EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1533-1542 ISSN 1307-5543 – ejpam.com Published by New York Business Global δp(Λ, p)-open sets in topological spaces Chawalit Boonpok1, Montri Thongmoon1,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand Abstract. This paper deals with the notion of δp(Λ, p)-open sets. Some properties of δp(Λ, p)- open sets and δp(Λ, p)-closed sets are investigated. Moreover, several characterizations of δp(Λ, p)- D1 spaces and δp(Λ, p)-R0 spaces are established. 2020 Mathematics Subject Classifications: 54A05, 54D10 Key Words and Phrases: δp(Λ, p)-open set, δp(Λ, p)-D1 space, δp(Λ, p)-R0 space 1. Introduction The concept of δ-open sets was first introduced by Veličko [10]. In 1982, Mashhour et al. [7] introduced and investigated the notion of preopen sets which is weaker than the notion of open sets in topological spaces. Raychaudhuri and Mukherjee [8] introduced and studied the notions of δ-preopen sets and δ-closure. The class of δ-preopen sets is larger than that of preopen sets. In 1996, Raychaudhuri and Mukherjee [9] introduced and investigated the concept of δp-closed spaces. Caldas et al. [3] introduced some weak separation axioms by utilizing the notions of δ-preopen sets and the δ-preclosure operator. Furthermore, Caldas et al. [3] showed that (δ, p)-T1 spaces, (δ, p)-R0 spaces and (δ, p)-symmetric spaces are all equivalent. In 2003, Caldas et al. [5] investigated some weak separation axioms by utilizing δ-semiopen sets and the δ-semiclosure operator. In 2005, Caldas et al. [4] investigated the notion of δ-Λs-semiclosed sets which is defined as the intersection of a δ-Λs-set and a δ-semiclosed set. In [2], the present authors introduced the notions of (Λ, p)-open sets and (Λ, p)-closed sets which are defined by utilizing the notions of Λp-sets and preclosed sets. Quite recently, Boonpok and Viriyapong [1] investigated some characterizations of (Λ, s)- R0 topological spaces. In this paper, we introduced the concept of δp(Λ, p)-open sets. Moreover, some properties of δp(Λ, p)-open sets and δp(Λ, p)-closed sets are discussed. In particular, several characterizations of δp(Λ, p)-D1 spaces and δp(Λ, p)-R0 spaces are investigated. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4733 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), montri.t@msu.ac.th (M. Thongmoon) https://www.ejpam.com 1533 © 2023 EJPAM All rights reserved. C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1533-1542 1534 2. Preliminaries Throughout the present paper, spaces (X, τ) and (Y, σ) (or simply X and Y ) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. For a subset A of a topological space (X, τ), Cl(A) and Int(A), represent