EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1434-1447 ISSN 1307-5543 – ejpam.com Published by New York Business Global Characterizations of some 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 is concerned with the concepts of some topological spaces. Firstly, we introduce the notions of δs(Λ, p)-open sets. Some properties concerning δs(Λ, p)-open sets are discussed. Secondly, the concept of s(Λ, p)-connected spaces is introduced. Moreover, we give several characterizations of s(Λ, p)-connected spaces by utilizing δs(Λ, p)-open sets. Thirdly, we apply the notion of s(Λ, p)-open sets to present and study new classes of spaces called s(Λ, p)-regular spaces and s(Λ, p)-normal spaces. Especially, some characterizations of s(Λ, p)-regular spaces and s(Λ, p)-normal spaces are established. Fourthly, we introduce and investigate the concepts of s(Λ, p)-T2 spaces and s(Λ, p)-Urysohn spaces. Finally, the notion of S(Λ, p)-closed spaces is studied. Basic properties and characterizations of S(Λ, p)-closed spaces are considered. 2020 Mathematics Subject Classifications: 54A05, 54D10 Key Words and Phrases: δs(Λ, p)-open set, s(Λ, p)-connected space, s(Λ, p)-regular space, s(Λ, p)-normal space, s(Λ, p)-T2 space, s(Λ, p)-Urysohn space, S(Λ, p)-closed space 1. Introduction In 1968, Veličko [14] introduced δ-open sets, which are stronger than open sets. In 1982, Mashhour et al. [9] introduced and investigated the notion of preopen sets which is weaker than the notion of open sets in topological spaces. In 1993, Raychaudhuri and Mukherjee [11] introduced and studied the notions of δ-preopen sets and δ-closures. The class of δ-preopen sets is larger than that of preopen sets. In 1996, Raychaudhuri and Mukherjee [12] introduced and investigated the concept of δp-closed spaces. In 2005, Caldas et al. [4] introduced some weak separation axioms by utilizing the notions of δ-preopen sets and the δ-preclosure operator. Caldas et al. [4] showed that (δ, p)-T1 spaces, (δ, p)-R0 spaces and (δ, p)-symmetric spaces are all equivalent. Moreover, Caldas et al. [6] investigated some weak separation axioms by utilizing δ-semiopen sets and the δ-semiclosure operator. Caldas et al. [5] 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 2011, Buadong et al. [1] introduced and investigated some separation axioms in generalized topology and minimal structure ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4737 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), montri.t@msu.ac.th (M. Thongmoon) https://www.ejpam.com 1434 © 2023 EJPAM All rights reserved. C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1434-1447 1435 spaces. Dungthaisong et al. [7] studied some properties of pairwise µ-T 1 2 -spaces. Torton et al. [13] introduced and investigated the notions of µ(m,n)-regular spaces and µ(m,n)- normal spaces. In [3], 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. This paper is organized as follows: in Section 2 is devoted to basic definitions and preliminaries. In Section 3, we introduce the notions of δs(Λ, p)-open sets and δs(Λ, p)- closed sets in topological spaces. Moreover, some characterizations of δs(Λ, p)-T0 spaces, δs(Λ, p)-T1 spaces and δs(Λ, p)-symmetric spaces are investigated. In Section 4, the notion of s(Λ, p)-connected spaces is introduced. Several characterizations of s(Λ, p)-connected spaces are obtained. In Section 5, we introduce the concepts of s(Λ, p)-regular spaces and s(Λ, p)-normal spaces. Furthermore, we give some characterizations of s(Λ, p)-regular spaces and s(Λ, p)-normal spaces by utilizing δs(Λ, p)-open sets. Basic properties and characterizations of s(Λ, p)-T2 spaces and s(Λ, p)-Urysohn spaces are discussed in Section 6. In the last Section 7, we define the notion of S(Λ, p)-closed spaces. Characterizations and properties concerning S(Λ, p)-closed spaces are considered. 