1, 0.9⟩, ⟨y, 0, 0.3, 1⟩} and the N -point y0.1,1,0.9. Then, y0.1,1,0.9 ∈ Cl(N) = U c, but U ∈ τ(y0.1,1,0.9) and U ⊓ N = ∅̃, which implies that y0.1,1,0.9 does not belong to Clp(N). Proposition 8. Let (X, τ) be a N -topological space and N,M ∈ N ′(X). Then, the following conditions hold: (1) N ⊑ Clp(N). (2) Clp(Clp(N)) = Clp(N). (3) If N ⊑M , then Clp(N) ⊑ Clp(M). (4) Clp(N ⊓M) ⊑ Clp(N) ⊓ Clp(M). (5) Clp(N ⊔M) = Clp(N) ⊔ Clp(M). (6) Clp(∅̃) = ∅̃. (7) Clp(X̃) = X̃. Proof. (1) It is obvious from Definition 10. (2) The inclusion Clp(N) ⊑ Clp(Clp(N)) is an immediate consequence of part (1). For the other inclusion, suppose that xa,b,c ∈ Clp(Clp(N)) and let O ∈ τ(xa,b,c). Then, O ⊓ Clp(N) ̸= ∅̃, which implies that there exists a N -point yu,v,w ∈ Clp(N) and O ∈ τ(yu,v,w). Thus, O ⊓N ̸= ∅̃ and hence, xa,b,c ∈ Clp(N). (3) Suppose that N,M ∈ N ′(X) are such that N ⊑ M . If xa,b,c /∈ Clp(M), then there exists U ∈ τ(xa,b,c) such that U ⊓M = ∅̃, which implies that U ⊓ N ⊑ U ⊓M = ∅̃ and so, U ⊓N = ∅̃, which proves that xa,b,c /∈ Clp(N). Therefore, Clp(N) ⊑ Clp(M) whenever N ⊑M . (4) The proof follows from part (3). (5) The inclusion Clp(N) ⊔ Clp(M) ⊑ Clp(N ⊔M) is an immediate consequence of part (1). Suppose that xa,b,c /∈ Clp(N) ⊔ Clp(M). Then, there exist O,U ∈ τ(xa,b,c) such that O ⊓N = ∅̃ and U ⊓M = ∅̃. Putting V = O ⊓ U , we have V ∈ τ(xa,b,c) and V ⊓ (N ⊔M) = (V ⊓N) ⊔ (V ⊓M) = (O ⊓ U ⊓N) ⊔ (O ⊓ U ⊓M) ⊑ (O ⊓N) ⊔ (U ⊓M) = ∅̃ ⊔ ∅̃ = ∅̃, which implies that V ⊓ (N ⊔M) = ∅̃ and hence, xa,b,c /∈ Clp(N ⊔M). (6) and (7) are deduced from the Definition 10. J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 11 of 15 In the following example, we show that the converse of the part (4) of Proposition 8 is not true in general. Example 4. Let (X, τ) be the N -topological space given in Example 3. Consider the N -sets N = {⟨x, 0.1, 1, 0.9⟩, ⟨y, 0, 0.3, 1⟩} and M = {⟨x, 0, 0.3, 1⟩, ⟨y, 0.1, 1, 0.9⟩}. Then, M ⊓N = ∅̃ and so (by part (6) of Proposition 8) Clp(M ⊓N) = ∅̃. On the other hand, as X̃ is the only N -open set to which the N -point x0.3,1,0.7 belongs and X̃⊓N ̸= ∅̃, X̃⊓M ̸= ∅̃, we obtain that x0.3,1,0.7 ∈ Clp(M) ⊓ Clp(N), which implies that Clp(M) ⊓ Clp(N) ̸= ∅̃. Therefore, the inclusion Clp(M) ⊓ Clp(N) ⊑ Clp(M ⊓N) is not satisfied. Remark 4. From Proposition 8 we infer that Clp satisfies the conditions of Definition 9. Thus, by Proposition 7, we get that the collection τp = {N ∈ N ′(X) : Clp(N c) = N c} is a N -topology on X and Clp is the N -closure in the N -topological space (X, τp). We say that a N -set N is N -τp-open if N ∈ τp. The complement of a N -τp-open set, we will call it a N -τp-closed set. Observe