the closure and the interior of A, respectively. A subset A of a topological space (X, τ) is said to be preopen [7] if A ⊆ Int(Cl(A)). The complement of a preopen set is called preclosed. The family of all preopen sets of a topological space (X, τ) is denoted by PO(X, τ). A subset Λp(A) [6] is defined as follows: Λp(A) = ∩{U | A ⊆ U,U ∈ PO(X, τ)}. A subset A of a topological space (X, τ) is called a Λp-set [1] (pre-Λ-set [6]) if A = Λp(A). A subset A of a topological space (X, τ) is called (Λ, p)-closed [1] if A = T ∩C, where T is a Λp-set and C is a preclosed set. The complement of a (Λ, p)-closed set is called (Λ, p)-open. The family of all (Λ, p)-open (resp. (Λ, p)-closed) sets in a topological space (X, τ) is denoted by ΛpO(X, τ) (resp. ΛpC(X, τ)). Let A be a subset of a topological space (X, τ). A point x ∈ X is called a (Λ, p)-cluster point [1] of A if A ∩ U ̸= ∅ for every (Λ, p)-open set U of X containing x. The set of all (Λ, p)-cluster points of A is called the (Λ, p)-closure [1] of A and is denoted by A(Λ,p). The union of all (Λ, p)-open sets contained in A is called the (Λ, p)-interior [1] of A and is denoted by A(Λ,p). A subset A of a topological space (X, τ) is said to be p(Λ, p)-open [1] if A ⊆ [A(Λ,p)](Λ,p). The complement of a p(Λ, p)-open set is said to be p(Λ, p)-closed. 3. δp(Λ, p)-open sets In this section, we introduced the concept of δp(Λ, p)-open sets. Moreover, some prop- erties of δp(Λ, p)-open sets and δp(Λ, p)-closed sets are investigated. Furthermore, several characterizations of δp(Λ, p)-D1 spaces and δp(Λ, p)-R0 spaces are discussed. Definition 1. Let A be a subset of a topological space (X, τ). A point x of X is called a δ(Λ, p)-cluster point of A if A∩[V (Λ,p)](Λ,p) ̸= ∅ for every (Λ, p)-open set V of X containing x. The set of all δ(Λ, p)-cluster points of A is called the δ(Λ, p)-closure of A and is denoted by Aδ(Λ,p). If A = Aδ(Λ,p), then A is said to be δ(Λ, p)-closed. The complement of a δ(Λ, p)- closed set is said to be δ(Λ, p)-open. The union of all δ(Λ, p)-open sets contained in A is called the δ(Λ, p)-interior of A and is denoted by Aδ(Λ,p). Definition 2. A subset A of a topological space (X, τ) is said to be δp(Λ, p)-open if A ⊆ [A(Λ,p)]δ(Λ,p). The complement of a δp(Λ, p)-open set is said to be δp(Λ, p)-closed. The family of all δp(Λ, p)-open (resp. δp(Λ, p)-closed) sets in a topological space (X, τ) is denoted by δp(Λ, p)O(X, τ) (resp. δp(Λ, p)C(X, τ)). Let A be a subset of a topological space (X, τ). The intersection of all δp(Λ, p)-closed sets containing A is called the δp(Λ, p)- closure of A and is denoted by Aδp(Λ,p). Lemma 1. For the δp(Λ, p)-closure of subsets A, B in a topological space (X, τ), the following