EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 2, 2022, 415-436 ISSN 1307-5543 – ejpam.com Published by New York Business Global On (Λ, p)-closed sets and the related notions in topological spaces Chawalit Boonpok1, Chokchai Viriyapong1,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand Abstract. This article deals with the concepts of Λp-sets and (Λ, p)-closed sets which are defined by utilizing the notions of preopen sets and preclosed sets. We also introduce and characterize some new low separation axioms. Characterizations of Λp-R0 spaces are given. Moreover, we introduce the concept of weakly (Λ, p)-continuous functions. In particular, several characterizations of weakly (Λ, p)-continuous functions are established. 2020 Mathematics Subject Classifications: 54A05, 54C08, 54D10 Key Words and Phrases: Λp-set, (Λ, p)-closed set, Λp-R0 space, weakly (Λ, p)-continuous function 1. Introduction In 1982, Mashhour et al. [16] introduced the notion of preopen sets which is also known under the name of locally dense sets [7] in the literature. Since then, this notion received wide usage in general topology. Kar and Bhattachryya [12] introduced new separation axioms pre-T0, pre-T1 and pre-T2 by using preopen sets due to Mashhour et al. [16]. Caldas [3] and Jafari [11] introduced independently the notions of p-D-sets and a separation axiom p-D1 which is strictly between pre-T0 and pre-T1. Caldas et al. [4] introduced two new classes of topological spaceS called pre-R0 and pre-R1 spaces in terms of concept of preopen sets and investigated some of their fundamental properties. Mashhour et al. [15] introduced and studied the concept of supra topological spaces by dropping a finite intersection condition of topological spaces. El-Shafei et al. [9] defined some concepts on supra topological spaces using supra preopen sets and investigated main properties. Al-shami et al. [2] introduced and investigated new separation axioms, namely supra semi Ti-spaces (i = 0, 1, 2, 3, 4). In [1], the present author introduce the version of complete Hausdorffness and complete regularity on supra topological spaces and discussed their ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4274 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), chokchai.v@msu.ac.th (C. Viriyapong) https://www.ejpam.com 415 © 2022 EJPAM All rights reserved. C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 416 fundamental properties. Cammaroto and Noiri [6] defined Λm-sets and generalized Λm- sets in an m-spaces (X,m) which is equivalent to a generalized topological spaces [14] and investigated properties of several low separation axioms of topologies constructed by the families of these sets. Ganster et al. [10] introduced the notions of a pre-Λ-set and a pre-V -set in a topological space and studied the fundamental properties of pre-Λ- sets and pre-V -sets. Caldas et al. [5] introduced and studied two new weak separation axioms called Λθ-R0 and Λθ-R1 spaces by using the notions of (Λ, θ)-open sets and (Λ, θ)- closure operators. The concept of weak continuity due to Levine [13] is one of the most important weak forms of continuity in topological spaces. Rose [18] introduced the notion of subweakly continuous functions and investigated the relationships between subweak continuity and weak continuity. Popa and Noiri [17] introduced the concept of weakly (τ,m)-continuous functions as functions from a topological space into a set satisfying some minimal conditions and investigated several characterizations of such functions. The paper is organized as follows. In section 3, we obtain fundamental properties of Λp-sets and investigate low separation axioms of an Alexandorff spaces (X,Λp). In section 4, we introduce the concept of (Λ, p)-closed sets and investigate properties of several low separation axioms of topologies constructed by the families of these sets. In section 5, we investigate some characterizations of Λp-R0 spaces. In the last section, we introduce the concept of weakly (Λ, p)-continuous functions and investigate several characterizations of such functions. 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 [16] (resp. preclosed [16]) if A ⊆ Int(Cl(A)) (resp. Cl(Int(A)) ⊆ A). By PO(X, τ) and PC(X, τ), we denote the collection of all preopen sets and the collection of all preclosed sets of a topological space (X, τ), respectively. The intersection of all preclosed sets containig A is called the preclosure [8] of A and is denoted by pCl(A). Definition 1. A topological space (X, τ) is said to be: (1) pre-T0 [12] if, for each pair of distinct points of X, there exists a preopen set con- taining one of the points but not the other; (2) pre-T1 [12] if, for each pair of distinct points x and y of X, there exists a pair of preopen sets one containing x but not y and the other containing y but not x; (3) pre-R0 [4] if every preopen set contains the preclosure of each of its singletons. Definition 2. Let A be a subset of a topological space (X, τ). A subset Λp(A) [10] is defined as follows: Λp(A) = ∩{O ∈ PO(X, τ)|A ⊆ O}. C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 417 Lemma 1. [10] For subsets A, B and Ai(i ∈ I) of a topological space (X, τ), the following properties hold: (1) A ⊆ Λp(A). (2) If A ⊆ B, then Λp(A) ⊆ Λp(B). (3) Λp(Λp(A)) = Λp(A). (4) Λp(∩{Ai | i ∈ I}) ⊆ ∩{Λp(Ai) | i ∈ I}. (5) Λp(∪{Ai | i ∈ I}) = ∪{Λp(Ai) | i ∈ I}. 