that a N -set M is τp-closed if and only if Clp(M) =M . Remark 5. If we restrict the definition of N -ideal to the collection N ′(X), then we can correct some results given in [12], as we show in the following: (1) If (X, τ) is a N -topological space and L is a N -ideal on X, then ∅̃⋆ = ∅̃, because for every N -point xa,b,c ∈ N ′(X) and every U ∈ τ(xa,b,c), ∅̃ ⊓ U = ∅̃ ∈ L. (2) For each N ∈ N ′(X), Cl⋆(N) is defined as the neutrosophic union of N with N⋆; that is, Cl⋆(N) = N ⊔ N⋆ (see [12]). Observe that, by Proposition 7, we have Cl⋆ is a N -closure operator. Using this fact, we denote by τ⋆ (or τ⋆(L)) to the N -topology generated by Cl⋆, that is, τ⋆ = {N ∈ N ′(X) : Cl⋆(N c) = N c}. (3) If L = {∅̃}, then for each N ∈ N ′(X), N⋆ = Clp(N) and hence, Cl⋆(N) = N ⊔N⋆ = N ⊔ Clp(N) = Clp(N), i.e. Cl⋆(N) = Clp(N). Therefore, τ⋆({∅̃}) = τp. (4) By using the result of the previous part, we correct part (3) of Theorem 1 as follows: N⋆ = Clp(N ⋆) ⊑ Clp(N) (N⋆ is a N -τp-closed set). The following is the proof of the above statement. Since {∅̃} ⊆ L for each N -ideal L on X, we have N⋆(L) ⊆ N⋆({∅̃}) = Clp(N) for each N ∈ N ′(X). Suppose that xa,b,c ∈ Clp(N ⋆) and let U ∈ τ(xa,b,c) arbitrary. Then, U ⊓N⋆ ̸= ∅̃ and so, there exists a N -point yu,v,w ∈ U ⊓ N⋆, which implies that yu,v,w ∈ U and yu,v,w ∈ N⋆. Since U ∈ τ(yu,v,w), it follows that U ⊓N ̸∈ L and so xa,b,c ∈ N⋆. On the other hand, since N⋆ ⊑ Clp(N ⋆), we conclude that N⋆ = Clp(N ⋆). (5) If L = N ′(X), then for any N ∈ N ′(X), N⋆ = ∅̃ and so, Cl⋆(N) = N ⊔ ∅̃ = N . Therefore, τ⋆(N ′(X)) = N ′(X). J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 12 of 15 (6) Since N⋆ = Clp(N ⋆) ⊑ Clp(N), we have Cl⋆(N) ⊑ Clp(N) for each N ∈ N ′(X). Hence, if N is a N -τp-closed set, then Cl⋆(N) ⊑ Clp(N) = N , which implies that Cl⋆(N) = N and so, N is N -τ⋆-closed. Thus, every N -τp-open set is N -τ⋆-open; i.e. τp ⊆ τ⋆. 4. N-point-kernel In 2017, S. Jafari and N. Rajesh [13] introduced the notion of N -kernel in a N - topological space (X, τ) as follows: the N -kernel of N ∈ N (X), denoted by Ker(N), is defined as Ker(N) = l {U ∈ N (X) : N ⊑ U and U ∈ τ}. In this section, we intro- duce and study the concept of N -point-kernel of N ∈ N ′(X) by using N -points, which is distinct from the notion given in [13], as we can see in Examples 5 and 6 below. Definition 11. Let (X, τ) be a N -topological space and N ∈ N ′(X). The N -point-kernel of N , denoted by Kerp(N), is defined as Kerp(N) = ⊔ {xa,b,c ∈ N ′(X) : F ⊓N ̸= ∅̃ for every F ∈ τ c(xa,b,c)}, where τ c(xa,b,c) = {F ∈ τ c : xa,b,c ∈ F}. Remark 6. GivenN ∈ N ′(X) we can see that, in general, is not