properties hold: C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1533-1542 1535 (1) If A ⊆ B, then Aδp(Λ,p) ⊆ Bδp(Λ,p). (2) A is δp(Λ, p)-closed in (X, τ) if and only if A = Aδp(Λ,p). (3) Aδp(Λ,p) is δp(Λ, p)-closed, that is, Aδp(Λ,p) = [Aδp(Λ,p)]δp(Λ,p). (4) x ∈ Aδp(Λ,p) if and only if A ∩ V ̸= ∅ for every V ∈ δp(Λ, p)O(X, τ) containing x. Lemma 2. For a family {Aγ | γ ∈ ∇} of a topological space (X, τ), the following properties hold: (1) [∩{Aγ | γ ∈ ∇}]δp(Λ,p) ⊆ ∩{Aδp(Λ,p) γ | γ ∈ ∇}. (2) [∪{Aγ | γ ∈ ∇}]δp(Λ,p) ⊇ ∪{Aδp(Λ,p) γ | γ ∈ ∇}. Definition 3. A subset A of a topological space (X, τ) is called a δp(Λ, p)D-set if there exist δp(Λ, p)-open sets U and V such that U ̸= X and A = U − V . Definition 4. A topological space (X, τ) is said to be: (i) δp(Λ, p)-T1 if for any distinct pair of points x and y of X, there exist a δp(Λ, p)-open set U of X containing x but not y and a δp(Λ, p)-open set V of X containing y but not x; (ii) δp(Λ, p)-D1 if for any distinct pair of points x and y of X, there exist a δp(Λ, p)D-set U of X containing x but not y and a δp(Λ, p)D-set V of X containing y but not x. Definition 5. A subset N of a topological space (X, τ) is called a δp(Λ, p)-neighborhood of a point x ∈ X if there exists a δp(Λ, p)-open set U such that x ∈ U ⊆ N . Definition 6. Let (X, τ) be a topological space. A point x ∈ X which has only X as the δp(Λ, p)-neighbourhood is called a δp(Λ, p)-neat point. Lemma 3. Let (X, τ) be a topological space. For each point x ∈ X, {x} is p(Λ, p)-open or p(Λ, p)-closed. Theorem 1. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is δp(Λ, p)-D1; (2) (X, τ) has no δp(Λ, p)-neat point. Proof. (1) ⇒ (2): Since (X, τ) is δp(Λ, p)-D1, so each point x of X is contained in a δp(Λ, p)D-set G = U − V and thus in U , where U and V are δp(Λ, p)-open sets. By definition U ̸= X. This implies that x is not a δp(Λ, p)-neat point. (2) ⇒ (1): By Lemma 3 for each distinct pair of points x, y ∈ X, at least one of them, x(say) has a δp(Λ, p)-neighborhood U containing x and not y. Thus, U which is different from X is a δp(Λ, p)D-set. If X has no δp(Λ, p)-neat point, then y is not a δp(Λ, p)-neat point. This means that there exists a δp(Λ, p)-neighborhood V of y such that V ̸= X. Thus, y ∈ V − U but not y and V − U is a δp(Λ, p)D-set. This shows that (X, τ) is δp(Λ, p)-D1. C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1533-1542 1536 Definition 7. A function f : (X, τ) → (Y, σ) is called δp(Λ, p)-continuous if, for each x ∈ X and each δp(Λ, p)-open set V of Y containing f(x), there exists a δp(Λ, p)-open set U of X containing x such that f(U) ⊆ V . Lemma 4. A function f : (X, τ) → (Y, σ) is δp(Λ, p)-continuous if and only if f−1(V ) is δp(Λ, p)-open in X for every δp(Λ, p)-open set V of Y . Theorem 2. If f : (X, τ) → (Y, σ) is a δp(Λ, p)-continuous surjective function and B is a δp(Λ, p)D-set in Y , then f−1(B) is a δp(Λ, p)D-set in X. Proof. Let B be a δp(Λ, p)D-set in Y . Then, there exist δp(Λ, p)-open sets U and V in Y such that B = U − V and U ̸= Y . By the δp(Λ, p)-continuity of f , f−1(U) and f−1(V ) are δp(Λ, p)-open in X. Since U ̸= Y , we have f−1(U) ̸= X. Thus, f−1(B) = f−1(U)− f−1(V ) is a δp(Λ, p)D-set. Theorem 3. If (Y, σ) is a δp(Λ, p)-D1 space and f : (X, τ) → (Y, σ) is a δp(Λ, p)- continuous bijection, then (X, τ) is δp(Λ, p)-D1. Proof. Suppose that (Y, σ) is a δp(Λ, p)-D1 space. Let x and y be any pair of distinct points in X. Since f is injective and (Y, σ) is δp(Λ, p)-D1, there exist δp(Λ, p)D-sets U and V of Y containing f(x) and f(y), respectively, such that f(y) ̸∈ U and f(x) ̸∈ V . By Theorem 2, f−1(U) and f−1(V ) are δp(Λ, p)D-sets in X containing x and y, respectively, such that y ̸∈ f−1(U) and x ̸∈ f−1(V ). This shows that (X, τ) is δp(Λ, p)-D1. Definition 8. A topological space (X, τ) is called δp(Λ, p)-R0 if for each δp(Λ, p)-open set U and each x ∈ U , {x}δp(Λ,p) ⊆ U . Theorem 4. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is δp(Λ, p)-R0. (2) For each δp(Λ, p)-closed set F and each x ∈ X−F , there exists U ∈ δp(Λ, p)O(X, τ) such that F ⊆ U and x ̸∈ U . (3) For each δp(Λ, p)-closed set F and each x ∈ X − F , F ∩ {x}δp(Λ,p) = ∅. (4) For any distinct points x, y in X, {x}δp(Λ,p) = {y}δp(Λ,p) or {x}δp(Λ,p)∩{y}δp(Λ,p) = ∅. Proof. (1) ⇒ (2): Let F be a δp(Λ, p)-closed set and x ∈ X − F . Since (X, τ) is δp(Λ, p)-R0, we have {x}δp(Λ,p) ⊆ X − F . Put U = X − {x}δp(Λ,p). Thus, by Lemma 1, U ∈ δp(Λ, p)O(X, τ), F ⊆ U and x ̸∈ U . (2) ⇒ (3): Let F be a δp(Λ, p)-closed set and x ∈ X − F . By (2), there exists U ∈ δp(Λ, p)O(X, τ) such that F ⊆ U and x ̸∈ U . Since U ∈ δp(Λ, p)O(X, τ), U ∩ {x}δp(Λ,p) = ∅ and hence F ∩ {x}δp(Λ,p) = ∅. C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1533-1542 1537 (3) ⇒ (4): Let x and y be distinct points of X. Suppose that {x}δp(Λ,p)∩{y}δp(Λ,p) ̸= ∅. By (3), we have x ∈ {y}δp(Λ,p) and y ∈ {x}δp(Λ,p). By Lemma 1, {x}δp(Λ,p) ⊆ {y}δp(Λ,p) ⊆ {x}δp(Λ,p) and hence {x}δp(Λ,p) = {y}δp(Λ,p). (4) ⇒ (1): Let V ∈ δp(Λ, p)O(X, τ) and x ∈ V . For each y ̸∈ V , V ∩ {y}δp(Λ,p) = ∅ and hence x ̸∈ {y}δp(Λ,p). Thus, {x}δp(Λ,p) ̸= {y}δp(Λ,p). By (4), for each y ̸∈ V , {x}δp(Λ,p) ∩ {y}δp(Λ,p) = ∅. Since X − V is δp(Λ, p)-closed, y ∈ {y}δp(Λ,p) ⊆ X − V and ∪y∈X−V {y}δp(Λ,p) = X − V . Thus, {x}δp(Λ,p) ∩ (X − V ) = {x}δp(Λ,p) ∩ [∪y∈X−V {y}δp(Λ,p)] = ∪y∈X−V [{x}δp(Λ,p) ∩ {y}δp(Λ,p)] = ∅ and hence {x}δp(Λ,p) ⊆ V . This shows that (X, τ) is δp(Λ, p)-R0. Corollary 