EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2581-2596 ISSN 1307-5543 – ejpam.com Published by New York Business Global Properties of generalized δp(Λ, s)-closed sets Chawalit Boonpok1, Napassanan Srisarakham1,∗ 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 concept of generalized δp(Λ, s)-closed sets. Especially, some properties of generalized δp(Λ, s)-closed sets are discussed. Moreover, we apply the notion of generalized δp(Λ, s)-closed sets to present and study new classes of spaces called δp(Λ, s)-T 1 2 -spaces and δp(Λ, s)-normal spaces. Several properties and characterizations concerning δp(Λ, s)-T 1 2 -spaces and δp(Λ, s)-normal spaces are established. 2020 Mathematics Subject Classifications: 54A05, 54D10 Key Words and Phrases: δp(Λ, s)-open set, generalized δp(Λ, s)-closed set 1. Introduction In 1970, Levine [11] introduced the concept of generalized closed sets in topological spaces and defined the notion of a T 1 2 -space to be one in which the closed sets and the generalized closed sets coincide. Dunham and Levine [9] investigated the further properties of generalized closed sets. The concept of generalized closed sets has been modified and studied by using weaker forms of open sets such as α-open sets [13], semi-open sets [10], preopen sets [12] and semi-preopen sets [1]. Levine [10] introduced the concept of semi- open sets which is weaker than the concept of open sets in topological spaces. Veličko [19] introduced δ-open sets, which are stronger than open sets. Park et al. [14] have offered new notion called δ-semiopen sets which are stronger than semi-open sets but weaker than δ-open sets and investigated the relationships between several types of these open sets. Caldas and Dontchev [4] introduced and investigated the notions of Λs-sets and Vs-sets in topological spaces. Moreover, Caldas et al. [7] investigated some weak separation axioms by utilizing δ-semiopen sets and the δ-semiclosure operator. Caldas et al. [6] investigated the notion of δ-Λs-semiclosed sets which is defined as the intersection of a δ-Λs-set and a δ-semiclosed set. Mashhour et al. [12] introduced and studied the concept of preopen sets. Raychaudhuri and Mukherjee [15] introduced the notions of δ-preopen sets and δ- preclosure. The class of δ-preopen sets is larger than that of preopen sets. Caldas et al. [5] ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4736 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), napassanan.sri@msu.ac.th (N. Srisarakham) https://www.ejpam.com 2581 © 2023 EJPAM All rights reserved. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (4) (2023), 2581-2596 2582 introduced some weak separation axioms by utilizing the notions of δ-preopen sets and the δ-preclosure operator. Buadong et al. [2] introduced and studied some separation axioms in generalized topology and minimal structure spaces. Dungthaisong et al. [8] investigated some properties of pairwise µ-T 1 2 -spaces. Torton et al. [18] introduced and studied the notions of µ(m,n)-regular spaces and µ(m,n)-normal spaces. Viriyapong and Boonpok [20] defined and investigated the notion of generalized (Λ, p)-closed sets in topological spaces. In [3], the present authors introduced and investigated the concept of (Λ, s)-closed sets by utilizing the notions of Λs-sets and semi-closed sets. In this paper, we introduce the concept of generalized δp(Λ, s)-closed sets. Moreover, some properties of generalized δp(Λ, s)-closed sets are discussed. In particular, we give several characterizations of δp(Λ, s)-T 1 2 -spaces and δp(Λ, s)-normal spaces by utilizing the concept of generalized δp(Λ, s)-closed sets. 