trueKer(N) ̸= Kerp(N). Example 5. Let (X, τ) be the N -topological space given in Example 2. Consider the N - set N = {⟨x, 0.1, 0.8, 0.9⟩, ⟨y, 0.4, 0.9, 0.6⟩} and the N -point x0.4,0.3,0.6. Then, Ker(N) = O and X̃ is the only N -closed set to which x0.4,0.3,0.6 belongs. Since X̃ ⊓ N ̸= ∅̃, it follows that x0.4,0.3,0.6 belongs to Kerp(N), but x0.4,0.3,0.6 does not belong to Ker(N) = {⟨x, 0.5, 0.5, 0.5⟩, ⟨y, 0.4, 0.4, 0.6⟩}. Example 6. Let X = {x, y} and O,U ∈ N ′(X) such that O = {⟨x, 0.8, 0.6, 0.2⟩, ⟨y, 0.6, 0.4, 0.4⟩}, U = {⟨x, 1, 0.7, 0⟩, ⟨y, 0.9, 0, 0.1⟩}. Consider the N -topology on X given by τ = {∅̃, X̃, O, U,O ⊓ U,O ⊔ U}. The collection of all N -closed sets on X is τ c = { X̃, ∅̃, Oc, U c, (O ⊓ U)c, (O ⊔ U)c } , where Oc = {⟨x, 0.2, 0.4, 0.8⟩, ⟨y, 0.4, 0.6, 0.6⟩}, U c = {⟨x, 0, 0.3, 1⟩, ⟨y, 0.1, 1, 0.9⟩}, (O ⊓ U)c = {⟨x, 0.2, 0.3, 0.8⟩, ⟨y, 0.4, 0.6, 0.6⟩}, (O ⊔ U)c = {⟨x, 0, 0.4, 1⟩, ⟨y, 0.1, 1, 0.9⟩}. J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 13 of 15 Consider the N -set N = {⟨x, 0.1, 1, 0.9⟩, ⟨y, 0, 0.3, 1⟩} and the N -point y0.1,1,0.9. Then, y0.1,1,0.9 ∈ Ker(N) = U , but U c ∈ τ c(y0.1,1,0.9) and U c ⊓ N = ∅̃, which implies that y0.1,1,0.9 does not belong to Kerp(N). Proposition 9. Let N,M ∈ N ′(X) and {Nα : α ∈ ∆} ⊆ N ′(X). Then, the following properties hold: (1) N ⊑ Kerp(N). (2) Si N ⊑M , entonces Kerp(N) ⊑ Kerp(M). (3) Kerp(Kerp(N)) = Kerp(N). (4) Kerp ( ⊔ α∈∆ Nα ) = ⊔ α∈∆ Kerp (Nα). (5) Kerp ( d α∈∆ Nα ) ⊑ d α∈∆ Kerp (Nα). (6) Kerp(∅̃) = ∅̃. (7) Kerp(X̃) = X̃. Proof. (1) It is clear from Definition 11. (2) Suppose that xa,b,c /∈ Kerp(M). Then, there exists F ∈ τ c(xa,b,c) such that F ⊓M = ∅̃. SinceN ⊑M , we have F⊓N ⊑ F⊓M = ∅̃ and so, F⊓N = ∅̃. Therefore, xa,b,c /∈ Kerp(N). (3) By part (1), we have Kerp(N) ⊑ Kerp(Kerp(N)). To demonstrate the opposite inclu- sion, suppose that xa,b,c ∈ Kerp(Kerp(N)) and let F ∈ τ c(xa,b,c). Then, F ⊓Kerp(N) ̸= ∅̃, which implies that there exists a N -point yu,v,w ∈ Kerp(N) and F ∈ τ c(yu,v,w). Thus, F ⊓N ̸= ∅̃ and hence, xa,b,c ∈ Kerp(N). (4) SinceNα ⊑ ⊔ α∈∆ Nα for each α ∈ △, by part (2) it follows thatKerp(Nα) ⊑ Kerp (⊔ α∈∆ Nα ) for each α ∈ △. Therefore, ⊔ α∈∆ Kerp(Nα) ⊑ Kerp (⊔ α∈∆ Nα ) . For the other inclu- sion, suppose that xa,b,c /∈ ⊔ α∈∆ Kerp(Nα). Then, x /∈ Kerp(Nα) for every α ∈ ∆ and so, there exists Fα ∈ τ c(xa,b,c) such that Fα ⊓ Nα = ∅̃ for each α ∈ ∆. Putting F = l α∈∆ Fα, we have F ∈ τ c(xa,b,c) and F ⊓ (⊔ α∈∆ Nα ) = ⊔ α∈∆ (F ⊓Nα) ⊑ ⊔ α∈∆ (Fα⊓Nα) = ∅̃. Thus, F ⊓ (⊔ α∈∆ Nα ) = ∅̃ and hence, xa,b,c /∈ Kerp (⊔ α∈∆ Nα ) , which shows that Kerp (⊔ α∈∆ Nα ) ⊑ ⊔ α∈∆ Kerp(Nα). J. Sanabria, E. Rosas, C. Granados / Eur. J. Pure Appl. Math, 