3. Λp-sets and a topological space (X,Λp) In this section, we obtain fundamental properties of Λp-sets and investigate low sepa- ration axioms of an Alexandorff space (X,Λp). Definition 3. A subset A of a topological space (X, τ) is called a Λp-set (pre-Λ-set [10]) if A = Λp(A). The family of all Λp-sets of (X, τ) is denoted by Λp(X, τ) (or simply Λp). Lemma 2. For a subset A of a topological space (X, τ), the following properties hold: (1) Λp(A) is a Λp-set. (2) If A is preopen, then A is a Λp-set. Proof. This follows readily from Lemma 1. Lemma 3. [10] For subsets A and Ai(i ∈ I) of a topological space (X, τ), the following properties hold: (1) ∅ and X are pre-Λ-sets. (2) Every union of pre-Λ-sets is a pre-Λp-set. (3) Every intersection of pre-Λ-sets is a pre-Λ-set. Theorem 1. For a topological space (X, τ), the pair (X,Λp) is an Alexandroff space. Proof. This is an immediate consequence of Lemma 3. Theorem 2. Let (X, τ) be a topological space. Then, Λp = ΛΛp. Proof. By Lemma 2, PO(X, τ) ⊆ Λp. Let A be any subset of X. Then, ΛΛp = ∩{U | A ⊆ U,U ∈ Λp} ⊆ {U | A ⊆ U,U ∈ PO(X, τ)} = Λp(A). Thus, ΛΛp(A) ⊆ Λp(A). Now, we suppose that x 6∈ ΛΛp(A). Then, there exists U ∈ Λp such that A ⊆ U and x 6∈ U . Since x 6∈ U , there exists V ∈ PO(X, τ) such that U ⊆ V and x 6∈ V . Therefore, x 6∈ Λp(A). This shows that ΛΛp(A) ⊇ Λp(A) and hence Λp(A) = ΛΛp(A). C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 418 Theorem 3. A topological space (X, τ) is pre-R0 if and only if the topological space (X,Λp) is R0. Proof. Let V ∈ Λp and let x ∈ V . Then, x ∈ Λp(V ) = ∩{U | V ⊆ U,U ∈ PO(X, τ)} and x ∈ U for any U ∈ PO(X, τ) containing V . Since (X, τ) is pre-R0, pCl({x}) ⊆ U for every U ∈ PO(X, τ) containing V . Thus, pCl({x}) ⊆ ∩{U | V ⊆ U,U ∈ PO(X, τ)} = Λp(V ) = V. Since PO(X, τ) ⊆ Λp, Λp-Cl({x}) ⊆ pCl({x}) ⊆ V , where Λp-Cl({x}) denotes the closure of the singleton {x} in the topological space (X,Λp). This shows that (X,Λp) is R0. Conversely, suppose that (X,Λp) is R0. Let V ∈ Λp and x ∈ V . Since PO(X, τ) ⊆ Λp, we have Λp-Cl({x}) ⊆ V . Since X − Λp-Cl({x}) ∈ Λp, X − Λp-Cl({x}) = ∩{U | X − Λp-Cl({x}) ⊆ U,U ∈ PO(X, τ)}. Then, there exists U ∈ PO(X, τ) such that X − Λp-Cl({x}) ⊆ U and x 6∈ U . Thus, x ∈ X − U ⊆ Λp-Cl({x}) ⊆ V . Since X − U is preclosed, pCl({x}) ⊆ X − U ⊆ V . Consequently, we obtain (X, τ) is pre-R0. Theorem 4. A topological space (X, τ) is pre-T0 if and only if the topological space (X,Λp) is T0. Proof. This is obvious since PO(X, τ) ⊆ Λp. Conversely, let x and y be any pair of distinct points of X. Since (X,Λp) is T0, there exists V ∈ Λp such that either x ∈ V and y 6∈ V or x 6∈ V and y ∈ V . In case x ∈ V and y 6∈ V , there exists U ∈ PO(X, τ) such that V ⊆ U and y 6∈ U . However, since x ∈ V, x ∈ U . In case x 6∈ V and y ∈ V , similarly there exists U ∈ PO(X, τ) such that x 6∈ U and y ∈ U . Thus, (X, τ) is pre-T0. Lemma 4. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is pre-T1; (2) For each x ∈ X, the singleton {x} is preclosed in (X, τ). (3) For each x ∈ X, the singleton {x} is a Λp-set. Proof. (1) ⇒ (2): Let y be any point of X and let x ∈ X − {y}. Then, there exists a preopen set Vx such that x ∈ Vx and y 6∈ Vx. Thus, X −{y} = ∪x∈X−{y}Vx and hence the singleton {y} is preclosed in X. (2) ⇒ (3): Let x be any point of X and let y ∈ X − {x}. Then, x ∈ (X − {y}) ∈ PO(X, τ) and Λp({x}) ⊆ X − {y}. Therefore, y 6∈ Λp({x}) and hence Λp({x}) ⊆ {x}. Thus, Λp({x}) = {x}. This shows that {x} is a Λp-set. C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 419 (3) ⇒ (1): Suppose that the singleton {x} is a Λp-set for each x ∈ X. Let x and y be any distinct points. Then, y 6∈ Λp({x}) and there exists a preopen set Ux such that x ∈ Ux and y 6∈ Ux. Similarly, x 6∈ Λp({y}) and there exists a preopen set Uy such that y ∈ Uy and x 6∈ Uy. This shows that (X, τ) is pre-T1. Theorem 5. A topological space (X, τ) is pre-T1 if and only if the topological space (X,Λp) is discrete. Proof. Suppose that (X, τ) is pre-T1. Let x ∈ X. By Lemma 4, {x} is a Λp-set and hence {x} is open in (X,Λp). Thus, every subset of X is open in (X,Λp). This shows that (X,Λp) is discrete. Conversely, suppose that a topological space (X,Λp) is discrete. For any point x ∈ X, {x} is open in (X,Λp) and hence {x} is a Λp-set, by Lemma 4, we have (X, τ) is pre-T1. Corollary 1. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is pre-T1; (2) (X, τ) is pre-R0 and pre-T0; (3) (X,Λp) is R0 and T0; (4) (X,Λp) is T1; (5) (X,Λp) is discrete. Proof. (1) ⇒ (2): By Lemma 4, every pre-T1 space is pre-R0 and pre-T0. (2) ⇒ (1): Since (X, τ) is pre-T0, for any distinct point x, y of X, there exists a preopen set U of X such that x ∈ U and y 6∈ U . Hence, pCl({x}) ⊆ U since (X, τ) is pre-R0. Thus, x 6∈ X − pCl({x}) and hence y ∈ X − U ⊆ X − pCl({x}) ∈ PO(X, τ). This shows that (X, τ) is pre-T1. (2) ⇔ (3): This is an immediate consequence of Theorem 3 and Theorem 4. (3) ⇔ (4): This proof is obvious. (5) ⇔ (1): This is an immediate consequence of Theorem 5. 4. (Λ, p)-closed sets In this section, we introduce the notion of (Λ, p)-closed sets in topological spaces. Moreover, some properties of (Λ, p)-closed sets are discussed. Definition 4. A subset A of a topological space (X, τ) is called (Λ, p)-closed if A = T ∩C, where T is a Λp-set and C is a preclosed set. The collection of all (Λ, p)-closed sets in a topological space (X, τ) is denoted by ΛpC(X, τ). Theorem 6. For a subset A of a topological space (X, τ), the following properties are equivalent: C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 420 (1) A is (Λ, p)-closed; (2) A = T ∩ pCl(A), where T is a Λp-set; (3) A = Λp(A) ∩ pCl(A). Proof. (1) ⇒ (2): Let A = T ∩ C,where T is a Λp-set and C is a preclosed set. Since A ⊆ C, we have pCl(A) ⊆ C and hence A = T ∩ C ⊇ T ∩ pCl(A) ⊇ A. Consequently, we obtain A = T ∩ pCl(A). (2) ⇒ (3): Let A = T ∩pCl(A), where T is a Λp-set. Since A ⊆ T , Λp(A) ⊆ Λp(T ) = T and hence A ⊆ Λp(A) ∩ pCl(A) ⊆ T ∩ pCl(A) = A. Thus, A = Λp(A) ∩ pCl(A). (3) ⇒ (1): Since Λp(A) is a Λp-set, pCl(A) is preclosed and A = Λp(A)∩ pCl(A). This shows that A is (Λ, p)-closed. Definition 5. A subset A of a topological space (X, τ) is said to be (Λ, p)-open if the complement of A is (Λ, p)-closed. The collection of all (Λ, p)-open sets in a topological space (X, τ) is denoted by ΛpO(X, τ). Theorem 7. For a subset Aγ(γ ∈ Γ) of a topological space (X, τ), the following properties hold: (1) If Aγ is (Λ, p)-closed for each γ ∈ Γ, then ∩{Aγ | γ ∈ Γ} is (Λ, p)-closed. (2) If Aγ is (Λ, p)-open for each γ ∈ Γ, then ∪{Aγ | γ ∈ Γ} is (Λ, p)-open. Proof. (1) Suppose that Aγ is (Λ, p)-closed for each γ ∈ Γ. Then, for each γ, there exist a Λp-set Tγ and a preclosed set Cγ such that Aγ = Tγ ∩ Cγ . Thus, ∩γ∈ΓAγ = ∩γ∈Γ(Tγ ∩ Cγ) = (∩γ∈ΓTγ) ∩ (∩γ∈ΓCγ). Since ∩γ∈ΓCγ is a preclosed set and by Lemma 3, we have ∩γ∈ΓTγ is a Λp-set. This shows that ∩γ∈ΓAγ is (Λ, p)-closed. (2) Let Aγ be (Λ, p)-open for each γ ∈ Γ. Then, X − Aγ is (Λ, p)-closed, by (1), we have X − ∪γ∈ΓAγ = ∩γ∈Γ(X −Aγ) is (Λ, p)-closed and hence ∪γ∈ΓAγ is (Λ, p)-open. Theorem 8. Let (X, τ) be a pre-R0 space. For each x ∈ X, {x} is (Λ, p)-closed if and only if {x} is preclosed. Proof. Suppose that {x} is a (Λ, p)-closed set. By Theorem 6, {x} = Λp({x}) ∩ pCl({x}). For any preopen set U containing x, pCl({x}) ⊆ U and hence pCl({x}) ⊆ Λp({x}). Thus, {x} = Λp({x}) ∩ pCl({x}) ⊇ pCl({x}). This shows that {x} is preclosed. Conversely, suppose that {x} is a preclosed set. Since {x} ⊆ Λp({x}), we have Λp({x}) ∩ pCl({x}) = Λp({x}) ∩ {x} = {x}, by Theorem 6, {x} is (Λ, p)-closed. C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 421 Theorem 9. A topological space (X, τ) is pre-T0 if and only if for each x ∈ X, the singleton {x} is (Λ, p)-closed. Proof. Suppose that (X, τ) is pre-T0. For each x ∈ X, it is obvious that {x} ⊆ Λp({x}) ∩ pCl({x}). If y 6= x, (i) there exists a preopen set Vx such that y 6∈ Vx and x ∈ Vx or (ii) there exists a preopen set Vy such that x 6∈ Vy and y ∈ Vy. In case of (i), y 6∈ Λp({x}) and y 6∈ Λp({x}) ∩ pCl({x}). Thus, {x} ⊇ Λp({x}) ∩ pCl({x}). In case (ii), y 6∈ pCl({x}) and y 6∈ Λp({x}) ∩ pCl({x}). This shows that {x} ⊇ Λp({x}) ∩ pCl({x}). Consequently, we obtain {x} = Λp({x}) ∩ pCl({x}). Conversely, suppose that (X, τ) is not pre-T0. There exist two distinct points x, y such that (i) y ∈ Vx for every preopen set Vx containing x and (ii) x ∈ Vy for every preopen set Vy containing y. From (i) and (ii), we obtain y ∈ Λp({x}) and y ∈ pCl({x}), respectively. Therefore, we have y ∈ Λp({x}) ∩ pCl({x}). By Theorem 6, {x} = Λp({x}) ∩ pCl({x}) since {x} is (Λ, p)-closed. This is contrary to x 6= y. Definition 6. Let A be a subset of a topological space (X, τ). A point x ∈ X is called a (Λ, p)-cluster point of A if A ∩ U 6= ∅ 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 of A and is denoted by A(Λ,p). Lemma 5. For subsets A,B of a topological space (X, τ), the following properties hold: (1) A ⊆ A(Λ,p) and [A(Λ,p)](Λ,p) = A(Λ,p). (2) If A ⊆ B, then A(Λ,p) ⊆ B(Λ,p). (3) A(Λ,p) = ∩{F |A ⊆ F and F is (Λ, p)-closed}. (4) A(Λ,p) is (Λ, p)-closed. (5) A is (Λ, p)-closed if and only if A = A(Λ,p). Remark 1. Every Λp-set is (Λ, p)-closed. The converse of Remark 1 is not true in general as shown by the following example. Example 1. Let X = {−2,−1} with a topology τ = {∅, {−2}, X}. Then, {−1} is a (Λ, p)-closed set, but {−1} is not a Λp-set. Lemma 6. For a subset A of a topological space (X, τ), the following properties hold: (1) If A is preclosed, then A is (Λ, p)-closed. (2) A is (Λ, p)-closed if and only if A = Λp(A) ∩A(Λ,p). C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 422 Proof. (1) It is sufficient to observe that A = X∩A, where the whole set X is a Λp-set. (2) Let A be a (Λ, p)-closed set. Then, there exist a Λp-set T and a preclosed set C such that A = T ∩ C. Since A ⊆ T , we have A ⊆ Λp(A) ⊆ Λp(T ) = T . Since C is preclosed, by (1), C is (Λ, p)-closed. Since A ⊆ C, A ⊆ A(Λ,p) ⊆ C(Λ,p) = C and hence A ⊆ Λp(A) ∩A(Λ,p) ⊆ T ∩ C = A. Thus, A = Λp(A) ∩A(Λ,p). Conversely, let A = Λp(A) ∩ A(Λ,p). Since Λp(A) is a Λp-set, by Remark 1, Λp(A) is (Λ, p)-closed. Since A(Λ,p) is (Λ, p)-closed, by Theroem 7(1), Λp(A)∩A(Λ,p) is (Λ, p)-closed and hence A is (Λ, p)-closed. The following example shows that the converse of Lemma 6(1) is not true in general. Example 2. Let X = {−2,−1, 0, 1, 2} with a topology τ = {∅, {−2}, {2}, {−2, 2}, X}. Then, {−2, 2} is (Λ, p)-closed, but {−2, 2} is not preclosed. Definition 7. Let A be a subset of a topological space (X, τ). A subset Λ(Λ,p)(A) is defined as follows: Λ(Λ,p)(A) = ∩{U ∈ ΛpO(X, τ) | A ⊆ U}. Lemma 7. For subsets A,B of a topological space (X, τ), the following properties hold: (1) A ⊆ Λ(Λ,p)(A). (2) If A ⊆ B, then Λ(Λ,p)(A) ⊆ Λ(Λ,p)(B). (3) Λ(Λ,p)[Λ(Λ,p)(A)] = Λ(Λ,p)(A); (4) If A is (Λ, p)-open, then Λ(Λ,p)(A) = A. Lemma 8. Let (X, τ) be a topological space and let x, y ∈ X. Then, y ∈ Λ(Λ,p)({x}) if and only if x ∈ {y}(Λ,p). Proof. Let y 6∈ Λ(Λ,p)({x}). Then, there exists a (Λ, p)-open set V containing x such that y 6∈ V . Hence, x 6∈ {y}(Λ,p). The converse is similarly shown. A subset Nx of a topological space (X, τ) is said to be (Λ, p)-neighbourhood of a point x ∈ X if there exists a (Λ, p)-open set U such that x ∈ U ⊆ Nx. Lemma 9. A subset A of a topological space (X, τ) is (Λ, p)-open if and only if A is (Λ, p)-neighbourhood of each x ∈ A. Definition 8. Let A be a subset of a topological space (X, τ). A subset 〈x〉p is defined as follows: 〈x〉p = Λ(Λ,p)({x}) ∩ {x}(Λ,p). Theorem 10. For a topological space (X, τ), the following properties hold: (1) Λ(Λ,p)(A) = {x ∈ X | A ∩ {x}(Λ,p) 6= ∅} for each subset A of X. (2) For each x ∈ X, Λ(Λ,p)(〈x〉p) = Λ(Λ,p)({x}). C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 423 (3) For each x ∈ X, (〈x〉p)(Λ,p) = {x}(Λ,p). (4) If U is (Λ, p)-open and x ∈ U , then 〈x〉p ⊆ U . (5) If F is (Λ, p)-closed and x ∈ F , then 〈x〉p ⊆ F . Proof. (1) Suppose that A ∩ {x}(Λ,p) = ∅. Then, we have x 6∈ X − {x}(Λ,p) which is a (Λ, p)-open set containing A. Therefore, x 6∈ Λ(Λ,p)(A). Consequently, we have Λ(Λ,p)(A) ⊆ {x ∈ X | A ∩ {x}(Λ,p) 6= ∅}. Next, let x ∈ X such that A ∩ {x}(Λ,p) 6= ∅ and suppose that x 6∈ Λ(Λ,p)(A). Then, there exists a (Λ, p)-open set U containing A and x 6∈ U . Let y ∈ A ∩ {x}(Λ,p). Hence, U is a (Λ, p)-neighbourhood of y which does not contain x. By this contradiction x ∈ Λ(Λ,p)(A). (2) Let x ∈ X. Then, we have {x} ⊆ {x}(Λ,p) ∩ Λ(Λ,p)({x}) = 〈x〉p. By Lemma 7, Λ(Λ,p)({x}) ⊆ Λ(Λ,p)(〈x〉p). Next, we show the opposite implication. Suppose that y 6∈ Λ(Λ,p)({x}). Then, there exists a (Λ, p)-open set V such that x ∈ V and y 6∈ V . Since 〈x〉p ⊆ Λ(Λ,p)({x}) ⊆ Λ(Λ,p)(V ) = V , we have Λ(Λ,p)(〈x〉p) ⊆ V . Since y 6∈ V , we have y 6∈ Λ(Λ,p)(〈x〉p). Thus, Λ(Λ,p)(〈x〉p) ⊆ Λ(Λ,p)({x}) and hence Λ(Λ,p)({x}) = Λ(Λ,p)(〈x〉p). (3) By the definition of 〈x〉p, we have {x} ⊆ 〈x〉p and {x}(Λ,p) ⊆ (〈x〉p)(Λ,p) by Lemma 5. On the other hand, we have 〈x〉p ⊆ {x}(Λ,p) and (〈x〉p)(Λ,p) ⊆ ({x}(Λ,p))(Λ,p) = {x}(Λ,p). Thus, (〈x〉p)(Λ,p) = {x}(Λ,p). (4) Let U be a (Λ, p)-open set and let x ∈ U . By Lemma 7, Λ(Λ,p)({x}) ⊆ U and hence 〈x〉p ⊆ U . (5) Let F be a (Λ, p)-closed set and let x ∈ F . By Lemma 5, we have 〈x〉p = {x}(Λ,p) ∩ Λ(Λ,p)({x}) ⊆ {x}(Λ,p) ⊆ F (Λ,p) = F. Lemma 10. For any points x and y in a topological space (X, τ), the following properties are equivalent: (1) Λ(Λ,p)({x}) 6= Λ(Λ,p)({y}); (2) {x}(Λ,p) 6= {y}(Λ,p). Proof. (1) ⇒ (2): Suppose that Λ(Λ,p)({x}) 6= Λ(Λ,p)({y}). There exists a point z ∈ X such that z ∈ Λ(Λ,p)({x}) and z 6∈ Λ(Λ,p)({y}) or z ∈ Λ(Λ,p)({y}) and z 6∈ Λ(Λ,p)({x}). We prove only the first case being the second analogous. From z ∈ Λ(Λ,p)({x}) it follows that {x} ∩ {z}(Λ,p) 6= ∅ which implies x ∈ {z}(Λ,p). By z 6∈ Λ(Λ,p)({y}), {y} ∩ {z}(Λ,p) = ∅. Since x ∈ {z}(Λ,p), {x}(Λ,p) ⊆ {z}(Λ,p) and {y} ∩ {x}(Λ,p) = ∅. Therefore, it follows that {x}(Λ,p) 6= {y}(Λ,p). Thus, Λ(Λ,p)({x}) 6= Λ(Λ,p)({y}) implies that {x}(Λ,p) 6= {y}(Λ,p). (2) ⇒ (1): Suppose that {x}(Λ,p) 6= {y}(Λ,p). Then, there exists a point z ∈ X such that z ∈ {x}(Λ,p) and z 6∈ {y}(Λ,p) or z ∈ {y}(Λ,p) and z 6∈ {x}(Λ,p). We prove only the first case being the second analogous. It follows that there exists a (Λ, p)-open set containing z and therefore x but not y, namely, y 6∈ Λ(Λ,p)({x}) and hence Λ(Λ,p)({x}) 6= Λ(Λ,p)({y}). C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 424 Lemma 11. For any points x and y in a topological space (X, τ), the following properties hold: (1) y ∈ Λ(Λ,p)({x}) if and only if x ∈ {y}(Λ,p); (2) Λ(Λ,p)({x}) = Λ(Λ,p)({y}) if and only if {x}(Λ,p) = {y}(Λ,p). Proof. (1) Let x 6∈ {y}(Λ,p). Then, there exists a (Λ, p)-open set U such that x ∈ U and y 6∈ U . Therefore, y 6∈ Λ(Λ,p)({x}). The converse is similarly shown. (2) Suppose that Λ(Λ,p)({x}) = Λ(Λ,p)({y}) for any points x and y in X. Since x ∈ Λ(Λ,p)({x}), x ∈ Λ(Λ,p)({y}) and by (1), y ∈ {x}(Λ,p). By Lemma 5, {y}(Λ,p) ⊆ {x}(Λ,p). Similarly, we have {x}(Λ,p) ⊆ {y}(Λ,p) and hence {x}(Λ,p) = {y}(Λ,p). Conversely, suppose that {x}(Λ,p) = {y}(Λ,p). Since x ∈ {x}(Λ,p), x ∈ {y}(Λ,p) and by (1), y ∈ Λ(Λ,p)({x}). By Lemma 7, Λ(Λ,p)({y}) ⊆ Λ(Λ,p)(Λ(Λ,p)({x})) = Λ(Λ,p)({x}). Similarly, we have Λ(Λ,p)({x}) ⊆ Λ(Λ,p)({y}) and hence Λ(Λ,p)({x}) = Λ(Λ,p)({y}). 5. Characterizations of Λp-R0 spaces In this section, we introduce the concept of Λp-R0 spaces. Moreover, some characteri- zations of Λp-R0 spaces are investigated. Definition 9. A topological space (X, τ) is called a Λp-R0 space if, for each (Λ, p)-open set U and each x ∈ U , {x}(Λ,p) ⊆ U . Theorem 11. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is Λp-R0; (2) for each (Λ, p)-closed set F and each x ∈ X − F , there exists a (Λ, p)-open set U such that F ⊆ U and x 6∈ U ; (3) for each (Λ, p)-closed set F and each x ∈ X − F , F ∩ {x}(Λ,p) = ∅; (4) for each x, y ∈ X, {x}(Λ,p) = {y}(Λ,p) or {x}(Λ,p) ∩ {y}(Λ,p) = ∅. Proof. (1) ⇒ (2): Let F be a (Λ, p)-closed set and let x ∈ X − F . Then, we have {x}(Λ,p) ⊆ X −F . Let U = X −{x}(Λ,p), then U is a (Λ, p)-open set such that F ⊆ U and x 6∈ U . (2) ⇒ (3): Let F be a (Λ, p)-closed set and let x ∈ X − F . There exists a (Λ, p)-open set U such that F ⊆ U and x 6∈ U . Thus, U ∩ {x}(Λ,p) = ∅ and hence F ∩ {x}(Λ,p) = ∅. (3) ⇒ (4): Let x, y be distinct points of X. Suppose that {x}(Λ,p) 6= {y}(Λ,p). By (3), x ∈ {y}(Λ,p) and y ∈ {x}(Λ,p). Thus, {x}(Λ,p) ⊆ {y}(Λ,p) ⊆ {x}(Λ,p) and hence {x}(Λ,p) = {y}(Λ,p). C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 425 (4) ⇒ (1): Let U be a (Λ, p)-open set and let x ∈ U . For each y 6∈ U , we have U ∩ {y}(Λ,p) = ∅ and hence x 6∈ {y}(Λ,p). Therefore, {y}(Λ,p) 6= {x}(Λ,p). By (4), {x}(Λ,p) ∩ {y}(Λ,p) = ∅. Since X − U is (Λ, p)-closed, y ∈ {y}(Λ,p) ⊆ X − U and ∪y∈X−U{y}(Λ,p) = X − U . Thus, {x}(Λ,p) ∩ (X − U) = {x}(Λ,p) ∩ [∪y∈X−U{y}(Λ,p)] = ∪y∈X−U [{x}(Λ,p) ∩ {y}(Λ,p)] = ∅ and hence {x}(Λ,p) ⊆ U . This shows that (X, τ) is Λp-R0. Corollary 2. A topological space (X, τ) is Λp-R0 if and only if, for each x, y ∈ X, {x}(Λ,p) 6= {y}(Λ,p) implies {x}(Λ,p) ∩ {y}(Λ,p) = ∅. Proof. This is obvious by Theorem 11. Conversely, let U be a (Λ, p)-open set and let x ∈ U . If y 6∈ U , then U ∩ {y}(Λ,p) = ∅. Thus, x 6∈ {y}(Λ,p) and {x}(Λ,p) 6= {y}(Λ,p). By the hypothesis, {x}(Λ,p) ∩ {y}(Λ,p) = ∅ and hence y 6∈ {x}(Λ,p). Therefore, {x}(Λ,p) ⊆ U . Thus, (X, τ) is Λp-R0. Theorem 12. A topological space (X, τ) is Λp-R0 if and only if, for each x, y ∈ X, Λ(Λ,p)({x}) 