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. Let A be a subset of a topological space (X, τ). The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. A subset A of a topological space (X, τ) is called semi-open [10] if A ⊆ Cl(Int(A)). The complement of a semi-open set is called semi- closed. The family of all semi-open (resp. semi-closed) sets in a topological space (X, τ) is denoted by SO(X, τ) (resp. SC(X, τ)). A subset AΛs [4] (resp. AVs) is defined as follows: AΛs = ∩{U | U ⊇ A, U ∈ SO(X, τ)} (resp. AVs = ∪{F | F ⊆ A, F ∈ SC(X, τ)}). A subset A of a topological space (X, τ) is called a Λs-set (resp. Vs-set) [4] if A = AΛs (resp. A = AVs). A subset A of a topological space (X, τ) is called (Λ, s)-closed [3] if A = T ∩C, where T is a Λs-set and C is a semi-closed set. The complement of a (Λ, s)-closed set is called (Λ, s)-open. The family of all (Λ, s)-closed (resp. (Λ, s)-open) sets in a topological space (X, τ) is denoted by ΛsC(X, τ) (resp. ΛsO(X, τ)). Let A be a subset of a topological space (X, τ). A point x ∈ X is called a (Λ, s)-cluster point [3] of A if for every (Λ, s)-open set U of X containing x we have A ∩ U ̸= ∅. The set of all (Λ, s)-cluster points of A is called the (Λ, s)-closure [3] of A and is denoted by A(Λ,s). The union of all (Λ, s)-open sets contained in A is called the (Λ, s)-interior [3] of A and is denoted by A(Λ,s). Let A be a subset of a topological space (X, τ). A point x of X is called a δ(Λ, s)-cluster point [16] of A if A ∩ [V (Λ,s)](Λ,s) ̸= ∅ for every (Λ, s)-open set V of X containing x. The set of all δ(Λ, s)-cluster points of A is called the δ(Λ, s)-closure [16] of A and is denoted by Aδ(Λ,s). If A = Aδ(Λ,s), then A is said to be δ(Λ, s)-closed [16]. The complement of a δ(Λ, s)-closed set is said to be δ(Λ, s)-open. The union of all δ(Λ, s)-open sets contained in A is called the δ(Λ, s)-interior [16] of A and is denoted by Aδ(Λ,s). Definition 1. [16] A subset A of a topological space (X, τ) is said to be δp(Λ, s)-open if A ⊆ [A(Λ,s)]δ(Λ,s). The complement of a δp(Λ, s)-open set is said to be δp(Λ, s)-closed. The family of all δp(Λ, s)-open (resp. δp(Λ, s)-closed) sets in a topological space (X, τ) is denoted by δp(Λ, s)O(X, τ) (resp. δp(Λ, s)C(X, τ)). Let A be a subset of a topological C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (4) (2023), 2581-2596 2583 space (X, τ). The intersection of all δp(Λ, s)-closed sets containing A is called the δp(Λ, s)- closure of A and is denoted by Aδp(Λ,s). Lemma 1. [16] For the δp(Λ, s)-closure of subsets A, B in a topological space (X, τ), the following properties hold: (1) If A ⊆ B, then Aδp(Λ,s) ⊆ Bδp(Λ,s). (2) A is δp(Λ, s)-closed in (X, τ) if and only if A = Aδp(Λ,s). (3) Aδp(Λ,s) is δp(Λ, s)-closed, that is, Aδp(Λ,s) = [Aδp(Λ,s)]δp(Λ,s). (4) x ∈ Aδp(Λ,s) if and only if A ∩ V ̸= ∅ for every V ∈ δp(Λ, s)O(X, τ) containing x. Lemma 2. [16] For a family {Aγ | γ ∈ ∇} of a topological space (X, τ), the following properties hold: (1) [∩{Aγ | γ ∈ ∇}]δp(Λ,s) ⊆ ∩{Aδp(Λ,s) γ | γ ∈ ∇}. (2) [∪{Aγ | γ ∈ ∇}]δp(Λ,s) ⊇ ∪{Aδp(Λ,s) γ | γ ∈ ∇}. Definition 2. Let A be a subset of a topological space (X, τ). The union of all (Λ, sp)-open sets contained in A is called the δp(Λ, s)-interior of A and is denoted by Aδp(Λ,s). Lemma 3. For subsets A and B of a topological space (X, τ), the following properties hold: (1) Aδp(Λ,s) ⊆ A and [Aδp(Λ,s)]δp(Λ,s) = Aδp(Λ,s). (2) If A ⊆ B, then Aδp(Λ,s) ⊆ Bδp(Λ,s). (3) Aδp(Λ,s) is δp(Λ, s)-open. (4) A is δp(Λ, s)-open if and only if Aδp(Λ,s) = A. (5) [X −A]δp(Λ,s) = X −Aδp(Λ,s). (6) [X −A]δp(Λ,s) = X −Aδp(Λ,s). 3. Generalized δp(Λ, s)-closed sets We begin this section by introducing the concept of generalized δp(Λ, s)-closed sets. Definition 3. A subset A of a topological space (X, τ) is said to be generalized δp(Λ, s)- closed (briefly, g-δp(Λ, s)-closed) if Aδp(Λ,s) ⊆ U whenever A ⊆ U and U is δp(Λ, s)-open in (X, τ). The complement of a generalized δp(Λ, s)-closed set is said to be generalized δp(Λ, s)-open (briefly, g-δp(Λ, s)-open). Theorem 1. A subset A of a topological space (X, τ) is g-δp(Λ, s)-closed if and only if Aδp(Λ,s) −A contains no nonempty δp(Λ, s)-closed set. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (4) (2023), 2581-2596 2584 Proof. Let F be a δp(Λ, s)-closed subset of Aδp(Λ,s) − A. Since A ⊆ X − F and A is g-δp(Λ, s)-closed, Aδp(Λ,s) ⊆ X − F and hence F ⊆ X −Aδp(Λ,s). Thus, F ⊆ Aδp(Λ,s) ∩ [X −Aδp(Λ,s)] = ∅ and F is empty. Conversely, suppose that A ⊆ U and U is δp(Λ, s)-open. If Aδp(Λ,s) ⊈ U , then Aδp(Λ,s) ∩ (X − U) is a nonempty δp(Λ, s)-closed subset of Aδp(Λ,s) −A. Corollary 1. Let A be a g-δp(Λ, s)-closed subset of a topological space (X, τ). Then, A is δp(Λ, s)-closed if and only if Aδp(Λ,s) −A is δp(Λ, s)-closed. Proof. If A is a δp(Λ, s)-closed set, then Aδp(Λ,s) −A = ∅. Conversely, suppose that Aδp(Λ,s) − A is δp(Λ, s)-closed. Since A is g-δp(Λ, s)-closed and Aδp(Λ,s) − A is a δp(Λ, s)-closed subset of itself, by Theorem 1, Aδp(Λ,s) − A = ∅ and hence Aδp(Λ,s) = A. Theorem 2. For a subset A of a topological space (X, τ), the following properties hold: (1) If A is δp(Λ, s)-closed, then A is g-δp(Λ, s)-closed. (2) If A is g-δp(Λ, s)-closed and δp(Λ, s)-open, then A is δp(Λ, s)-closed. (3) If A is g-δp(Λ, s)-closed and A ⊆ B ⊆ Aδp(Λ,s), then B is g-δp(Λ, s)-closed. Proof. (1) Let A be δp(Λ, s)-closed and A ⊆ U ∈ δp(Λ, s)O(X, τ). Then, by Lemma 1, Aδp(Λ,s) = A ⊆ U and hence A is g-δp(Λ, s)-closed. (2) Let A be g-δp(Λ, s)-closed and δp(Λ, s)-open. Then, Aδp(Λ,s) = A and by Lemma 1, A is δp(Λ, s)-closed. (3) Let B ⊆ U and U ∈ δp(Λ, s)O(X, τ). Since A ⊆ U and A is g-δp(Λ, s)-closed, we have Aδp(Λ,s) ⊆ U . Since A ⊆ B ⊆ Aδp(Λ,s), by Lemma 1, Aδp(Λ,s) = Bδp(Λ,s) and hence Bδp(Λ,s) ⊆ U . Thus, B is g-δp(Λ, s)-closed. Corollary 2. For a subset A of a topological space (X, τ), the following properties hold: (1) If A is δp(Λ, s)-open, then A is g-δp(Λ, s)-open. (2) If A is g-δp(Λ, s)-open and δp(Λ, s)-closed, then A is δp(Λ, s)-open. (3) If A is g-δp(Λ, s)-open and Aδp(Λ,s) ⊆ B ⊆ A, then B is g-δp(Λ, s)-open. Proof. This follows from Theorem 2. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (4) (2023), 2581-2596 2585 Definition 4. Let A be a subset of a topological space (X, τ). The δp(Λ, s)-frontier of A, δp(Λ, s)Fr(A), is defined as follows: δp(Λ, s)Fr(A) = Aδp(Λ,s) ∩ [X −A]δp(Λ,s). Theorem 3. Let A be a subset of a topological space (X, τ). If A is g-δp(Λ, s)-closed and A ⊆ V ∈ δp(Λ, s)O(X, τ), then δp(Λ, s)Fr(V ) ⊆ [X −A]δp(Λ,s). Proof. Let A be g-δp(Λ, s)-closed and A ⊆ V ∈ δp(Λ, s)O(X, τ). Then, Aδp(Λ,s) ⊆ V . Let x ∈ δp(Λ, s)Fr(V ). Since V ∈ δp(Λ, s)O(X, τ), we have δp(Λ, s)Fr(V ) = V δp(Λ,s)−V . Thus, x ̸∈ V and hence x ̸∈ Aδp(Λ,s). Therefore, x ∈ [X − A]δp(Λ,s). This shows that δp(Λ, s)Fr(V ) ⊆ [X −A]δp(Λ,s). Theorem 4. Let (X, τ) be a topological space. For each x ∈ X, either {x} is δp(Λ, s)- closed or g-δp(Λ, s)-open. Proof. Suppose that {x} is not δp(Λ, s)-closed. Then, X−{x} is not δp(Λ, s)-open and the only δp(Λ, s)-open set containing X−{x} is X itself. Therefore, [X−{x}]δp(Λ,s) ⊆ X. Thus, X − {x} is g-δp(Λ, s)-closed and hence {x} is g-δp(Λ, s)-open. Theorem 5. Let A be a subset of a topological space (X, τ). Then, A is g-δp(Λ, s)-open if and only if F ⊆ Aδp(Λ,s) whenever F ⊆ A and F is δp(Λ, s)-closed. Proof. Suppose that A is a g-δp(Λ, s)-open set. Let F be a δp(Λ, s)-closed set and F ⊆ A. Then, X − A ⊆ X − F ∈ δp(Λ, s)O(X, τ) and X − A is g-δp(Λ, s)-closed. Thus, X −Aδp(Λ,s) = [X −A]δp(Λ,s) ⊆ X − F and hence F ⊆ Aδp(Λ,s). Conversely, let X −A ⊆ U and U ∈ δp(Λ, s)O(X, τ). Then, X −U ⊆ A and X −U is δp(Λ, s)-closed. By the hypothesis, X − U ⊆ Aδp(Λ,s) and hence [X −A]δp(Λ,s) = X −Aδp(Λ,s) ⊆ U. Thus, X −A is g-δp(Λ, s)-closed. This shows that A is g-δp(Λ, s)-open. Lemma 4. Let A be a subset of a topological space (X, τ). If G ∈ δp(Λ, s)O(X, τ) and A ∩G = ∅, then Aδp(Λ,s) ∩G = ∅. Theorem 6. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A is g-δp(Λ, s)-closed. (2) Aδp(Λ,s) −A contains no nonempty δp(Λ, s)-closed set. (3) Aδp(Λ,s) −A is g-δp(Λ, s)-open. Proof. (1) ⇒ (2): This follows from Theorem 1. (2) ⇒ (3): Let F be a δp(Λ, s)-closed set and F ⊆ Aδp(Λ,s) − A. By (2), we have F = ∅ and F ⊆ [Aδp(Λ,s) − A]δp(Λ,s). It follows from Theorem 5 that Aδp(Λ,s) − A is g-δp(Λ, s)-open. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (4) (2023), 2581-2596 2586 (3) ⇒ (1): Suppose that A ⊆ U and U ∈ δp(Λ, s)O(X, τ). Then, Aδp(Λ,s) − U ⊆ Aδp(Λ,s) −A. By (3), we have Aδp(Λ,s) − A is g-δp(Λ, s)-open. Since Aδp(Λ,s) − U is δp(Λ, s)-closed, by Theorem 5, Aδp(Λ,s) − U ⊆ [Aδp(Λ,s) − A]δp(Λ,s) = ∅. Thus, Aδp(Λ,s) ⊆ U and hence A is g-δp(Λ, s)-closed. Now, the proof of [Aδp(Λ,s) − A]δp(Λ,s) = ∅ is given as follows. Suppose that [Aδp(Λ,s) − A]δp(Λ,s) ̸= ∅. Then, there exists x ∈ [Aδp(Λ,s) − A]δp(Λ,s) and hence there exists G ∈ δp(Λ, s)O(X, τ) such that x ∈ G ⊆ Aδp(Λ,s) − A. Since G ⊆ X − A, we have G ∩ A = ∅, by Lemma 4, G ∩ Aδp(Λ,s) = ∅ and hence G ⊆ X − Aδp(Λ,s). Thus, G ⊆ [X −Aδp(Λ,s)] ∩Aδp(Λ,s) = ∅. This is a contradiction. Theorem 7. A subset A of a topological space (X, τ) is g-δp(Λ, s)-closed if and only if F ∩Aδp(Λ,s) = ∅ whenever A ∩ F = ∅ and F is δp(Λ, s)-closed. Proof. Suppose that A is a δp(Λ, s)-closed set. Let F be a δp(Λ, s)-closed set and A ∩ F = ∅. Then, A ⊆ X − F ∈ δp(Λ, s)O(X, τ) and Aδp(Λ,s) ⊆ X − F . Thus, F ∩Aδp(Λ,s) = ∅. Conversely, let A ⊆ U and U ∈ δp(Λ, s)O(X, τ). Then, A∩ (X −U) = ∅ and X −U is δp(Λ, s)-closed. By the hypothesis, (X −U) ∩Aδp(Λ,s) = ∅ and hence Aδp(Λ,s) ⊆ U . Thus, A is g-δp(Λ, s)-closed. Theorem 8. A subset A of a topological space (X, τ) is g-δp(Λ, s)-closed if and only if A ∩ {x}δp(Λ,s) ̸= ∅ for every x ∈ Aδp(Λ,s). Proof. Let A be a g-δp(Λ, s)-closed set and suppose that there exists x ∈ Aδp(Λ,s) such that A ∩ {x}δp(Λ,s) = ∅. Thus, A ⊆ X − {x}δp(Λ,s) and hence Aδp(Λ,s) ⊆ X − {x}δp(Λ,s). Therefore, x ̸∈ Aδp(Λ,s), which is a contradiction. Conversely, suppose that the condition of the theorem holds and let U be any δp(Λ, s)- open set containing A. Let x ∈ Aδp(Λ,s). By the hypothesis, A ∩ Aδp(Λ,s) ̸= ∅, so there exists y ∈ A ∩ {x}δp(Λ,s) and hence y ∈ A ⊆ U . Thus, {x} ∩ U ̸= ∅. Therefore, x ∈ U , which implies that Aδp(Λ,s) ⊆ U . This shows that A is g-δp(Λ, s)-closed. Corollary 3. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A is g-δp(Λ, s)-open. (2) A−Aδp(Λ,s) does not contain any nonempty δp(Λ, s)-closed set. (3) (X −A) ∩ {x}δp(Λ,s) ̸= ∅ for every x ∈ A−Aδp(Λ,s). Theorem 9. For a topological space (X, τ), the following properties are equivalent: C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (4) (2023), 2581-2596 2587 (1) For every δp(Λ, s)-open set U of X, U δp(Λ,s) ⊆ U . (2) Every subset of X is g-δp(Λ, s)-closed. Proof. (1) ⇒ (2): Let A be any subset of X and A ⊆ U ∈ δ(Λ, s)O(X, τ). By (1), U δp(Λ,s) ⊆ U and hence Aδp(Λ,s) ⊆ U δp(Λ,s) ⊆ U . Thus, A is g-δp(Λ, s)-closed. (2) ⇒ (1): Let U ∈ δp(Λ, s)O(X, τ). By (2), U is g-δp(Λ, s)-closed and hence U δp(Λ,s) ⊆ U. Theorem 10. A subset A of a topological space (X, τ) is g-δp(Λ, s)-open if and only if U = X whenever U is δp(Λ, s)-open and (X −A) ∩Aδp(Λ,s) ⊆ U . Proof. Suppose that A is g-δp(Λ, s)-open and U ∈ δp(Λ, s)O(X, τ) such that (X −A) ∩Aδp(Λ,s) ⊆ U. Thus, X − U ⊆ [X − Aδp(Λ,s)] ∩ A and hence X − U ⊆ [X − A]δp(Λ,s) − (X − A). Since X −A is g-δp(Λ, s)-closed and X − U is δp(Λ, s)-closed, by Theorem 1, X − U = ∅. This shows that X = U . Conversely, suppose that F ⊆ A and F is δp(Λ, s)-closed. By Lemma 3, (X −A) ∪Aδp(Λ,s) ⊆ (X − F ) ∪Aδp(Λ,s) ∈ δp(Λ, s)O(X, τ). By the hypothesis, we have X = (X − F ) ∪Aδp(Λ,s) and hence F = F ∩ [(X − F ) ∪Aδp(Λ,s)] = F ∩Aδp(Λ,s) ⊆ Aδp(Λ,s). It follows from Theorem 5 that A is g-δp(Λ, s)-open. Theorem 11. Let A be a subset of a topological space (X, τ). If A is g-δp(Λ, s)-open and Aδp(Λ,s) ⊆ B ⊆ A, then B is g-δp(Λ, s)-open. Proof. We have X − A ⊆ X − B ⊆ X − Aδp(Λ,s) = [X − A]δp(Λ,s). Since X − A is g-δp(Λ, s)-closed, it follows from Theorem 2 that X −B is g-δp(Λ, s)-closed and hence B is g-δp(Λ, s)-open. Definition 5. A subset A of a topological space (X, τ) is said to be locally δp(Λ, s)-closed if A = U ∩ F , where U ∈ δp(Λ, s)O(X, τ) and F is a δp(Λ, s)-closed set. Lemma 5. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A is locally δp(Λ, s)-closed; (2) A = U ∩Aδp(Λ,s) for some U ∈ δp(Λ, s)O(X, τ); C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (4) (2023), 2581-2596 2588 (3) Aδp(Λ,s) −A is δp(Λ, s)-closed; (4) [A ∪ (X −Aδp(Λ,s))] ∈ δp(Λ, s)O(X, τ); (5) A ⊆ [A ∪ [X −Aδp(Λ,s)]]δp(Λ,s). Proof. (1) ⇒ (2): Let A = U ∩F , where U ∈ δp(Λ, s)O(X, τ) and F is δp(Λ, s)-closed. Since A ⊆ F , we have Aδp(Λ,s) ⊆ F δp(Λ,s) = F . Since A ⊆ U , A ⊆ U∩Aδp(Λ,s) ⊆ U∩F = A. Thus, A = U ∩Aδp(Λ,s). (2) ⇒ (3): Suppose that A = U ∩ Aδp(Λ,s) for some U ∈ δp(Λ, s)O(X, τ). Then, we have Aδp(Λ,s)−A = (X− [U ∩Aδp(Λ,s)])∩Aδp(Λ,s) = (X−U)∩Aδp(Λ,s). Thus, Aδp(Λ,s)−A is δp(Λ, s)-closed. (3) ⇒ (4): Since X− (Aδp(Λ,s)−A) = (X−Aδp(Λ,s))∪A and by (3), A∪ (X−Aδp(Λ,s)) is δp(Λ, s)-open. (4) ⇒ (5): By (4), A ⊆ A ∪ (X −Aδp(Λ,s)) = [A ∪ (X −Aδp(Λ,s))]δp(Λ,s). (5) ⇒ (1): We put U = [A ∪ (X − Aδp(Λ,s))]δp(Λ,s). Then, U is δp(Λ, s)-open and A = A ∩ U ⊆ U ∩ Aδp(Λ,s) ⊆ [A ∪ (X − Aδp(Λ,s))] ∩ Aδp(Λ,s) = A ∩ Aδp(Λ,s) = A. Thus, A = U ∩ Aδp(Λ,s), where U ∈ δp(Λ, s)O(X, τ) and Aδp(Λ,s) is δp(Λ, s)-closed. This shows that A is locally δp(Λ, s)-closed. Theorem 12. A subset A of a topological space (X, τ) is δp(Λ, s)-closed if and only if A is locally δp(Λ, s)-closed and g-δp(Λ, s)-closed. Proof. Let A be a δp(Λ, s)-closed set. By Theorem 2, A is g-δp(Λ, s)-closed. Since X is δp(Λ, s)-open and A = X ∩A, A is locally δp(Λ, s)-closed. Conversely, suppose that A is locally δp(Λ, s)-closed and g-δp(Λ, s)-closed. Since A is locally δp(Λ, s)-closed, by Lemma 5, A ⊆ [A ∪ [X −Aδp(Λ,s)]]δp(Λ,s). Since [A ∪ [X −Aδp(Λ,s)]]δp(Λ,s) ∈ δp(Λ, s)O(X, τ) and A is g-δp(Λ, s)-closed, we have Aδp(Λ,s) ⊆ [A∪ [X−Aδp(Λ,s)]]δp(Λ,s) ⊆ A∪ [X−Aδp(Λ,s)] and hence Aδp(Λ,s) = A. Thus, by Lemma 1, A is δp(Λ, s)-closed. Definition 6. [17] Let A be a subset of a topological space (X, τ). A subset δp(Λ, s)Ker(A) is defined as follows: δp(Λ, s)Ker(A) = ∩{U | A ⊆ U,U ∈ δp(Λ, s)O(X, τ)}. Lemma 6. For subsets A,B of a topological space (X, τ), the following properties hold: (1) A ⊆ δp(Λ, s)Ker(A). (2) If A ⊆ B, then δp(Λ, s)Ker(A) ⊆ δp(Λ, s)Ker(B). (3) δp(Λ, s)Ker[δp(Λ, s)Ker(A)] = δp(Λ, s)Ker(A). (4) If A is δp(Λ, s)-open, δp(Λ, s)Ker(A) = A. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (4) (2023), 2581-2596 2589 A subset Nx of a topological space (X, τ) is said to be a δp(Λ, s)-neighbourhood [16] of a point x ∈ X if there exists a δp(Λ, s)-open set U such that x ∈ U ⊆ Nx. Lemma 7. A subset A of a topological space (X, τ) is δp(Λ, s)-open in X if and only if A is a δp(Λ, s)-neighbourhood of each point of A. Definition 7. Let (X, τ) be a topological space and x ∈ X. A subset ⟨x⟩δp(Λ,s) is defined as follows: ⟨x⟩δp(Λ,s) = δp(Λ, s)Ker({x}) ∩ {x}δp(Λ,s). Theorem 13. For a topological space (X, τ), the following properties hold: (1) Λδp(Λ,s)(A) = {x ∈ X | A ∩ {x}δp(Λ,s) ̸= ∅} for each subset A of X. (2) For each x ∈ X, δp(Λ, s)Ker(⟨x⟩δp(Λ,s)) = δp(Λ, s)Ker({x}). (3) For each x ∈ X, (⟨x⟩δp(Λ,s))δp(Λ,s) = {x}δp(Λ,s). (4) If U is δp(Λ, s)-open in X and x ∈ U , then ⟨x⟩δp(Λ,s) ⊆ U . (5) If F is δp(Λ, s)-closed in X and x ∈ F , then ⟨x⟩δp(Λ,s) ⊆ F . Proof. (1) Suppose that A ∩ {x}δp(Λ,s) = ∅. Then, x ̸∈ X − {x}δp(Λ,s) which is a δp(Λ, s)-open set containing A. Thus, x ̸∈ δp(Λ, s)Ker(A) and hence δp(Λ, s)Ker(A) ⊆ {x ∈ X | A ∩ {x}δp(Λ,s) ̸= ∅}. Next, let x ∈ X such that A∩{x}δp(Λ,s) ̸= ∅ and suppose that x ̸∈ δp(Λ, s)Ker(A). Then, there exists