1. A topological space (X, τ) is δp(Λ, p)-R0 if and only if, for any points x and y in X, {x}δp(Λ,p) ̸= {y}δp(Λ,p) implies {x}δp(Λ,p) ∩ {y}δp(Λ,p) = ∅. Proof. This is obvious by Theorem 4. Conversely, let U ∈ δp(Λ, p)O(X, τ) and x ∈ U . If y ̸∈ U , then U∩{y}δp(Λ,p) = ∅. Thus, x ̸∈ {y}δp(Λ,p) and {x}δp(Λ,p) ̸= {y}δp(Λ,p). By the hypothesis, {x}δp(Λ,p) ∩ {y}δp(Λ,p) = ∅ and hence y ̸∈ {x}δp(Λ,p). This shows that {x}δp(Λ,p) ⊆ U . Thus, (X, τ) is δp(Λ, p)-R0. Definition 9. Let A be a subset of a topological space (X, τ). The δp(Λ, p)-kernel of A, denoted by δp(Λ, p)Ker(A), is defined to be the set δp(Λ, p)Ker(A) = ∩{U ∈ δp(Λ, p)O(X, τ) | A ⊆ U}. Lemma 5. For subsets A,B of a topological space (X, τ), the following properties hold: (1) A ⊆ δp(Λ, p)Ker(A). (2) If A ⊆ B, then δp(Λ, p)Ker(A) ⊆ δp(Λ, p)Ker(B). (3) δp(Λ, p)Ker(δp(Λ, sp)Ker(A)) = δp(Λ, p)Ker(A). (4) If A is δp(Λ, p)-open, δp(Λ, p)Ker(A) = A. Theorem 5. For any points x and y in a topological space (X, τ), the following properties are equivalent: C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1533-1542 1538 (1) δp(Λ, p)Ker({x}) ̸= δp(Λ, p)Ker({y}). (2) {x}δp(Λ,p) ̸= {y}δp(Λ,p). Proof. (1) ⇒ (2): Suppose that δp(Λ, p)Ker({x}) ̸= δp(Λ, p)Ker({y}). Then, there exists a point z ∈ X such that z ∈ δp(Λ, p)Ker({x}) and z ̸∈ δp(Λ, p)Ker({y}) or z ∈ δp(Λ, p)Ker({y}) and z ̸∈ δp(Λ, p)Ker({x}). We prove only the first case being the second analogous. From z ∈ δp(Λ, p)Ker({x}) it follows that {x} ∩ {z}δp(Λ,p) ̸= ∅ which implies x ∈ {z}δp(Λ,p). By z ̸∈ δp(Λ, p)Ker({y}), we have {y} ∩ {z}δp(Λ,p) = ∅. Since x ∈ {z}δp(Λ,p), {x}δp(Λ,p) ⊆ {z}δp(Λ,p) and {y} ∩ {x}δp(Λ,p) = ∅. Therefore, {x}δp(Λ,p) ̸= {y}δp(Λ,p). Thus, δp(Λ, p)Ker({x}) ̸= δp(Λ, p)Ker({y}) implies that {x}δp(Λ,p) ̸= {y}δp(Λ,p). (2) ⇒ (1): Suppose that {x}δp(Λ,p) ̸= {y}δp(Λ,p). There exists a point z ∈ X such that z ∈ {x}δp(Λ,p) and z ̸∈ {y}δp(Λ,p) or z ∈ {y}δp(Λ,p) and z ̸∈ {x}δp(Λ,p). We prove only the first case being the second analogous. It follows that there exists a δp(Λ, p)- open set containing z and therefore x but not y, namely, y ̸∈ δp(Λ, p)Ker({x}) and thus δp(Λ, p)Ker({x}) ̸= δp(Λ, p)Ker({y}). Lemma 6. Let (X, τ) be a topological space and x, y ∈ X. Then, the following properties hold: (1) y ∈ δp(Λ, p)Ker({x}) if and only if x ∈ {y}δp(Λ,p). (2) δp(Λ, p)Ker({x}) = δp(Λ, p)Ker({y}) if and only if {x}δp(Λ,p) = {y}δp(Λ,p). Proof. (1) Let x ̸∈ {y}δp(Λ,p). Then, there exists U ∈ δp(Λ, p)O(X, τ) such that x ∈ U and y ̸∈ U . Thus, y ̸∈ δp(Λ, p)Ker({x}). The converse is similarly shown. (2) Suppose that δp(Λ, p)Ker({x}) = δp(Λ, p)Ker({y}) for any x, y ∈ X. Since x ∈ δp(Λ, p)Ker({x}), x ∈ δp(Λ, p)Ker({y}), by (1), y ∈ {x}δp(Λ,p). By Lemma 