6= Λ(Λ,p)({y}) implies Λ(Λ,p)({x}) ∩ Λ(Λ,p)({y}) = ∅. Proof. Suppose that Λ(Λ,p)({x}) ∩ Λ(Λ,p)({y}) 6= ∅. Let z ∈ Λ(Λ,p)({x}) ∩ Λ(Λ,p)({y}). Then, z ∈ Λ(Λ,p)({x}) and by Lemma 11, x ∈ {z}(Λ,p). Thus, x ∈ {z}(Λ,p) ∩ {x}(Λ,p) and by Corollary 2, {z}(Λ,p) = {x}(Λ,p). Similarly, we have {z}(Λ,p) = {y}(Λ,p) and by Lemma 11, Λ(Λ,p)({x}) = Λ(Λ,p)({y}). Conversely, we shows the sufficiency by using Corollary 2. Suppose that {x}(Λ,p) 6= {y}(Λ,p). By Lemma 11, Λ(Λ,p)({x}) 6= Λ(Λ,p)({y}) and hence Λ(Λ,p)({x}) ∩ Λ(Λ,p)({y}) = ∅. There- fore, {x}(Λ,p) ∩ {y}(Λ,p) = ∅. In fact, assume z ∈ {x}(Λ,p) ∩ {y}(Λ,p). Then, z ∈ {x}(Λ,p) implies x ∈ Λ(Λ,p)({z}) and hence x ∈ Λ(Λ,p)({z}) ∩ Λ(Λ,p)({x}). By the hypothesis, Λ(Λ,p)({z}) = Λ(Λ,p)({x}) and by Lemma 11, {z}(Λ,p) = {x}(Λ,p). Similarly, we have {z}(Λ,p) = {y}(Λ,p) and hence {x}(Λ,p) = {y}(Λ,p). This contradicts that {x}(Λ,p) 6= {y}(Λ,p). Thus, {x}(Λ,p) ∩ {y}(Λ,p) = ∅. This shows that (X, τ) is Λp-R0. Theorem 13. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is Λp-R0; (2) x ∈ {y}(Λ,p) if and only if y ∈ {x}(Λ,p). Proof. (1) ⇒ (2): Suppose that (X, τ) is Λp-R0. Let x ∈ {y}(Λ,p). By Lemma 11, y ∈ Λ(Λ,p)({x}) and hence Λ(Λ,p)({x}) ∩ Λ(Λ,p)({y}) 6= ∅. By Theorem 12, we have Λ(Λ,p)({x}) = Λ(Λ,p)({y}) C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 426 and hence x ∈ Λ(Λ,p)({y}). Thus, y ∈ {x}(Λ,p) by Lemma 11. The converse is similarly shown. (2) ⇒ (1): Let U be a (Λ, p)-open set and let x ∈ U . If y 6∈ U , then {y}(Λ,p) ∩ U = ∅. Thus, x 6∈ {y}(Λ,p) and hence y 6∈ {x}(Λ,p). This implies that {x}(Λ,p) ⊆ U . Therefore, (X, τ) is Λp-R0. Theorem 14. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is Λp-R0; (2) for each nonempty subset A of X and each (Λ, p)-open set U such that A ∩ U 6= ∅, there exists a (Λ, p)-closed set F such that A ∩ F 6= ∅ and F ⊆ U ; (3) F = Λ(Λ,p)(F ) for each (Λ, p)-closed set F ; (4) {x}(Λ,p) = Λ(Λ,p)({x}) for each x ∈ X. Proof. (1) ⇒ (2): Let A be a nonempty subset of X and let U ∈ ΛpO(X, τ) such that A ∩U 6= ∅. Then, there exists x ∈ A ∩U and hence {x}(Λ,p) ⊆ U . Put F = {x}(Λ,p), then F is (Λ, p)-closed, A ∩ F 6= ∅ and F ⊆ U . (2) ⇒ (3): Let F be a (Λ, p)-closed set. By Lemma 7, we have F ⊆ Λ(Λ,p)(F ). Next, we show F ⊇ Λ(Λ,p)(F ). Let x 6∈ F . Then, x ∈ (X − F ) ∈ ΛpO(X, τ) and by (2), there exists a (Λ, p)-closed set K such that x ∈ K and K ⊆ X−F . Now, put U = X−K. Then, F ⊆ U ∈ ΛpO(X, τ) and x 6∈ U . Thus, x 6∈ Λ(Λ,p)(F ). This shows that F ⊇ Λ(Λ,p)(F ). (3) ⇒ (4): Let x ∈ X and let y 6∈ Λ(Λ,p)({x}). There exists a (Λ, p)-open set U such that x ∈ U and y 6∈ U . Therefore, {y}(Λ,p) ∩ U = ∅. By (3), we have Λ(Λ,p)({y}(Λ,p)) ∩ U = ∅. Since x 6∈ Λ(Λ,p)({y}(Λ,p)), there exists a (Λ, p)-open set G such that {y}(Λ,p) ⊆ G and x 6∈ G. Hence, {x}(Λ,p) ∩ G = ∅. Since y ∈ G, we have y 6∈ {x}(Λ,p) and hence {x}(Λ,p) ⊆ Λ(Λ,p)({x}). Moreover, {x}(Λ,p) ⊆ Λ(Λ,p)({x}) ⊆ Λ(Λ,p)({x}(Λ,p)) = {x}(Λ,p). Consequently, we obtain {x}(Λ,p) = Λ(Λ,p)({x}). (4) ⇒ (5): The proof is obvious. (5) ⇒ (1): Let U ∈ ΛpO(X, τ) and let x ∈ U . If y 6∈ U , then {y}(Λ,p) ∩ U = ∅ and x 6∈ {y}(Λ,p). By Lemma 11, y 6∈ Λ(Λ,p)({x}) and by (5), y 6∈ {x}(Λ,p). Thus, {x}(Λ,p) ⊆ U and hence (X, τ) is Λp-R0. Corollary 3. A topological space (X, τ) is Λp-R0 if and only if {x}(Λ,p) ⊆ Λ(Λ,p)({x}) for each x ∈ X. Proof. This is obvious by Theorem 14. Conversely, let x ∈ {y}(Λ,p). By Lemma 11, we have y ∈ Λ(Λ,p)({x}) and hence y ∈ {x}(Λ,p). Similarly, if y ∈ {x}(Λ,p), then x ∈ {y}(Λ,p). It follows from Theorem 13 that (X, τ) is Λp-R0. C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 427 Theorem 15. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is Λp-R0; (2) 〈x〉p = {x}(Λ,p) for each x ∈ X; (3) 〈x〉p is (Λ, p)-closed for each x ∈ X. Proof. (1) ⇒ (2): By Theorem 14, {x}(Λ,p) = Λ(Λ,p)({x}) for each x ∈ X and hence {x}(Λ,p) = {x}(Λ,p) ∩ Λ(Λ,p)({x}) = 〈x〉p. (2) ⇒ (1): Since {x}(Λ,p) = 〈x〉p for each x ∈ X, we have {x}(Λ,p) ⊆ Λ(Λ,p)({x}). By Corollary 3, (X, τ) is Λp-R0. (2) ⇔ (3): This is a consequence of Lemma 7. 6. Characterizations of weakly (Λ, p)-continuous functions In this section, we introduce the notion of weakly (Λ, p)-continuous functions and obtain several characterizations of weakly (Λ, p)-continuous functions. Definition 10. Let A be a subset of a topological space (X, τ). The union of all (Λ, p)-open sets contained in A is called the (Λ, p)-interior of A and is denoted by A(Λ,p). Lemma 12. Let A and B be subsets of a topological space (X, τ). For the (Λ, p)-interior, the following properties hold: (1) A(Λ,p) ⊆ A and [A(Λ,p)](Λ,p) = A(Λ,p). (2) If A ⊆ B, then A(Λ,p) ⊆ B(Λ,p). (3) A(Λ,p) = ∪{G | G ⊆ A and G is (Λ, p)-open}. (4) A(Λ,p) is (Λ, p)-open. (5) A is (Λ, p)-open if and only if A(Λ,p) = A. (6) [X −A](Λ,p) = X −A(Λ,p). Definition 11. A function f : (X, τ) → (Y, σ) is said to be weakly (Λ, p)-continuous at a point x ∈ X if, for each (Λ, p)-open set V containing f(x), there exists a (Λ, p)-open set U containing x such that f(U) ⊆ V (Λ,p). A function f : (X, τ) → (Y, σ) is said to be (Λ, p)-continuous if f has this property at each point x ∈ X. Theorem 16. A function f : (X, τ) → (Y, σ) is weakly (Λ, p)-continuous at x ∈ X if and only if for each (Λ, p)-open set V containing f(x), x ∈ [f−1(V (Λ,p))](Λ,p). C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 428 Proof. Let V be a (Λ, p)-open set containing f(x). Then, there exists a (Λ, p)-open set U containing x such that f(U) ⊆ V (Λ,p) and hence x ∈ U ⊆ f−1(V (Λ,p)). Thus, x ∈ [f−1(V (Λ,p))](Λ,p). Conversely, let V be a (Λ, p)-open set containing f(x). By the hypothesis, we have x ∈ [f−1(V (Λ,p))](Λ,p). There exists a (Λ, p)-open set U such that x ∈ U ⊆ f−1(V (Λ,p)); hence f(U) ⊆ V (Λ,p). This shows that