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 said to be preopen [9] 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) [8] 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 [3] (pre-Λ-set [8]) if A = Λp(A). A subset A of a topological space (X, τ) is called (Λ, p)-closed [3] 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 [3] 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 [3] of A and is denoted by A(Λ,p). The union of all (Λ, p)-open sets of X contained in A is called the (Λ, p)-interior [3] of A and is denoted by A(Λ,p). A subset A of a topological space (X, τ) is said to be α(Λ, p)-open (resp. p(Λ, p)-open, s(Λ, p)- open, β(Λ, p)-open, r(Λ, p)-open [3]) if A ⊆ [[A(Λ,p)] (Λ,p)](Λ,p) (resp. A ⊆ [A(Λ,p)](Λ,p), A ⊆ [A(Λ,p)] (Λ,p), A ⊆ [[A(Λ,p)](Λ,p)] (Λ,p), A = [A(Λ,p)](Λ,p)). The family of all α(Λ, p)-open (resp. p(Λ, p)-open, s(Λ, p)-open, β(Λ, p)-open, r(Λ, p)-open) sets in a topological space (X, τ) is denoted by α(Λ, p)O(X, τ) (resp. p(Λ, p)O(X, τ), s(Λ, p)O(X, τ), β(Λ, p)O(X, τ), r(Λ, p)O(X, τ)). The complement of a p(Λ, p)-open (resp. s(Λ, p)-open, α(Λ, p)-open, β(Λ, p)-open, r(Λ, p)-open) set is said to be p(Λ, p)-closed (resp. s(Λ, p)-closed, α(Λ, p)- closed, β(Λ, p)-closed, r(Λ, p)-closed). Let A be a subset of a topological space (X, τ). The intersection of all s(Λ, p)-closed sets of X containing A is called the s(Λ, p)-closure C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1434-1447 1436 of A and is denoted by As(Λ,p). A point x of X is called a δ(Λ, p)-cluster point [2] 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 [2] of A and is denoted by Aδ(Λ,p). If A = Aδ(Λ,p), then A is said to be δ(Λ, p)-closed [2]. 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 of X contained in A is called the δ(Λ, p)-interior [2] of A and is denoted by Aδ(Λ,p). 3. δs(Λ, p)-open sets In this section, we introduce the notion of δs(Λ, p)-open sets. Moreover, some char- acterizations of δs(Λ, p)-T0 spaces, δs(Λ, p)-T1 spaces and δs(Λ, p)-symmetric spaces are discussed. Definition 1. A subset A of a topological space (X, τ) is said to be δs(Λ, p)-open if A ⊆ [A(Λ,p)] δ(Λ,p). The complement of a δs(Λ, p)-open set is said to be δs(Λ, p)-closed. The family of all δs(Λ, p)-open (resp. δs(Λ, p)-closed) sets in a topological space (X, τ) is denoted by δs(Λ, p)O(X, τ) (resp. δs(Λ, p)C(X, τ)). Definition 2. Let A be a subset of a topological space (X, τ). A point x of X is called a δs(Λ, p)-cluster point of A if A ∩ U ̸= ∅ for every δs(Λ, s)-open set U of X containing x. The set of all δs(Λ, p)-cluster points of A is called the δs(Λ, p)-closure of A and is denoted by Aδs(Λ,p). Lemma 1. The intersection of arbitrary collection of δs(Λ, s)-closed sets in (X, τ) is δs(Λ, p)-closed. Corollary 1. Let A be a subset of a topological space (X, τ). Then, Aδs(Λ,p) = ∩{F ∈ δs(Λ, p)C(X, τ) | A ⊆ F}. Lemma 2. For the δs(Λ, p)-closure of subsets A, B in a topological space (X, τ), the following properties hold: (1) A is δs(Λ, p)-closed in (X, τ) if and only if A = Aδs(Λ,p). (2) If A ⊆ B, then Aδs(Λ,p) ⊆ Bδs(Λ,p). (3) Aδs(Λ,p) is δs(Λ, p)-closed, that is, Aδs(Λ,p) = [Aδs(Λ,p)]δs(Λ,p). Lemma 3. For a family {Aγ | γ ∈ ∇} of a topological space (X, τ), the following properties hold: (1) [∩{Aγ | γ ∈ ∇}]δs(Λ,p) ⊆ ∩{Aδs(Λ,p) γ | γ ∈ ∇}. (2) [∪{Aγ | γ ∈ ∇}]δs(Λ,p) ⊇ ∪{Aδs(Λ,p) γ | γ ∈ ∇}. C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1434-1447 1437 Definition 3. A subset A of a topological space (X, τ) is called s(Λ, p)-regular if A is s(Λ, p)-open and s(Λ, p)-closed. The family of all s(Λ, p)-regular sets in a topological space (X, τ) is denoted by s(Λ, p)r(X, τ). Lemma 4. For a subset A of a topological space (X, τ), the following properties hold: (1) If A is a s(Λ, p)-regular set, then A is δs(Λ, p)-open. (2) If A is a δs(Λ, p)-open set, then A is s(Λ, p)-open. (3) If A is a s(Λ, p)-open set, then As(Λ,p) is s(Λ, p)-regular. Definition 4. Let A be a subset of a topological space (X, τ). A point x of X is called a θs(Λ, p)-cluster point of A if A∩U s(Λ,p) ̸= ∅ for every s(Λ, p)-open set U of X containing x. The set of all θs(Λ, p)-cluster points of A is called the θs(Λ, p)-closure of A, denoted by Aθs(Λ,p). A subset A of a topological space (X, τ) is said to be θs(Λ, p)-closed if A = Aθs(Λ,p). The complement of a θs(Λ, p)-closed set is said to be θs(Λ, p)-open. Lemma 5. Let (X, τ) be a topological space. Then, V θs(Λ,p) = V δs(Λ,p) = V s(Λ,p) for each V ∈ s(Λ, p)O(X, τ). Definition 5. A topological space (X, τ) is called δs(Λ, p)-T0 if, for any distinct pair of points in X, there exists a δs(Λ, p)-open set containing one of the points but not the other. Theorem 1. A topological space (X, τ) is δs(Λ, p)-T0 if and only if for each point of distinct points x, y of X, {x}δs(Λ,p) ̸= {y}δs(Λ,p). Proof. Suppose that x, y ∈ X, x ̸= y and {x}δs(Λ,p) ̸= {y}δs(Λ,p). Let z be a point of X such that z ∈ {x}δs(Λ,p) but z ̸∈ {y}δs(Λ,p). We claim that x ̸∈ {y}δs(Λ,p). For, if x ∈ {y}δs(Λ,p), then {x}δs(Λ,p) ⊆ {y}δs(Λ,p) and this contradicts the fact that z ̸∈ {y}δs(Λ,p). Thus, x belongs to the δs(Λ, p)-open set X − {y}δs(Λ,p) to which y does not belong. Conversely, let (X, τ) be a δs(Λ, p)-T0 space and x, y be any two distinct points of X. Then, there exists a δs(Λ, p)-open set U containing x or y, say x but not y. Then, X −U is a δs(Λ, p)-closed set which does not contain x but contains y. Thus, {y}δs(Λ,p) ⊆ X −U and hence x ̸∈ {y}δs(Λ,p). This shows that {x}δs(Λ,p) ̸= {y}δs(Λ,p). Definition 6. A topological space (X, τ) is called δs(Λ, p)-T1 if, for any distinct pair of points x and y in X, there exist a δs(Λ, p)-open set U of X containing x but not y and a δs(Λ, p)-open set V of X containing y but not x. Theorem 2. A topological space (X, τ) is δs(Λ, p)-T1 if and only if the singletons are δs(Λ, p)-closed sets. C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1434-1447 1438 Proof. Suppose that (X, τ) is δs(Λ, p)-T1 and x be any point of X. Let y ∈ X − {x}. Then, x ̸= y and so there exists a δs(Λ, p)-open set Vy such that y ∈ Vy but x ̸∈ Vy. Therefore, y ∈ Vy ⊆ X − {x}. Thus, X − {x} = ∪{Vy | y ∈ (X − {x})} which is δs(Λ, p)-open. Conversely, suppose that {z} is δs(Λ, p)-closed for each z ∈ X. Let x, y ∈ X with x ̸= y. Now x ̸= y implies y ∈ X − {x}. Thus, X − {x} is a δs(Λ, p)-open set containing y but not containing x. Similarly, X − {y} is a δs(Λ, p)-open set containing x but not containing y. This shows that (X, τ) is a δs(Λ, p)-T1 space. Definition 7. A topological space (X, τ) is called δs(Λ, p)-symmetric if, for each x and y in X, x ∈ {y}δs(Λ,p) implies y ∈ {y}δs(Λ,p). Lemma 6. Let (X, τ) be a topological space. For each point x ∈ X, {x} is s(Λ, p)-open or s(Λ, p)-closed. Theorem 3. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is δs(Λ, p)-symmetric. (2) For each x ∈ X, {x} is δs(Λ, p)-closed. (3) (X, τ) is δs(Λ, p)-T1. Proof. (1) ⇒ (2): Suppose that (X, τ) is δs(Λ, p)-symmetric. Let x be any point of X and y be any distinct point from x. By Lemma 6, {y} is s(Λ, p)-open or s(Λ, p)-closed in (X, τ). (i) In case {y} is s(Λ, p)-open, put Vy = {y}, then Vy ∈ δs(Λ, p)O(X, τ). (ii) In case {y} is s(Λ, p)-closed, x ̸∈ {y} = {y}s(Λ,p) and x ̸∈ {y}δs(Λ,p). By (1), y ̸∈ {x}δs(Λ,p). Now put Vy = X − {x}δs(Λ,p). Then, x ̸∈ Vy, y ∈ Vy and Vy ∈ δs(Λ, p)O(X, τ). Thus, X − {x} = ∪ y∈X−{x} Vy ∈ δs(Λ, p)O(X, τ) and hence {x} is δs(Λ, p)-closed. (2) ⇒ (3): Suppose that {z} is δs(Λ, p)-closed for each z ∈ X. Let x, y ∈ X with x ̸= y. Now x ̸= y implies y ∈ X −{x}. Thus, X −{x} is a δs(Λ, p)-open set containing y but not containing x. Similarly, we have X − {y} is a δs(Λ, p)-open set containing x but not containing y. This shows that (X, τ) is δs(Λ, p)-T1. (3) ⇒ (1): Suppose that y ̸∈ {x}δs(Λ,p). Then, since x ̸= y, by (3) there exists a δs(Λ, p)-open set U containing x such that y ̸∈ U and hence x ̸∈ {y}δs(Λ,p). This shows that x ∈ {y}δs(Λ,p) implies y ∈ {x}δs(Λ,p). Thus, (X, τ) is δs(Λ, p)-symmetric. Definition 8. A subset A of a topological space (X, τ) is called generalized δs(Λ, p)-closed (briefly g-δs(Λ, p)-closed) if Aδs(Λ,p) ⊆ U whenever A ⊆ U and U is δs(Λ, p)-open in (X, τ). Theorem 4. A subset A of a topological space (X, τ) is g-δs(Λ, p)-closed if and only if Aδs(Λ,p) −A contains no nonempty δs(Λ, p)-closed set. C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1434-1447 1439 Proof. Let F be a δs(Λ, p)-closed subset of Aδs(Λ,p) − A. Since A ⊆ X − F and A is g-δs(Λ, p)-closed, Aδs(Λ,p) ⊆ X − F and hence F ⊆ X −Aδs(Λ,p). Thus, F ⊆ Aδs(Λ,p) ∩ [X −Aδs(Λ,p)] = ∅ and F is empty. Conversely, suppose that A ⊆ U and U is δs(Λ, p)-open. If Aδs(Λ,p) ⊈ U , then Aδs(Λ,p) ∩ (X − U) is a nonempty δs(Λ, p)-closed subset of Aδs(Λ,p) −A. Theorem 5. A subset A of a topological space (X, τ) is g-δs(Λ, p)-closed if and only if F ∩Aδs(Λ,p) = ∅ whenever A ∩ F = ∅ and F is δs(Λ, p)-closed. Proof. Suppose that A is a δs(Λ, p)-closed set. Let F be a δs(Λ, p)-closed set and A ∩ F = ∅. Then, A ⊆ X − F ∈ δs(Λ, p)O(X, τ) and Aδs(Λ,p) ⊆ X − F . Thus, F ∩Aδs(Λ,p) = ∅. Conversely, let A ⊆ U and U ∈ δs(Λ, p)O(X, τ). Then, A∩ (X −U) = ∅ and X −U is δs(Λ, p)-closed. By the hypothesis, (X −U) ∩Aδs(Λ,p) = ∅ and hence Aδs(Λ,p) ⊆ U . Thus, A is g-δs(Λ, p)-closed. Theorem 6. A subset A of a topological space (X, τ) is g-δs(Λ, p)-closed if and only if A ∩ {x}δs(Λ,p) ̸= ∅ for every x ∈ Aδs(Λ,p). Proof. Let A be a g-δs(Λ, p)-closed set and suppose that there exists x ∈ Aδs(Λ,p) such that A ∩ {x}δs(Λ,p) = ∅. Thus, A ⊆ X − {x}δs(Λ,p) and hence Aδs(Λ,p) ⊆ X − {x}δs(Λ,p). Therefore, x ̸∈ Aδs(Λ,p), which is a contradiction. Conversely, suppose that the condition of the theorem holds and let U be any δs(Λ, p)- open set containing A. Let x ∈ Aδs(Λ,p). By the hypothesis, A ∩ Aδs(Λ,p) ̸= ∅, so there exists y ∈ A ∩ {x}δs(Λ,p) and hence y ∈ A ⊆ U . Thus, {x} ∩ U ̸= ∅. Therefore, x ∈ U , which implies that Aδs(Λ,p) ⊆ U . This shows that A is g-δs(Λ, p)-closed. Theorem 7. A topological space (X, τ) is δs(Λ, p)-symmetric if and only