a δp(Λ, s)-open set U containing A and x ̸∈ U . Let y ∈ A∩ {x}δp(Λ,s). There- fore, U is a δp(Λ, s)-neighbourhood of y which does not contain x. By this contradiction x ∈ δp(Λ, s)Ker(A). (2) Let x ∈ X. Then, we have {x} ⊆ {x}δp(Λ,s) ∩ δp(Λ, s)Ker({x}) = ⟨x⟩δp(Λ,s). By Lemma 6, we obtain δp(Λ, s)Ker({x}) ⊆ δp(Λ, s)Ker(⟨x⟩δp(Λ,s)). Next, we show the opposite implication. Suppose that y ̸∈ δp(Λ, s)Ker({x}). Then, there exists a δp(Λ, s)- open set V such that x ∈ V and y ̸∈ V . Since ⟨x⟩δp(Λ,s) ⊆ δp(Λ, s)Ker({x}) ⊆ δp(Λ, s)Ker(V ) = V, we have δp(Λ, s)Ker(⟨x⟩δp(Λ,s)) ⊆ V . Since y ̸∈ V , y ̸∈ δp(Λ, s)Ker(⟨x⟩δp(Λ,s)). Thus, δp(Λ, s)Ker(⟨x⟩δp(Λ,s)) ⊆ δp(Λ, s)Ker({x}) and hence δp(Λ, s)Ker({x}) = δp(Λ, s)Ker(⟨x⟩δp(Λ,s)). (3) By the definition of ⟨x⟩δp(Λ,s), we have {x} ⊆ ⟨x⟩δp(Λ,s) and {x}δp(Λ,s) ⊆ (⟨x⟩δp(Λ,s))δp(Λ,s) by Lemma 1. On the other hand, we have ⟨x⟩δp(Λ,s) ⊆ {x}δp(Λ,s) and (⟨x⟩δp(Λ,s))δp(Λ,s) ⊆ ({x}δp(Λ,s))δp(Λ,s) = {x}δp(Λ,s). C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (4) (2023), 2581-2596 2590 Thus, (⟨x⟩δp(Λ,s))δp(Λ,s) ⊆ {x}δp(Λ,s). (4) Since x ∈ U and U is a δp(Λ, s)-open set, we have δp(Λ, s)Ker({x}) ⊆ U . Thus, ⟨x⟩δp(Λ,s) ⊆ U . (5) Since x ∈ F and F is a δp(Λ, s)-closed set, ⟨x⟩δp(Λ,s) = {x}δp(Λ,s) ∩ δp(Λ, s)Ker({x}) ⊆ {x}δp(Λ,s) ⊆ F δp(Λ,s) = F. Theorem 14. For any points x and y in a topological space (X, τ), the following properties are equivalent: (1) δp(Λ, s)Ker({x}) ̸= δp(Λ, s)Ker({y}). (2) {x}δp(Λ,s) ̸= {y}δp(Λ,s). Proof. (1) ⇒ (2): Suppose that δp(Λ, s)Ker({x}) ̸= δp(Λ, s)Ker({y}). Then, there exists a point z ∈ X such that z ∈ δp(Λ, s)Ker({x}) and z ̸∈ δp(Λ, s)Ker({y}) or z ∈ δp(Λ, s)Ker({y}) and z ̸∈ δp(Λ, s)Ker({x}). We prove only the first case being the second analogous. From z ∈ δp(Λ, s)Ker({x}) it follows that {x} ∩ {z}δp(Λ,s) ̸= ∅ which implies x ∈ {z}δp(Λ,s). By z ̸∈ δp(Λ, s)Ker({y}), we have {y} ∩ {z}δp(Λ,s) = ∅. Since x ∈ {z}δp(Λ,s), {x}δp(Λ,s) ⊆ {z}δp(Λ,s) and {y} ∩ {x}δp(Λ,s) = ∅. Therefore, {x}δp(Λ,s) ̸= {y}δp(Λ,s). Thus, δp(Λ, s)Ker({x}) ̸= δp(Λ, s)Ker({y}) implies that {x}δp(Λ,s) ̸= {y}δp(Λ,s). (2) ⇒ (1): Suppose that {x}δp(Λ,s) ̸= {y}δp(Λ,s). There exists a point z ∈ X such that z ∈ {x}δp(Λ,s) and z ̸∈ {y}δp(Λ,s) or z ∈ {y}δp(Λ,s) and z ̸∈ {x}δp(Λ,s). We prove only the first case being the second analogous. It follows that there exists a δp(Λ, s)- open set containing z and therefore x but not y, namely, y ̸∈ δp(Λ, s)Ker({x}) and thus δp(Λ, s)Ker({x}) ̸= δp(Λ, s)Ker({y}). Theorem 15. Let (X, τ) be a topological space and x, y ∈ X. Then, the following prop- erties hold: (1) y ∈ δp(Λ, s)Ker({x}) if and only if x ∈ {y}δp(Λ,s). (2) δp(Λ, s)Ker({x}) = δp(Λ, s)Ker({y}) if and only if {x}δp(Λ,s) = {y}δp(Λ,s). C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (4) (2023), 2581-2596 2591 Proof. (1) Let x ̸∈ {y}δp(Λ,s). Then, there exists U ∈ δp(Λ, s)O(X, τ) such that x ∈ U and y ̸∈ U . Thus, y ̸∈ δp(Λ, s)Ker({x}). The converse is similarly shown. (2) Suppose that δp(Λ, s)Ker({x}) = δp(Λ, s)Ker({y}) for any x, y ∈ X. Since x ∈ δp(Λ, s)Ker({x}), x ∈ δp(Λ, s)Ker({y}), by (1), y ∈ {x}δp(Λ,s). By Lemma 1, {y}δp(Λ,s) ⊆ {x}δp(Λ,s). Similarly, we have {x}δp(Λ,s) ⊆ {y}δp(Λ,s) and hence {x}δp(Λ,s) = {y}δp(Λ,s). Conversely, suppose that {x}δp(Λ,s) = {y}δp(Λ,s). Since x ∈ {x}δp(Λ,s), x ∈ {y}δp(Λ,s) and by (1), y ∈ δp(Λ, s)Ker({x}). By Lemma 6, δp(Λ, s)Ker({y}) ⊆ δp(Λ, s)Ker(δp(Λ, s)Ker({x})) = δp(Λ, s)Ker({x}). Similarly, we have δp(Λ, s)Ker({x}) ⊆ δp(Λ, s)Ker({y}) and hence δp(Λ, s)Ker({x}) = δp(Λ, s)Ker({y}). Definition 8. A subset A of a topological space (X, τ) is called a Λδp(Λ,s)-set if A = δp(Λ, s)Ker(A). The family of all Λδp(Λ,s)-sets of a topological space (X, τ) is denoted by Λδp(Λ,s)(X, τ) (or simply Λδp(Λ,s)). Definition 9. A subset A of a topological space (X, τ) is called a generalized Λδp(Λ,s)-set (briefly g-Λδp(Λ,s)-set) if δp(Λ, s)Ker(A) ⊆ F