f is weakly (Λ, p)-continuous at x ∈ X. Theorem 17. A function f : (X, τ) → (Y, σ) is weakly (Λ, p)-continuous if and only if f−1(V ) ⊆ [f−1(V (Λ,p))](Λ,p) for every (Λ, p)-open set V of Y . Proof. Let V be any (Λ, p)-open set of Y and let x ∈ f−1(V ). Then f(x) ∈ V . Since f is weakly (Λ, p)-continuous at x, by Theorem 16, x ∈ [f−1(V (Λ,p))](Λ,p) and hence f−1(V ) ⊆ [f−1(V (Λ,p))](Λ,p). Conversely, let x ∈ X and let V be any (Λ, p)-open set of Y containing f(x). Then, we have x ∈ f−1(V ) ⊆ [f−1(V (Λ,p))](Λ,p) and hence x ∈ [f−1(V (Λ,p))](Λ,p). Thus, f is weakly (Λ, p)-continuous by Theorem 16. Theorem 18. A function f : (X, τ) → (Y, σ) is weakly (Λ, p)-continuous if and only if [f−1(V )](Λ,p) ⊆ f−1(V (Λ,p)) for every (Λ, p)-open set V of Y . Proof. Let V be any (Λ, p)-open subset of Y and let x ∈ f−1(V ). There exists a (Λ, p)-open set U containing x such that f(U) ⊆ V (Λ,p). Since x ∈ U ⊆ f−1(V (Λ,p)), we have x ∈ [f−1(V (Λ,p))](Λ,p) and hence f−1(V ) ⊆ [f−1(V (Λ,p))](Λ,p). Conversely, let x ∈ X and let V be any (Λ, p)-open set containing f(x). Since V ∩ [Y − V (Λ,p)] = ∅, f(x) 6∈ [Y − V (Λ,p)](Λ,p) and hence x 6∈ f−1([Y − V (Λ,p)](Λ,p)). By the hypothesis, x 6∈ [f−1(Y − V (Λ,p))](Λ,p) = [X − f−1(V (Λ,p))](Λ,p) and there exists a (Λ, p)-open set U containing x such that U ∩ [X − f−1(V (Λ,p))] = ∅. Thus, f(U) ⊆ V (Λ,p). This shows that f is is weakly (Λ, p)-continuous. Theorem 19. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is weakly (Λ, p)-continuous; (2) f−1(U) ⊆ [f−1(U (Λ,p))](Λ,p) for every (Λ, p)-open subset U of Y ; (3) [f−1(F(Λ,p))] (Λ,p) ⊆ f−1(F ) for every (Λ, p)-closed subset F of Y ; (4) [f−1([A(Λ,p)](Λ,p))] (Λ,p) ⊆ f−1(A(Λ,p)) for every subset A of Y ; (5) f−1(A(Λ,p)) ⊆ [f−1([A(Λ,p)] (Λ,p))](Λ,p) for every subset A of Y ; C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 429 (6) [f−1(U)](Λ,p) ⊆ f−1(U (Λ,p)) for every (Λ, p)-open subset U of Y . Proof. (1) ⇒ (2): It follows from Theorem 17. (2) ⇒ (3): Let F be any (Λ, p)-closed subset of Y . Then, Y −F is (Λ, p)-open, by (2), f−1(Y − F ) ⊆ [f−1([Y − F ](Λ,p))](Λ,p) = [f−1(Y − F(Λ,p))](Λ,p) = X − [f−1(F(Λ,p))] (Λ,p). Thus, [f−1(F(Λ,p))] (Λ,p) ⊆ f−1(F ). (3) ⇒ (4): Let A be any subset of Y . Since A(Λ,p) is (Λ, p)-closed, by (3), [f−1([A(Λ,p)](Λ,p))] (Λ,p) ⊆ f−1(A(Λ,p)). (4) ⇒ (5): Let A be any subset of Y . By (4), we have f−1(A(Λ,p)) = X − f−1([Y −A](Λ,p)) ⊆ X − [f−1([[Y −A](Λ,p)](Λ,p))] (Λ,p) = [f−1([A(Λ,p)] (Λ,p))](Λ,p). Thus, we get the result. (5) ⇒ (6): Let U be any (Λ, p)-open subset of Y . Suppose that x 6∈ f−1(U (Λ,p)). Then, f(x) 6∈ U (Λ,p) and so there exists a (Λ, p)-open set V containing x such that U ∩ V = ∅. Thus, U ∩ V (Λ,p) = ∅. By (5), x ∈ f−1(V ) ⊆ [f−1(V (Λ,p))](Λ,p). There exists a (Λ, p)- open set W containing x such that x ∈ W ⊆ f−1(V (Λ,p)). Since U ∩ V (Λ,p) = ∅ and f(W ) ⊆ V (Λ,p), we have W ∩ f−1(U) = ∅ and hence x 6∈ [f−1(U)](Λ,p). This shows that [f−1(U)](Λ,p) ⊆ f−1(U (Λ,p)). (6) ⇒ (1): This is obvious from Theorem 18. Definition 12. A subset A of a topological space (X, τ) is said to be: (i) s(Λ, p)-open if A ⊆ [A(Λ,p)] (Λ,p); (ii) p(Λ, p)-open if A ⊆ [A(Λ,p)](Λ,p); (iii) β(Λ, p)-open if A ⊆ [[A(Λ,p)](Λ,p)] (Λ,p); (iv) r(Λ, p)-open if A = [A(Λ,p)](Λ,p). The complement of a s(Λ, p)-open (resp. p(Λ, p)-open, β(Λ, p)-open, r(Λ, p)-open) set is called s(Λ, p)-closed (resp. p(Λ, p)-closed, β(Λ, p)-closed, r(Λ, p)-closed). Theorem 20. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is weakly (Λ, p)-continuous; (2) [f−1(F(Λ,p))] (Λ,p) ⊆ f−1(F ) for every r(Λ, p)-closed subset F of Y ; (3) [f−1([U (Λ,p)](Λ,p))] (Λ,p) ⊆ f−1(U (Λ,p)) for every β(Λ, p)-open subset U of Y ; C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 430 (4) [f−1([U (Λ,p)](Λ,p))] (Λ,p) ⊆ f−1(U (Λ,p)) for every s(Λ, p)-open subset U of Y . Proof. (1) ⇒ (2): Let F be any r(Λ, p)-closed subset of Y . Then, F(Λ,p) is (Λ, p)-open, by Theorem 19, [f−1(F(Λ,p))] (Λ,p) ⊆ f−1([F(Λ,p)] (Λ,p)). Since F is r(Λ, p)-closed, we have [f−1(F(Λ,p))] (Λ,p) ⊆ f−1([F(Λ,p)] (Λ,p)) = f−1(F ). (2) ⇒ (3): Let U be any β(Λ, p)-open set. Then, U (Λ,p) ⊆ [[U (Λ,p)](Λ,p)] (Λ,p) ⊆ U (Λ,p) and hence U (Λ,p) is r(Λ, p)-closed. By (2), [f−1([U (Λ,p)](Λ,p))] (Λ,p) ⊆ f−1(U (Λ,p)). (3) ⇒ (4): The proof is obvious. (4) ⇒ (1): Let U be any (Λ, p)-open subset of Y . Then, we have U is s(Λ, p)-open and by (4), [f−1(U)](Λ,p) ⊆ [f−1([U(Λ,p)] (Λ,p))](Λ,p) ⊆ f−1(U (Λ,p)). Thus, f is weakly (Λ, p)- continuous by Theorem 19. Theorem 21. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is weakly (Λ, p)-continuous; (2) [f−1([U(Λ,p)] (Λ,p))](Λ,p) ⊆ f−1(U (Λ,p)) for every p(Λ, p)-open subset U of Y ; (3) [f−1(U)](Λ,p) ⊆ f−1(U (Λ,p)) for every p(Λ, p)-open subset U of Y ; (4) f−1(U) ⊆ [f−1(U (Λ,p))](Λ,p) for every p(Λ, p)-open subset U of Y . Proof. (1) ⇒ (2): Let U be any p(Λ, p)-open subset of Y . Then, we have U (Λ,p) = [[U (Λ,p)](Λ,p)] (Λ,p) and hence U (Λ,p) is r(Λ, p)-closed. By Theorem 20, [f−1([U (Λ,p)](Λ,p))] (Λ,p) ⊆ f−1(U (Λ,p)). (2) ⇒ (3): Let U be any p(Λ, p)-open subset of Y . Then, U ⊆ [U (Λ,p)](Λ,p) and by (2), we have [f−1(U)](Λ,p) ⊆ [f−1([U (Λ,p)](Λ,p))] (Λ,p) ⊆ f−1(U (Λ,p)). (3) ⇒ (4): Let U be any p(Λ, p)-open subset of Y . By (3), we have f−1(U) ⊆ f−1([U (Λ,p)](Λ,p)) = X − f−1([Y − U (Λ,p)](Λ,p)) = X − [f−1(Y − U (Λ,p))](Λ,p) = [f−1(U (Λ,p))](Λ,p). (4) ⇒ (1): Since every (Λ, p)-open set is p(Λ, p)-open, by (4) and Theorem 19, it follows that f is weakly (Λ, p)-continuous. Theorem 22. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is weakly (Λ, p)-continuous; (2) [f−1([A(Λ,p)](Λ,p))] (Λ,p) ⊆ f−1(A(Λ,p)) for every subset A of Y ; C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 431 (3) [f−1(F(Λ,p))] (Λ,p) ⊆ f−1(F ) for every r(Λ, p)-closed subset F of Y ; (4) [f−1(U)](Λ,p) ⊆ f−1(U (Λ,p)) for every (Λ, p)-open subset U of Y ; (5) f−1(U) ⊆ [f−1(U (Λ,p))](Λ,p) for every (Λ, p)-open subset U of Y ; (6) [f−1(U)](Λ,p) ⊆ f−1(U (Λ,p)) for every p(Λ, p)-open subset U of Y ; (7) f−1(U) ⊆ [f−1(U (Λ,p))](Λ,p) for every p(Λ, p)-open subset U of Y . Proof. (1) ⇒ (2): Let A be any subset of Y and let x ∈ X − f−1(A(Λ,p)). Then, f(x) ∈ Y −A(Λ,p) and there exists a (Λ, p)-open set U containing f(x) such that U ∩A = ∅ and hence U (Λ,p) ∩ [A(Λ,p)](Λ,p) = ∅. Since f is weakly (Λ, p)-continuous, there exists a (Λ, p)-open set W containing x such that f(W ) ⊆ U (Λ,p). Then W ∩ f−1([A(Λ,p)](Λ,p)) = ∅ and hence x ∈ X − [f−1([A(Λ,p)](Λ,p))] (Λ,p). This shows that [f−1([A(Λ,p)](Λ,p))] (Λ,p) ⊆ f−1(A(Λ,p)). (2) ⇒ (3): Let F be any r(Λ, p)-closed subset of Y . By (2), we have [f−1(F(Λ,p))] (Λ,p) = [f−1([[F(Λ,p)] (Λ,p)](Λ,p))] (Λ,p) ⊆ f−1([F(Λ,p)] (Λ,p)) = f−1(F ). (3) ⇒ (4): Let U be any (Λ, p)-open subset of Y . Since U (Λ,p) is r(Λ, p)-closed and by (3), [f−1(U)](Λ,p) ⊆ [f−1([U (Λ,p)](Λ,p))] (Λ,p) ⊆ f−1(U (Λ,p)). (4) ⇒ (5): Let U be any (Λ, p)-open subset of Y . Since Y − U (Λ,p) is (Λ, p)-open, by (4), X − [f−1(U (Λ,p))](Λ,p) = [f−1(Y −U (Λ,p))](Λ,p) ⊆ f−1([Y −U (Λ,p)](Λ,p)) ⊆ X − f−1(U) and hence f−1(U) ⊆ [f−1(U (Λ,p))](Λ,p). (5) ⇒ (1): Let x ∈ X and let U be any (Λ, p)-open subset of Y containing f(x). By (5), x ∈ f−1(U) ⊆ [f−1(U (Λ,p))](Λ,p). Put W = [f−1(U (Λ,p))](Λ,p). Thus, f(W ) ⊆ U (Λ,p) and hence f is weakly (Λ, p)-continuous at x. This shows that f is weakly (Λ, p)-continuous. (1) ⇒ (6): Let U be any p(Λ, p)-open subset of Y and let x ∈ X − f−1(U (Λ,p)). There exists a (Λ, p)-open set V containing f(x) such that V ∩U = ∅ and hence [V ∩U ](Λ,p) = ∅. Since U is (Λ, p)-open, we have U ∩ V (Λ,p) ⊆ [U ∩ V ](Λ,p) = ∅. Since f is weakly (Λ, p)- continuous and V is a (Λ, p)-open set containing f(x), there exists a (Λ, p)-open set W containing x such that f(W ) ⊆ V (Λ,p). Then, f(W ) ∩ U = ∅ and hence W ∩ f−1(U) = ∅. Thus, x ∈ X − [f−1(U)](Λ,p). This shows that [f−1(U)](Λ,p) ⊆ f−1(U (Λ,p)). (6) ⇒ (7): Let U be any p(Λ, p)-open subset of Y . Since Y −U (Λ,p) is (Λ, p)-open and by (6), we have X − [f−1(U (Λ,p))](Λ,p) = [f−1(Y − U (Λ,p))](Λ,p) ⊆ f−1([Y − U (Λ,p)](Λ,p)) ⊆ X − f−1(U) and hence f−1(U) ⊆ [f−1(U (Λ,p))](Λ,p). (7) ⇒ (1): Let x ∈ X and let V be any (Λ, p)-open subset of Y containing f(x). Then, V is p(Λ, p)-open, by (7), x ∈ f−1(V ) ⊆ [f−1(V (Λ,p))](Λ,p). Put U = [f−1(V (Λ,p))](Λ,p). Then, f(U) ⊆ V (Λ,p) and hence f is weakly (Λ, p)-continuous at x. Thus, f is weakly (Λ, p)-continuous. C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 432 Definition 13. A topological space (X, τ) is said to be Λp-T2 if, for any disjoint pair of points x and y in X, there exist (Λ, p)-open sets U and V such that x ∈ U, y ∈ V and U ∩ V = ∅. Definition 14. A topological space (X, τ) is said to be Λp-Urysohn if, for each distinct points x, y ∈ X, there exist (Λ, p)-open sets U and V containing x and y, respectively, such that U (Λ,p) ∩ V (Λ,p) = ∅. Theorem 23. If f : (X, τ) → (Y, σ) is a weakly (Λ, p)-continuous injection and (Y, σ) is Λp-Urysohn, then (X, τ) is Λp-T2. Proof. Let x, y be distinct points of X. Then, f(x) 6= f(y). Since (Y, σ) is Λp-Urysohn, there exist (Λ, p)-open sets U and V containing f(x) and f(y), respectively, such that U (Λ,p)∩V (Λ,p) = ∅. Since f is weakly (Λ, p)-continuous, there exist (Λ, p)-open sets G and W containing x and y, respectively, such that f(G) ⊆ U (Λ,p) and f(W ) ⊆ V (Λ,p). This shows that G ∩W = ∅. Thus, (X, τ) is Λp-T2. Theorem 24. If f : (X, τ) → (Y, σ) is weakly (Λ.p)-continuous and (Y, σ) is Λp-T2, then f has (Λ, p)-closed point inverses. Proof. Let y ∈ Y . We show that f−1(y) = {x ∈ X | f(x) = y} is (Λ, p)-closed, or equivalently G = {x ∈ X | f(x) 6= y} is (Λ, p)-open. Let x ∈ G. Since f(x) 6= y and (Y, σ) is Λp-T2, there exist disjoint (Λ, p)-open sets U and V such that f(x) ∈ U and y ∈ V . Since U ∩V = ∅, U (Λ,p)∩V = ∅ and hence y 6∈ U (Λ,p). Since f is weakly (Λ, p)-continuous, there exists a (Λ, p)-open set W containing x such that f(W ) ⊆ U (Λ,p). Now, suppose that W is not contained in G. Then, there exists a point z ∈ W such that f(z) = y. Since f(W ) ⊆ U (Λ,p), y = f(z) ∈ U (Λ,p). This is a contradiction. Therefore, W ⊆ G and by Lemma 9, G is (Λ, p)-open. Theorem 25. Let (X, τ) be a topological space. If for each pair of distinct points x1 and x2 in X, there exists a function f : (X, τ) → (Y, σ) such that (1) (Y, σ) is Λp-Urysohn, (2) f(x1) 6= f(x2) and (3) f is weakly (Λ, p)-continuous at x1 and x2, then (X, τ) is Λp-T2. Proof. Let x1, x2 be any distinct points of X. By the hypothesis, there exists a function f : (X, τ) → (Y, σ) which satisfies the conditions (1), (2) and (3). Let yi = f(xi) for i = 1, 2. Then, y1 6= y2. Since (Y, σ) is Λp-Urysohn, there exist (Λ, p)-open sets Vi in (Y, σ) containing yi such that V (Λ,p) 1 ∩ V (Λ,p) 2 = ∅. Since f is weakly (Λ, p)-continuous at x1 and x2, for i = 1, 2, there exist (Λ, p)-open sets Ui in (X, τ) containing xi such that f(Ui) ⊆ V (Λ,p) i . Hence, we get U1 ∩ U2 = ∅. This shows that (X, τ) is Λp-T2. C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 433 Corollary 4. If f : (X, τ) → (Y, σ) is a weakly (Λ, p)-continuous injection and (Y, σ) is Λp-Urysohn, then (X, τ) is Λp-T2. Definition 15. Let A be a subset of a topological space (X, τ). The θ(Λ, p)-closure of A, Aθ(Λ,p), is defined as follows: Aθ(Λ,p) = {x ∈ X | A ∩ U (Λ,p) 6= ∅ for each (Λ, p)-open set U containing x}. A subset A of a topological space (X, τ) is called θ(Λ, p)-closed if A = Aθ(Λ,p). The complement of a θ(Λ, p)-closed set is said to be θ(Λ, p)-open. Lemma 13. Let A be a subset of a topological