if {x} is g- δs(Λ, p)-closed for each x ∈ X. Proof. Suppose that x ∈ {y}δs(Λ,p) but y ∈ {x}δs(Λ,p). This means that the complement of {x}δs(Λ,p) contains y. Thus, the set {y} is a subset of the complement of {x}δs(Λ,p). This implies that {y}δs(Λ,p) is a subset of the complement of {x}δs(Λ,p). Now the complement of {x}δs(Λ,p) contains x which is a contradiction. Conversely, suppose that {x} ⊆ U ∈ δs(Λ, p)O(X, τ), but {x}δs(Λ,p) is not a subset of U . This means that {x}δs(Λ,p) and the complement of U are not disjoint. Let y belongs to their intersection. Now we have x ∈ {y}δs(Λ,p) which is a subset of the complement of U and x ̸∈ U . This is a contradiction. C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1434-1447 1440 4. Characterizations of s(Λ, p)-connected spaces We begin this section by introducing the concept of s(Λ, p)-connected spaces. Definition 9. A topological space (X, τ) is called s(Λ, p)-connected if X cannot be ex- pressed by the disjoint union of two nonempty s(Λ, p)-open sets. Theorem 8. For a topological space (X, τ), the following properties are equivalent: (1) V (Λ,p) = X for every nonempty (Λ, p)-open set V of X; (2) (X, τ) is s(Λ, p)-connected; (3) X cannot be expressed by the disjoint union of two nonempty δs(Λ, p)-open sets; (4) V δs(Λ,p) = X for every nonempty δs(Λ, p)-open set V of X. Proof. (1) ⇔ (2): The proof follows from Theorem 4.3 of [10]. (2) ⇒ (3): Suppose that there exist two nonempty δs(Λ, p)-open sets V1, V2 such that V1 ∩ V2 = ∅ and V1 ∪ V2 = X. Since δs(Λ, p)O(X, τ) ⊆ s(Λ, p)O(X, τ), this shows that (X, τ) is not s(Λ, p)-connected. (3) ⇒ (4): Suppose that V δs(Λ,p) ̸= X for some nonempty δs(Λ, p)-open set V of X. Then, X − V δs(Λ,p) ̸= ∅ and X = (X − V δs(Λ,p)) ∪ V δs(Λ,p). Since δs(Λ, p)O(X, τ) ⊆ s(Λ, p)r(X, τ), by Lemma 4 and 5, V δs(Λ,p) = V s(Λ,p) ∈ s(Λ, p)r(X, τ). Moreover, since s(Λ, p)r(X, τ) ⊆ δs(Λ, p)O(X, τ), (X − V δs(Λ,p)) and V δs(Λ,p) are δs(Λ, p)-open. (4) ⇒ (1): Let V be any nonempty (Λ, p)-open set of X. Then, V (Λ,p) is r(Λ, p)-closed and hence s(Λ, p)-regular. Thus, V (Λ,p) is δs(Λ, p)-open and X = [V (Λ,p)]δs(Λ,p) = [V (Λ,p)]s(Λ,p) = V (Λ,p). Theorem 9. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is s(Λ, p)-connected; (2) V δs(Λ,p) = X for every nonempty V ∈ β(Λ, p)O(X, τ); (3) V δs(Λ,p) = X for every nonempty V ∈ s(Λ, p)O(X, τ); (4) V δs(Λ,p) = X for every nonempty V ∈ p(Λ, p)O(X, τ); (5) V δs(Λ,p) = X for every nonempty V ∈ α(Λ, p)O(X, τ); (6) V δs(Λ,p) = X for every nonempty V ∈ ΛpO(X, τ). C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1434-1447 1441 Proof. (1) ⇒ (2): Let V be any nonempty β(Λ, p)-open set and U be any nonempty δs(Λ, p)-open set. Then, [V (Λ,p)](Λ,p) ̸= ∅ and U(Λ,p) ̸= ∅. Thus, by Theorem 8, ∅ ≠ U(Λ,p) ∩ [V (Λ,p)](Λ,p) ⊆ U ∩ [V (Λ,p)](Λ,p) ⊆ U ∩ (V ∪ [V (Λ,p)](Λ,p)) = U ∩ V s(Λ,p) ⊆ U ∩ V δs(Λ,p). Since U ∈ δs(Λ, p)O(X, τ), U ∩ V ̸= ∅. This shows that V δs(Λ,p) = X. (6) ⇒ (1): Let U, V be any nonempty δs(Λ, p)-open sets. Since δs(Λ, p)O(X, τ) ⊆ s(Λ, p)O(X, τ) and V(Λ,p) ̸= ∅, we have ∅ ≠ U ∩ V(Λ,p) ⊆ U ∩ V . This shows that V δs(Λ,p) = X for every nonempty V ∈ δs(Λ, p)O(X, τ). Thus, by Theorem 8, (X, τ) is s(Λ, p)-connected. Other implications are obvious since ΛpO(X, τ) ⊆ α(Λ, p)O(X, τ) ⊆ s(Λ, p)O(X, τ) ∩ p(Λ, p)O(X, τ) and s(Λ, p)O(X, τ) ∪ p(Λ, p)O(X, τ) ⊆ β(Λ, p)O(X, τ). Corollary 2. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is s(Λ, p)-connected; (2) U ∩ V ̸= ∅ for every nonempty sets U ∈ β(Λ, p)O(X, τ) and V ∈ δs(Λ, p)O(X, τ); (3) U ∩ V ̸= ∅ for every nonempty sets U ∈ p(Λ, p)O(X, τ) and V ∈ δs(Λ, p)O(X, τ); (4) U ∩ V ̸= ∅ for every nonempty sets U ∈ s(Λ, p)O(X, τ) and V ∈ δs(Λ, p)O(X, τ); (5) U ∩ V ̸= ∅ for every nonempty sets U ∈ α(Λ, p)O(X, τ) and V ∈ δs(Λ, p)O(X, τ); (6) U ∩ V ̸= ∅ for every nonempty sets U ∈ ΛpO(X, τ) and V ∈ δs(Λ, p)O(X, τ); (7) U ∩ V ̸= ∅ for every nonempty sets U ∈ δs(Λ, p)O(X, τ) and V ∈ δs(Λ, p)O(X, τ). Proof. This is immediate consequence of Theorem 8 and 9. C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1434-1447 1442 5. Characterizations of s(Λ, p)-regular spaces and s(Λ, p)-normal spaces In this section, we introduce the notions of s(Λ, p)-regular spaces and s(Λ, p)-normal spaces. Moreover, several characterizations of s(Λ, p)-regular spaces and s(Λ, p)-normal spaces are discussed. Definition 10. A topological space (X, τ) is said to be s(Λ, p)-regular if, for each s(Λ, p)- closed set F of X and each point x ̸∈ F , there exist U, V ∈ s(Λ, p)O(X, τ) such that x ∈ U , F ⊆ V and U ∩ V = ∅. Theorem 10. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is s(Λ, p)-regular. (2) For each s(Λ, p)-closed set F and each point x ̸∈ F , there exist U, V ∈ δs(Λ, p)O(X, τ) such that x ∈ U , F ⊆ V and U ∩ V = ∅. (3) For each point x ∈ X and each s(Λ, p)-open set V containing x, there exists U ∈ δs(Λ, p)O(X, τ) such that x ∈ U ⊆ U δs(Λ,p) ⊆ V . Proof. (1) ⇒ (2): Let F be a s(Λ, p)-closed set and x ̸∈ F . Then, there exist G,H ∈ s(Λ, p)O(X, τ) such that x ∈ G, F ⊆ H and G ∩H = ∅. By Lemma 4, Gs(Λ,p) is s(Λ, p)-regular and Gs(Λ,p)∩H = ∅. Thus, Gs(Λ,p)∩Hs(Λ,p) = ∅. Now, we put U = Gs(Λ,p) and V = Hs(Λ,p), then U and V are δs(Λ, p)-open sets such that x ∈ U , F ⊆ V and U ∩ V = ∅. (2) ⇒ (3): Let x ∈ X and V be any s(Λ, p)-open set containing x. Since x ̸∈ X − V , there exist U,G ∈ δs(Λ, p)O(X, τ) such that x ∈ U , X − V ⊆ G and U ∩ G = ∅. Since X −G is δs(Λ, p)-closed and U ⊆ X −G, x ∈ U ⊆ U δs(Λ,p) ⊆ X −G ⊆ V . (3) ⇒ (1): Let F be a s(Λ, p)-closed set and x ̸∈ F . Then, X − F is s(Λ, p)-open set containing x. By (3), there exists U ∈ δs(Λ, p)O(X, τ) such that x ∈ U ⊆ U δs(Λ,p) ⊆ X−F . Thus, x ∈ U , F ⊆ X − U δs(Λ,p) and U ∩ (X − U δs(Λ,p)) = ∅. Since δs(Λ, p)O(X, τ) ⊆ s(Λ, p)O(X, τ), (X, τ) is s(Λ, p)-regular. Definition 11. A topological space (X, τ) is said to be s(Λ, p)-normal if, for each disjoint s(Λ, p)-closed sets F and K of X, there exist U, V ∈ s(Λ, p)O(X, τ) such that F ⊆ U , K ⊆ V and U ∩ V = ∅. Theorem 11. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is s(Λ, p)-normal. C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1434-1447 1443 (2) For each disjoint s(Λ, p)-closed sets F and K of X, there exist U, V ∈ δs(Λ, p)O(X, τ) such that F ⊆ U , K ⊆ V and U ∩ V = ∅. (3) For each s(Λ, p)-closed set F and each s(Λ, p)-open set V containing F , there exists U ∈ δs(Λ, p)O(X, τ) such that F ⊆ U ⊆ U δs(Λ,p) ⊆ V . Proof. The proof is analogous to that of Theorem 10 and is omitted. 6. Characterizations of s(Λ, p)-T2 spaces and s(Λ, p)-Urysohn spaces In this section, we introduce the notions of s(Λ, p)-T2 spaces and s(Λ, p)-Urysohn spaces. Furthermore, some characterizations of s(Λ, p)-T2 spaces and s(Λ, p)-Urysohn spaces are investigated. Definition 12. A topological space (X, τ) is said to be s(Λ, p)-T2 if, for each pair of distinct points x, y ∈ X, there exist U, V ∈ s(Λ, p)O(X, τ) such that x ∈ U , y ∈ V and U ∩ V = ∅. Theorem 12. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is s(Λ, p)-T2. (2) For each pair of distinct points x, y ∈ X, there exist U, V ∈ s(Λ, p)r(X, τ) such that x ∈ U , y ∈ V and U ∩ V = ∅. (3) For each pair of distinct points x, y ∈ X, there exist U, V ∈ δs(Λ, p)O(X, τ) such that x ∈ U , y ∈ V and U δs(Λ,p) ∩ V δs(Λ,p) = ∅. (4) For each pair of distinct points x, y ∈ X, there exist U, V ∈ δs(Λ, p)O(X, τ) such that x ∈ U , y ∈ V and U s(Λ,p) ∩ V s(Λ,p) = ∅. (5) For each pair of distinct points x, y ∈ X, there exist U, V ∈ δs(Λ, p)O(X, τ) such that x ∈ U , y ∈ V and U ∩ V = ∅. Proof. (1) ⇒ (2): Suppose that (X, τ) is s(Λ, p)-T2. Then, for each pair of distinct points x, y ∈ X, there exist G,H ∈ s(Λ, p)O(X, τ) such that x ∈ G, y ∈ H and G∩H = ∅. Thus, Gs(Λ,p) ∩H = ∅. By Lemma 4, we have Gs(Λ,p) ∈ s(Λ, p)r(X, τ) and Gs(Λ,p) ∩Hs(Λ,p) = ∅. Now set U = Gs(Λ,p) and V = Hs(Λ,p). Then, U and V are s(Λ, p)-regular sets such that x ∈ U , y ∈ V and U ∩ V = ∅. (2) ⇒ (3): This is follows from the facts that s(Λ, p)r(X, τ) ⊆ δs(Λ, p)O(X, τ) and U δs(Λ,p) = U s(Λ,p) = U for every U ∈ s(Λ, p)r(X, τ). (3) ⇒ (4): This follows from the fact that U δs(Λ,p) = U s(Λ,p) for every U ∈ δs(Λ, p)O(X, τ). C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1434-1447 1444 (4) ⇒ (5): This is obvious. (5) ⇒ (1): This is obvious since δs(Λ, p)O(X, τ) ⊆ s(Λ, p)O(X, τ). Definition 13. A topological space (X, τ) is said to be s(Λ, p)-Urysohn if, for each pair of distinct points x, y ∈ X, there exist U, V ∈ s(Λ, p)O(X, τ) such that x ∈ U , y ∈ V and U (Λ,p) ∩ V (Λ,p) = ∅. Theorem 13. A topological space (X, τ) is s(Λ, p)-Urysohn if and only if for each pair of distinct points x, y of X, there exist U, V ∈ δs(Λ, p)O(X, τ) such that x ∈ U , y ∈ V and U (Λ,p) ∩ V (Λ,p) = ∅. Proof. Suppose that (X, τ) is s(Λ, p)-Urysohn. Then, for each pair of distinct points x, y of X, there exist U, V ∈ s(Λ, p)O(X, τ) such that x ∈ U, y ∈ V and U (Λ,p) ∩ V (Λ,p) = ∅. Since U ∈ s(Λ, p)O(X, τ), U (Λ,p) = [U(Λ,p)] (Λ,p) and U (Λ,p) is r(Λ, p)-closed. Thus, U (Λ,p), V (Λ,p) ∈ s(Λ, p)r(X, τ) ⊆ δs(Λ, p)O(X, τ). It is obvious that x ∈ U (Λ,p), y ∈ V (Λ,p) and [U (Λ,p)](Λ,p)∩[V (Λ,p)](Λ,p) = U (Λ,p)∩V (Λ,p) = ∅. Conversely, the proof is obvious since δs(Λ, p)O(X, τ) ⊆ s(Λ, p)O(X, τ). 7. Characterizations of S(Λ, p)-closed spaces In this section, we introduce the notion of S(Λ, p)-closed spaces. In particular, several characterizations of S(Λ, p)-closed spaces are discussed. Definition 14. A topological space (X, τ) is said to be S(Λ, p)-closed if, for every cover {Vγ | γ ∈ ∇} of X by s(Λ, p)-open sets of X, there exists a finite subset ∇0 of ∇ such that X = ∪ γ∈∇0 V s(Λ,p) γ . Theorem 14. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is S(Λ, p)-closed. (2) For every δs(Λ, p)-open cover {Vγ | γ ∈ ∇} of X, there exists a finite subset ∇0 of ∇ such that X = ∪ γ∈∇0 V s(Λ,p) γ . (3) For every δs(Λ, p)-open cover {Vγ | γ ∈ ∇} of X, there exists a finite subset ∇0 of ∇ such that X = ∪ γ∈∇0 V δs(Λ,p) γ . C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 16 (3) (2023), 1434-1447 1445 Proof. (1) ⇒ (2): Suppose that (X, τ) is S(Λ, p)-closed. Let {Vγ | γ ∈ ∇} be a δs(Λ, p)-open cover of X. By Lemma 4, δs(Λ, p)O(X, τ) ⊆ s(Λ, p)O(X, τ) and there exists a finite subset ∇0 of ∇ such that X = ∪ γ∈∇0 V s(Λ,p) γ . (2) ⇒ (3): Let {Vγ | γ ∈ ∇} be a δs(Λ, p)-open cover of X. By Lemma 4, δs(Λ, p)O(X, τ) ⊆ s(Λ, p)O(X, τ) and it follows from Lemma 5 that V δs(Λ,p) γ = V s(Λ,p) γ for each γ ∈ ∇. (3) ⇒ (1): Let {Vγ | γ ∈ ∇} be a s(Λ, p)-open cover of X. Then, X = ∪ γ∈∇0 V s(Λ,p) γ . By Lemma 4, V s(Λ,p) γ ∈ s(Λ, p)r(X, τ) ⊆ δs(Λ, p)O(X, τ) and there exists a finite subset ∇0 of ∇ such that X = ∪ γ∈∇0 [V s(Λ,p) γ ]δs(Λ,p). By Lemma 5, [V s(Λ,p) γ ]δs(Λ,p) = [V s(Λ,p) γ ]s(Λ,p) = V s(Λ,p) γ and hence X = ∪ γ∈∇0 V s(Λ,p) γ . Thus, (X, τ) is S(Λ, p)-closed. Theorem 15. A topological space (X, τ) is S(Λ, p)-closed if and only if for every θs(Λ, p)- open cover {Vγ | γ ∈ ∇} of X, there exists a finite subset ∇0 of ∇ such that X = ∪ γ∈∇0 Vγ. Proof. Let {Vγ | γ ∈ ∇} be a θs(Λ, p)-open cover of X. For each x ∈ X, there exists γ(x) ∈ ∇ such that x ∈ Vγ(x). Since Vγ(x) is θs(Λ, p)-open, there exists Gγ(x) ∈ s(Λ, p)O(X, τ) such that x ∈ Gγ(x) ⊆ G s(Λ,p) γ(x) ⊆ Vγ(x). Since {Gγ(x) | x ∈ X} is a s(Λ, p)-open cover of X, there exist finite points, say, x1, x2, ..., xn such that X = n ∪ i=1 G s(Λ,p) γ(xi) . Thus, X = n ∪ i=1 Vγ(xi). Conversely, let {Vγ | γ ∈ ∇} be a s(Λ, p)-open cover of X. By Lemma 4, {V s(Λ,p) γ | γ ∈ ∇} is a s(Λ, p)-regular cover of X and hence a θs(Λ, p)-open cover of X. Thus, there exists a finite subset ∇0 of ∇ such that X = ∪ γ∈∇0 V s(Λ,p) γ . This shows that (X, τ) is S(Λ, p)-closed. Acknowledgements This research project was financially supported by Mahasarakham University. REFERENCES 1446 References [1] S. Baudong, C. Viriyapong, and C. Boonpok. On generalized topology and minimal structure spaces. International Journal of Mathematical Analysis, 5(31):1507–1516, 2011. [2] C. Boonpok and M. Thongmoon. δp(Λ, p)-open sets in topological spaces. European Journal of Pure and Applied Mathematics, In Press, 2023. https://doi.org/10.29020/nybg.ejpam.v16i3.4737. [3] 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. [4] M. Caldas, T. Fukutake, S. Jafari, and T. Noiri. Some applications of δ-preopen sets in topological spaces. Bulletin of the Institute of Mathematics, Academia Sinica, 33(3):261–276, 2005. [5] M. Caldas, M. Ganster, D. N. Georgiou, S. Jafari, and T. Noiri. δ-semiopen sets in topological spaces. Topology Proceedings, 29(2):369–383, 2005. [6] M. Caldas, D. N. Georgiou, S. Jafari, and T. Noiri. More on δ-semiopen sets. Note di Matematica, 22(2):1–14, 2003. [7] W. Dungthaisong, C. Boonpok, and C. Viriyapong. Generalized closed sets in bigeneralized topological spaces. International Journal of Mathematical Analysis, 5(24):1175–1184, 2011. [8] M. Ganster, S. Jafari, and T. Noiri. On pre-Λ-sets and pre-V -sets. Acta Mathematica Hungarica, 95:337–343, 2002. [9] A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deeb. On precontinuous and weak precontinuous mappings. Proceedings of the Mathematical and Physical Society of Egypt, 53:47–53, 1982. [10] V. Pipitone and G. Russo. Spazi semiconnessi e spazi semiaperti. Rendiconti del Circolo Matematico di Palermo Series 2, 24:273–285, 1975. [11] S. Raychaudhuri and M. N. Mukherjee. On δ-almost continuity and δ-preopen sets. Bulletin of the Institute of Mathematics, Academia Sinica, 21:357–366, 1993. [12] S. Raychaudhuri and M. N. Mukherjee. δp-closedness for topological spaces. The Journal of the Indian Academy of Mathematics, 18:89–99, 1996. [13] P. Torton, C. Viriyapong, and C. Boonpok. Some separation axioms in bigeneralized topological spaces. International Journal of Mathematical Analysis, 6(56):2789–2796, 2012. REFERENCES 1447 [14] N. V. Veličko. H-closed topological spaces. American Mathematical Society Transla- tions, 78(2):102–118, 1968.