18 (3) (2025), 6339 14 of 15 (5) Since l α∈∆ Nα ⊑ Nα for each α ∈ ∆, by using part (2), we have Kerp ( l α∈∆ Nα ) ⊑ Kerp(Nα) for each α ∈ ∆, which implies that Kerp ( l α∈∆ Nα ) ⊑ l α∈∆ Kerp (Nα). (6) and (7) are immediate consequences of Definition 11. In the following example, we show that the converse of the part (4) of Proposition 9 is not true in general. Example 7. Let (X, τ) be the N -topological space given in Example 5. Consider the N -sets N = {⟨x, 0.1, 1, 0.9⟩, ⟨y, 0, 0.3, 1⟩} and M = {⟨x, 0, 0.3, 1⟩, ⟨y, 0.1, 1, 0.9⟩}. Then, M⊓N = ∅̃ and so (by part (6) of Proposition 9)Kerp(M⊓N) = ∅̃. On the other hand, as X̃ is the only N -closed set to which the N -point x0.3,1,0.7 belongs and X̃⊓N ̸= ∅̃, X̃⊓M ̸= ∅̃, we obtain that x0.3,1,0.7 ∈ Kerp(M)⊓Kerp(N), which implies that Kerp(M)⊓Kerp(N) ̸= ∅̃. Therefore, the inclusion Kerp(M) ⊓Kerp(N) ⊑ Kerp(M ⊓N) is not satisfied. Remark 7. By Proposition 9, we have Kerp satisfies the conditions of Definition 9 and by Proposition 7, we conclude that τk = {N ∈ N ′(X) : Kerp(N c) = N c} is a N -topology on X and Kerp is the N -closure in the N -topological space (X, τk). The elements of τk are called N -τk-open sets and their complements are said to be N -τk-closed sets. It is clear that M is N -τk-closed if and only if Kerp(M) =M . 5. Conclusion The concept ofN -set has been the cornerstone of neutrosophic science, which has found its place in contemporary research, since this science means development and applications of neutrosophic logic, set, measure, integral, probability, etc., and their applications in any field of knowledge. In this research it was possible to verify that, in some cases, neutrosophic set theory does not behave like classical set theory; for example, the union of a N -set with its neutrosophic complement is not equal to the neutrosophic universe and the neutrosophic empty set is not the only N -set that does not contain N -points. Also, the notions of closure and kernel of an N -set were introduced by means of the concept of N -point, the main properties of these notions were discussed and two new N -topologies related to the introduced notions were generated. The results presented here constitute a contribution to the theory of N -topological spaces and may be useful to extend this area of knowledge by developing new investigations involving N -functions as has been done in the works of S. Das and B.C. Tripathy [14], S. F. Matar and A.A. Hijab [15], P. Basker and B. Said [16]. References [1] F. Smarandache. Neutrosophic set - a generalization of the intuitionistic fuzzy set. 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