space (X, τ). Then, x ∈ A(Λ,p) if and only if U ∩A 6= ∅ for every (Λ, p)-open set U containing x. Lemma 14. For a subset A of a topological space (X, τ), the following properties hold: (1) If A is (Λ, p)-open in (X, τ), then A(Λ,p) = Aθ(Λ,p). (2) Aθ(Λ,p) is (Λ, p)-closed for every subset A of X. Proof. (1) In general, we have A(Λ,p) ⊆ Aθ(Λ,p). Suppose that x 6∈ A(Λ,p). By Lemma 13, there exists a (Λ, p)-open set U containing x such that U ∩A = ∅; hence A∩U (Λ,p) = ∅ since A is (Λ, p)-open. Thus, x 6∈ Aθ(Λ,p). Consequently, we obtain A(Λ,p) = Aθ(Λ,p). (2) Let x ∈ X − Aθ(Λ,p). Then, we have x 6∈ Aθ(Λ,p). There exists a (Λ, p)-open set Ux containing x such that A ∩ U (Λ,p) x = ∅ and hence Ux ∩ Aθ(Λ,p) = ∅. Therefore, x ∈ Ux ⊆ X − Aθ(Λ,p). Thus, X − Aθ(Λ,p) = ∪x∈X−Aθ(Λ,p)Ux and hence X − Aθ(Λ,p) is (Λ, p)-open. This shows that Aθ(Λ,p) is (Λ, p)-closed. Theorem 26. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is weakly (Λ, p)-continuous; (2) f(A(Λ,p)) ⊆ [f(A)]θ(Λ,p) for every subset A of X; (3) [f−1(B)](Λ,p) ⊆ f−1(Bθ(Λ,p)) for every subset B of Y ; (4) [f−1(V )](Λ,p) ⊆ f−1(V (Λ,p)) for every (Λ, p)-open subset V of Y . Proof. (1) ⇒ (2): Let A be any subset of X. Let x ∈ A(Λ,p) and V be any (Λ, p)-open set containing f(x). Since f is weakly (Λ, p)-continuous, there exists a (Λ, p)-open set U containing x such that f(U) ⊆ V (Λ,p). Since x ∈ A(Λ,p), we have U ∩A 6= ∅. It follows that ∅ 6= f(U) ∩ f(A) ⊆ V (Λ,p) ∩ f(A) and hence V (Λ,p) ∩ f(A) 6= ∅. Thus, f(x) ∈ [f(A)]θ(Λ,p). Consequently, we obtain f(A(Λ,p)) ⊆ [f(A)]θ(Λ,p). (2) ⇒ (3): Let B be any subset of Y . By (2), we have f([f−1(B)](Λ,p)) ⊆ [f(f−1(B))]θ(Λ,p) ⊆ Bθ(Λ,p) and hence [f−1(B)](Λ,p) ⊆ f−1(Bθ(Λ,p)). C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 415-436 434 (3) ⇒ (4): Let V be any (Λ, p)-open subset of Y . By Lemma 14, V (Λ,p) = V θ(Λ,p). Thus, the proof is obvious. (4) ⇒ (1): Let V be any (Λ, p)-open set containing f(x). Since V ∩ [Y − V (Λ,p)] = ∅, we have f(x) 6∈ [Y − V (Λ,p)](Λ,p) and hence x 6∈ f−1([Y − V (Λ,p)](Λ,p)). Since Y − V (Λ,p) is (Λ, p)-open, by (4), x 6∈ [f−1([Y − V (Λ,p)])](Λ,p) and there exists a (Λ, p)-open set U containing x such that U ∩f−1(Y −V (Λ,p)) = ∅; hence f(U)∩ [Y −V (Λ,p)] = ∅. This shows that f(U) ⊆ V (Λ,p). Thus, f is weakly (Λ, p)-continuous. Definition 16. A topological space (X, τ) is said to be Λp-regular if, for each (Λ, p)-closed set F and each x 6∈ F , there exist disjoint (Λ, p)-open sets U and V such that x ∈ U and F ⊆ V . Lemma 15. A topological space (X, τ) is Λp-regular if and only if for each x ∈ X and each (Λ, p)-open set U containing x, there exists a (Λ, p)-open set V such that x ∈ V ⊆ V (Λ,p) ⊆ U . Proof. Let x ∈ X and let U be a (Λ, p)-open set containing x. Then, x 6∈ X − U and X − U is (Λ, p)-closed. There exist disjoint (Λ, p)-open sets V and W such that x ∈ V and X − U ⊆ W . Thus, V ⊆ X − W ⊆ U . Since X − W is (Λ, p)-closed, we have V (Λ,p) ⊆ X −W ⊆ U and hence x ∈ V ⊆ V (Λ,p) ⊆ U . Conversely, let F be a (Λ, p)-closed set and let x 6∈ F . Then, x ∈ X −F . Since X −F is (Λ, p)-open, there exists a (Λ, p)-open set V such that x ∈ V ⊆ V (Λ,p) ⊆ X − F and hence F ⊆ X − V (Λ,p). This shows that (X, τ) is Λp-regular. Lemma 16. Let (X, τ) be a Λp-regular space. Then, the following properties hold: (1) A(Λ,p) = Aθ(Λ,p) for every subset A of X. (2) Every (Λ, p)-open set is θ(Λ, p)-open. Proof. (1) In general, we have A(Λ,p) ⊆ Aθ(Λ,p) for every subset A of X. Next, we show that Aθ(Λ,p) ⊆ A(Λ,p). Let x ∈ Aθ(Λ,p) and U be any (Λ, p)-open set containing x. By Lemma 15, there exists a (Λ, p)-open set V such that x ∈ V ⊆ V (Λ,p) ⊆ U . Since x ∈ Aθ(Λ,p), it follows that A ∩ V (Λ,p) 6= ∅ and hence U ∩ A 6= ∅. Thus, x ∈ A(Λ,p). Consequently, we obtain Aθ(Λ,p) ⊆ A(Λ,p). (2) Let V be a (Λ, p)-open set. By (1), we have X − V = [X − V ](Λ,p) = [X − V ]θ(Λ,p) and hence X − V is θ(Λ, p)-closed. Thus, V is θ(Λ, p)-open. Theorem 27. Let (Y, σ) be a Λp-regular space. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f−1(Bθ(Λ,p)) is θ(Λ, p)-closed in X for every subset B of Y ; (2) f is weakly (Λ, p)-continuous; (3) f−1(F ) is (Λ, p)-closed in X for every θ(Λ, p)-closed subset F of Y ; REFERENCES 435 (4) f−1(V ) is (Λ, p)-open in X for every θ(Λ, p)-open subset V of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . Then, [f−1(B)](Λ,p) ⊆ [f−1(Bθ(Λ,p))](Λ,p) = f−1(Bθ(Λ,p)), by Theorem 26, f is is weakly (Λ, p)-continuous. (2) ⇒ (3): Let F be any θ(Λ, p)-closed subset of Y . By Theorem 26, we have [f−1(F )](Λ,p) ⊆ f−1(F θ(Λ,p)) = f−1(F ) and hence f−1(F ) is (Λ, p)-closed in X. (3) ⇒ (4): Let V be any θ(Λ, p)-open subset of Y . Then, Y − V is θ(Λ, p)-closed, by (3), X − f−1(V ) = F−1(Y − V ) is (Λ, p)-closed in X. Thus, f−1(V ) is (Λ, p)-open. (4) ⇒ (1): Let B be any subset of Y . By Lemma 14, Bθ(Λ,p) is (Λ, p)-closed in Y and by Lemma 16, Y −Bθ(Λ,p) is θ(Λ, p)-open in Y . Thus, by (4), we have X − f−1(Bθ(Λ,p)) = f−1(Y −Bθ(Λ,p)) is (Λ, p)-open in X and hence f−1(Bθ(Λ,p)) is θ(Λ, p)-closed. 7. Conclusion Closedness and openness are fundamental with respect to the investigation of gen- eral topological spaces. Various types of generalizations of closed sets and open sets in topological spaces have been researched by many mathematicians. This article is de- voted to introducing and discussing the concepts of (Λ, p)-closed sets and (Λ, p)-open sets. Moreover, some characterizations of Λp-R0 spaces are explored. Additionally, several char- acterizations of weakly (Λ, p)-continuous functions are established. The ideas and results of this article may motivate further research. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] T. M. Al-shami. 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