whenever A ⊆ F and F is a δp(Λ, s)-closed set. Definition 10. A topological space (X, τ) is called a δp(Λ, s)-T 1 2 -space if every g-δp(Λ, s)- closed set of X is δp(Λ, s)-closed. Lemma 8. For a topological space (X, τ), the following properties hold: (1) For each x ∈ X, the singleton {x} is δp(Λ, s)-closed or X −{x} is g-δp(Λ, s)-closed. (2) For each x ∈ X, the singleton {x} is δp(Λ, s)-open or X − {x} is a g-Λδp(Λ,s)-set. Proof. (1) Let x ∈ X and the singleton {x} be not δp(Λ, s)-closed. Then, X − {x} is not δp(Λ, s)-open and X is the only δp(Λ, s)-open set which contains X − {x} and hence X − {x} is g-δp(Λ, s)-closed. (2) Let x ∈ X and the singleton {x} be not δp(Λ, s)-open. Then, X − {x} is not δp(Λ, s)-closed and X is the only δp(Λ, s)-closed set which contains X − {x} and hence X − {x} is a g-Λδp(Λ,s)-set. Theorem 16. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is a δp(Λ, s)-T 1 2 -space. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (4) (2023), 2581-2596 2592 (2) For each x ∈ X, the singleton {x} is δp(Λ, s)-open or δp(Λ, s)-closed. (3) Every g-Λδp(Λ,s)-set is a Λδp(Λ,s)-set. Proof. (1) ⇒ (2): By Lemma 8, for each x ∈ X, the singleton {x} is δp(Λ, s)-closed or X−{x} is g-δp(Λ, s)-closed. Since (X, τ) is a δp(Λ, s)-T 1 2 -space, X−{x} is δp(Λ, s)-closed and hence {x} is δp(Λ, s)-open in the latter case. Thus, the singleton {x} is δp(Λ, s)-open or δp(Λ, s)-closed. (2) ⇒ (3): Suppose that there exists a g-Λδp(Λ,s)-set A which is not a Λδp(Λ,s)-set. There exists x ∈ δp(Λ, s)Ker(A) such that x ̸∈ A. In case the singleton {x} is δp(Λ, s)-open, A ⊆ X − {x} and X − {x} is δp(Λ, s)-closed. Since A is a g-Λδp(Λ,s)-set, δp(Λ, s)Ker(A) ⊆ X − {x}. This is a contradiction. In case the singleton {x} is δp(Λ, s)-closed, A ⊆ X − {x} and X − {x} is δp(Λ, s)-open. By Lemma 6, δp(Λ, s)Ker(A) ⊆ δp(Λ, s)Ker(X − {x}) = X − {x}. This is a contradiction. Thus, every g-Λδp(Λ,s)-set is a Λδp(Λ,s)-set. (3) ⇒ (1): Suppose that (X, τ) is not a δp(Λ, s)-T 1 2 -space. Then, there exists a g- δp(Λ, s)-closed set A which is not δp(Λ, s)-closed. Since A is not δp(Λ, s)-closed, there exists a point x ∈ Aδp(Λ,s) such that x ̸∈ A. By Lemma 8, the singleton {x} is δp(Λ, s)- open or X − {x} is a Λδp(Λ,s)-set. (a) In case {x} is δp(Λ, s)-open, since x ∈ Aδp(Λ,s), {x} ∩ A ̸= ∅ and x ∈ A. This is a contradiction. (b) In case X − {x} is a Λδp(Λ,s)-set, if {x} is not δp(Λ, s)-closed, X − {x} is not δp(Λ, s)-open and δp(Λ, s)Ker(X − {x}) = X. Thus, X − {x} is not a Λδp(Λ,s)-set. This contradicts (3). If {x} is δp(Λ, s)-closed, A ⊆ X − {x} ∈ δp(Λ, s)O(X, τ) and A is g-δp(Λ, s)-closed. Hence, we have Aδp(Λ,s) ⊆ X − {x}. This contradicts that x ∈ Aδp(Λ,s). This shows that (X, τ) is a δp(Λ, s)-T 1 2 -space. Definition 11. A topological space (X, τ) is said to be δp(Λ, s)-normal if for any pair of disjoint δp(Λ, s)-closed sets F and H, there exist disjoint δp(Λ, s)-open sets U and V such that F ⊆ U and H ⊆ V . Lemma 9. Let (X, τ) be a topological space. If U is δp(Λ, s)-open in X, then U δp(Λ,s) ∩A ⊆ [U ∩A]δp(Λ,s) for every subset A of X. Theorem 17. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is δp(Λ, s)-normal. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (4) (2023), 2581-2596 2593 (2) For every pair of δp(Λ, s)-open sets U and V whose union is X, there exist δp(Λ, s)- closed sets F and H such that F ⊆ U , H ⊆ V and F ∪H = X. (3) For every δp(Λ, s)-closed set F and every δp(Λ, s)-open set G containing F , there exists a δp(Λ, s)-open set U such that F ⊆ U ⊆ U δp(Λ,s) ⊆ G. (4) For every pair of disjoint δp(Λ, s)-closed sets F and H, there exist disjoint δp(Λ, s)- open sets U and V such that F ⊆ U and H ⊆ V and U δp(Λ,s) ∩ V δp(Λ,s) = ∅. Proof. (1) ⇒ (2): Let U and V be any pair of δp(Λ, s)-open sets in X such that X = U ∪ V . Then, X − U and X − V are disjoint δp(Λ, s)-closed sets. Since (X, τ) is