1, {y}δp(Λ,p) ⊆ {x}δp(Λ,p). Similarly, we have {x}δp(Λ,p) ⊆ {y}δp(Λ,p) and hence {x}δp(Λ,p) = {y}δp(Λ,p). Conversely, suppose that {x}δp(Λ,p) = {y}δp(Λ,p). Since x ∈ {x}δp(Λ,p), we have x ∈ {y}δp(Λ,p) and by (1), y ∈ δp(Λ, p)Ker({x}). By Lemma 5, δp(Λ, p)Ker({y}) ⊆ δp(Λ, p)Ker(δp(Λ, p)Ker({x})) = δp(Λ, p)Ker({x}). Similarly, we have δp(Λ, p)Ker({x}) ⊆ δp(Λ, p)Ker({y}) and hence δp(Λ, p)Ker({x}) = δp(Λ, p)Ker({y}). C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1533-1542 1539 Theorem 6. A topological space (X, τ) is δp(Λ, p)-R0 if and only if for each points x and y in X, δp(Λ, p)Ker({x}) ̸= δp(Λ, p)Ker({y}) implies δp(Λ, p)Ker({x}) ∩ δp(Λ, p)Ker({y}) = ∅. Proof. Let (X, τ) be δp(Λ, p)-R0. Suppose that δp(Λ, p)Ker({x}) ∩ δp(Λ, p)Ker({y}) ̸= ∅. Let z ∈ δp(Λ, p)Ker({x})∩δp(Λ, p)Ker({y}). Then, z ∈ δp(Λ, p)Ker({x}) and by Lemma 6, x ∈ {z}δp(Λ,p). Thus, x ∈ {z}δp(Λ,p) ∩ {x}δp(Λ,p) and by Corollary 1, {z}δp(Λ,p) = {x}δp(Λ,p). Similarly, we have {z}δp(Λ,p) = {y}δp(Λ,p) and hence {x}δp(Λ,p) = {y}δp(Λ,p), by Lemma 6, δp(Λ, p)Ker({x}) = δp(Λ, p)Ker({y}). Conversely, we show the sufficiency by using Corollary 1. Suppose that {x}δp(Λ,p) ̸= {y}δp(Λ,p). By Lemma 6, δp(Λ, p)Ker({x}) ̸= δp(Λ, p)Ker({y}) and hence δp(Λ, p)Ker({x}) ∩ δp(Λ, p)Ker({y}) = ∅. Thus, {x}δp(Λ,p) ∩ {y}δp(Λ,p) = ∅. In fact, assume that z ∈ {x}δp(Λ,p) ∩ {y}δp(Λ,p). Then, z ∈ {x}δp(Λ,p) implies x ∈ δp(Λ, p)Ker({z}) and hence x ∈ δp(Λ, p)Ker({z}) ∩ δp(Λ, p)Ker({x}). By the hypothesis, δp(Λ, p)Ker({z}) = δp(Λ, p)Ker({x}) and by Lemma 6, {z}δp(Λ,p) = {x}δp(Λ,p). Similarly, we have {z}δp(Λ,p) = {y}δp(Λ,p) and hence {x}δp(Λ,p) = {y}δp(Λ,p). This contra- dicts that {x}δp(Λ,p) ̸= {y}δp(Λ,p). Thus, {x}δp(Λ,p)∩{y}δp(Λ,p) = ∅. This shows that (X, τ) is δp(Λ, p)-R0. Theorem 7. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is δp(Λ, p)-R0. (2) x ∈ {y}δp(Λ,p) if and only if y ∈ {x}δp(Λ,p). Proof. (1) ⇒ (2): Suppose that x ∈ {y}δp(Λ,p). By Lemma 6, y ∈ δp(Λ, p)Ker({x}) and hence δp(Λ, p)Ker({x}) ∩ δp(Λ, p)Ker({y}) ̸= ∅. By Theorem 6, δp(Λ, p)Ker({x}) = δp(Λ, p)Ker({y}) and hence x ∈ δp(Λ, p)Ker({y}). Thus, by Lemma 6, y ∈ {x}δp(Λ,p). The converse is similarly shown. (2) ⇒ (1): Let U ∈ δp(Λ, p)O(X, τ) and x ∈ U . If y ̸∈ U , then U ∩ {y}δp(Λ,p) = ∅. Thus, x ̸∈ {y}δp(Λ,p) and y ̸∈ {x}δp(Λ,p). This implies that {x}δp(Λ,p) ⊆ U . Therefore, (X, τ) is δp(Λ, p)-R0. C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1533-1542 1540 Theorem 8. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is δp(Λ, p)-R0. (2) For each nonempty subset A of X and each U ∈ δp(Λ, p)O(X, τ) such that A∩U ̸= ∅, there exists a δp(Λ, p)-closed set F such that A ∩ F ̸= ∅ and F ⊆ U . (3) F = δp(Λ, p)Ker(F ) for each δp(Λ, p)-closed set F . (4) {x}δp(Λ,p) = δp(Λ, p)Ker({x}) for each x ∈ X. (5) {x}δp(Λ,p) ⊆ δp(Λ, p)Ker({x}) for each x ∈ X. Proof. (1) ⇒ (2): Let A be a nonempty subset of X and U ∈ δp(Λ, p)O(X, τ) such that A ∩ U ̸= ∅. Then, there exists x ∈ A ∩ U and hence {x}δp(Λ,p) ⊆ U . Put F = {x}δp(Λ,p). Then, F is δp(Λ, p)-closed, A ∩ F ̸= ∅ and F ⊆ U . (2) ⇒ (3): Let F be any δp(Λ, p)-closed set of X. By Lemma 5, we have F ⊆ δp(Λ, p)Ker(F ). Next, we show F ⊇ δp(Λ, p)Ker(F ). Let x ̸∈ F . Then, x ∈ X − F ∈ δp(Λ, p)O(X, τ) and by (2), there exists a δp(Λ, p)-closed set K such that x ∈ K and K ⊆ X − F . Now, put U = X − K. Then, F ⊆ U ∈ δp(Λ, p)O(X, τ) and x ̸∈ U . Thus, x ̸∈ δp(Λ, p)Ker(F ). This shows that F ⊇ δp(Λ, p)Ker(F ). (3) ⇒ (4): Let x ∈ X and y ̸∈ δp(Λ, p)Ker({x}). There exists U ∈ δp(Λ, p)O(X, τ) such that x ∈ U and y ̸∈ U . Thus, U ∩ {y}δp(Λ,p) = ∅. By (3), U ∩ δp(Λ, p)Ker({y}δp(Λ,p)) = ∅. Since x ̸∈ δp(Λ, p)Ker({y}δp(Λ,p)), there exists V ∈ δp(Λ, p)O(X, τ) such that {y}δp(Λ,p) ⊆ V and x ̸∈ V . Thus, V ∩ {x}δp(Λ,p) = ∅. Since y ∈ V , y ̸∈ {x}δp(Λ,p) and hence {x}δp(Λ,p) ⊆ δp(Λ, p)Ker({x}). Moreover, {x}δp(Λ,p) ⊆ δp(Λ, p)Ker({x}) ⊆ δp(Λ, p)Ker({x}δp(Λ,p)) = {x}δp(Λ,p). This shows that {x}δp(Λ,p) = δp(Λ, p)Ker({x}). (4) ⇒ (5): The proof is obvious. (5) ⇒ (1): Let U ∈ δp(Λ, p)O(X, τ) and x ∈ U . If y ̸∈ U , then U ∩ {y}δp(Λ,p) = ∅ and x ̸∈ {y}δp(Λ,p). By Lemma 6, y ̸∈ δp(Λ, p)Ker({x}) and by (5), y ̸∈ {x}δp(Λ,p). Thus, {x}δp(Λ,p) ⊆ U and hence (X, τ) is δp(Λ, p)-R0. Corollary 2. A topological space (X, τ) is δp(Λ, p)-R0 if and only if δp(Λ, p)Ker({x}) ⊆ {x}δp(Λ,p) for each x ∈ X. REFERENCES 1541 Proof. This is obvious by Theorem 8. Conversely, let x ∈ {y}δp(Λ,p). Thus, by Lemma 6, y ∈ δp(Λ, p)Ker({x}) and hence y ∈ {x}δp(Λ,p). Similarly, if y ∈ {x}δp(Λ,p), then x ∈ {y}δp(Λ,p). It follows from Theorem 7 that (X, τ) is δp(Λ, p)-R0. Definition 10. Let (X, τ) be a topological space and x ∈ X. A subset ⟨x⟩δp(Λ,p) is defined as follows: ⟨x⟩δp(Λ,p) = δp(Λ, p)Ker({x}) ∩ {x}δp(Λ,p). Theorem 9. A topological space (X, τ) is δp(Λ, p)-R0 if and only if ⟨x⟩δp(Λ,p) = {x}δp(Λ,p) for each x ∈ X. Proof. Let x ∈ X. By Theorem 8, δp(Λ, p)Ker({x}) = {x}δp(Λ,sp). Thus, ⟨x⟩δp(Λ,p) = δp(Λ, p)Ker({x}) ∩ {x}δp(Λ,p) = {x}δp(Λ,p). Conversely, let x ∈ X. By the hypothesis, {x}δp(Λ,p) = ⟨x⟩δp(Λ,p) = δp(Λ, p)Ker({x}) ∩ {x}δp(Λ,p) ⊆ δp(Λ, p)Ker({x}). It follows from Theorem 8 that (X, τ) is δp(Λ, p)-R0. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] C. Boonpok and C. Viriyapong. On (Λ, p)-closed sets and the related notions in topological spaces. European Journal of Pure and Applied Mathematics, 15(2):415– 436, 2022. [2] C. Boonpok and C. Viriyapong. 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