δp(Λ, s)-normal, there exist disjoint δp(Λ, s)-open sets G and W such that X − U ⊆ G and X − V ⊆ W . Put F = X −G and H = X −W . Then, F and H are δp(Λ, s)-closed sets such that F ⊆ U , H ⊆ V and F ∪H = X. (2) ⇒ (3): Let F be a δp(Λ, s)-closed set and G be a δp(Λ, s)-open set containing F . Then, X − F and G are δp(Λ, s)-open sets whose union is X. Then by (2), there exist δp(Λ, s)-closed sets M and N such that M ⊆ X − F , N ⊆ G and M ∪ N = X. Then, F ⊆ X−M , X−G ⊆ X−N and (X−M)∩(X−N) = ∅. Put U = X−M and V = X−N . Then, U and V are disjoint δp(Λ, s)-open sets such that F ⊆ U ⊆ X − V ⊆ G. As X − V is a δp(Λ, s)-closed set, we have U δp(Λ,s) ⊆ X − V and F ⊆ U ⊆ U δp(Λ,s) ⊆ G. (3) ⇒ (4): Let F and H be two disjoint δp(Λ, s)-closed sets of X. Then, F ⊆ X −H and X − H is δp(Λ, s)-open, by (3), there exists a δp(Λ, s)-open set U of X such that F ⊆ U ⊆ U δp(Λ,s) ⊆ X − H. Put V = X − U δp(Λ,s). Then, U and V are disjoint δp(Λ, s)-open sets of X such that F ⊆ U , H ⊆ V and U δp(Λ,s) ∩ V δp(Λ,s) = ∅. (4) ⇒ (1): The proof is obvious. Theorem 18. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is δp(Λ, s)-normal. (2) For any pair of disjoint δp(Λ, s)-closed sets F and H, there exist disjoint g-δp(Λ, s)- open sets U and V such that F ⊆ U and H ⊆ V . (3) For each δp(Λ, s)-closed set F and each δp(Λ, s)-open set G containing F , there exists a g-δp(Λ, s)-open set U such that F ⊆ U ⊆ U δp(Λ,s) ⊆ G. (4) For each δp(Λ, s)-closed set F and each g-δp(Λ, s)-open set G containing F , there exists a δp(Λ, s)-open set U such that F ⊆ U ⊆ U δp(Λ,s) ⊆ Gδp(Λ,s). (5) For each δp(Λ, s)-closed set F and each g-δp(Λ, s)-open set G containing F , there exists a g-δp(Λ, s)-open set U such that F ⊆ U ⊆ U δp(Λ,s) ⊆ Gδp(Λ,s). (6) For each g-δp(Λ, s)-closed set F and each δp(Λ, s)-open set G containing F , there exists a δp(Λ, s)-open set U such that F δp(Λ,s) ⊆ U ⊆ U δp(Λ,s) ⊆ G. (7) For each g-δp(Λ, s)-closed set F and each δp(Λ, s)-open set G containing F , there exists a g-δp(Λ, s)-open set U such that F δp(Λ,s) ⊆ U ⊆ U δp(Λ,s) ⊆ G. REFERENCES 2594 Proof. (1) ⇒ (2): The proof is obvious. (2) ⇒ (3): Let F be a δp(Λ, s)-closed set and G be a δp(Λ, s)-open set containing F . Then, F and X − G are two disjoint δp(Λ, s)-closed sets. Hence by (2), there exist disjoint g-δp(Λ, s)-open sets U and V of X such that F ⊆ U and X − G ⊆ V . Since V is g-δp(Λ, s)-open and X − G is δp(Λ, s)-closed, by Theorem 5, X − G ⊆ Vδp(Λ,s). Thus, [X − V ]δp(Λ,s) = X − Vδp(Λ,s) ⊆ G and hence F ⊆ U ⊆ U δp(Λ,s) ⊆ G. (3) ⇒ (5): Let F be a δp(Λ, s)-closed set and G be a g-δp(Λ, s)-open set containing F . Since G is g-δp(Λ, s)-open and F is δp(Λ, s)-closed, by Theorem 5, F ⊆ Gδp(Λ,s). Thus, by (3), there exists a g-δp(Λ, s)-open set U such that F ⊆ U ⊆ U δp(Λ,s) ⊆ Gδp(Λ,s). (5) ⇒ (6): Let F be a g-δp(Λ, s)-closed set and G be a δp(Λ, s)-open set containing F . Then, we have F δp(Λ,s) ⊆ G. Since G is g-δp(Λ, s)-open and F δp(Λ,s) is δp(Λ, s)-closed, by (5), there exists a g-δp(Λ, s)-open set U such that F δp(Λ,s) ⊆ U ⊆ U δp(Λ,s) ⊆ G. Since U is g-δp(Λ, s)-open and F δp(Λ,s) ⊆ U , by Theorem 5, F δp(Λ,s) ⊆ Uδp(Λ,s). Put V = Uδp(Λ,s). Then, V is δp(Λ, s)-open and F δp(Λ,s) ⊆ V ⊆ V δp(Λ,s) = [Uδp(Λ,s)] δp(Λ,s) ⊆ U δp(Λ,s) ⊆ G. (6) ⇒ (4): Let F be a δp(Λ, s)-closed set and G be a g-δp(Λ, s)-open set containing F . Thus, by Theorem 5, F δp(Λ,s) = F ⊆ Gδp(Λ,s). Since F is g-δp(Λ, s)-closed and Gδp(Λ,s) is δp(Λ, s)-open, by (6), there exists a δp(Λ, s)-open set U such that F δp(Λ,s) ⊆ U ⊆ U δp(Λ,s) ⊆ Gδp(Λ,s). (4) ⇒ (5): The proof is obvious. (6) ⇒ (7) and (7) ⇒ (3): The proofs are obvious. (3) ⇒ (1): Let F and H be two disjoint δp(Λ, s)-closed sets of X. Then, F is a δp(Λ, s)-closed set and X − H is a δp(Λ, s)-open set containing F , by (3), there exists a g-δp(Λ, s)-open set U such that F ⊆ U ⊆ U δp(Λ,s) ⊆ X − H. 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