EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 2, 2022, 572-588 ISSN 1307-5543 – ejpam.com Published by New York Business Global (Λ, sp)-open sets in topological spaces Chawalit Boonpok1, Jeeranunt Khampakdee1,∗ 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 s(Λ, sp)-open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets, β(Λ, sp)-open sets and b(Λ, sp)-open sets. Some properties of s(Λ, sp)-open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets, β(Λ, sp)-open sets and b(Λ, sp)-open sets are discussed. In particular, the relationships between s(Λ, sp)-open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets, β(Λ, sp)-open sets, b(Λ, sp)-open sets and other related sets are established. Moreover, several characterizations of Λsp-extremally disconnected spaces are investigated. 2020 Mathematics Subject Classifications: 54A05, 54G05 Key Words and Phrases: (Λ, sp)-closed set, (Λ, sp)-open set, Λsp-extremally disconnected space 1. Introduction Semi-open sets, preopen sets, α-open sets, b-open sets and β-open sets play an impor- tant for the study and investigation in topological spaces. In 1963, Levine [6] introduced the concept of semi-open sets in topological spaces. After the work of Levine on semi- open sets, several mathematicians turned their attention to the generalizations of various concepts of topology by considering semi-open sets instead of open sets. While open sets are replaced by semi-open sets, new results are obtained in some occasions and in other occasions substantial generalizations are exhibited. In this direction, in 1975, Mahesh- wari and Prasad [7], used semi-open sets to define and investigate three new separation axioms called semi-T0, semi-T1 and semi-T2. Later, in 1987, Bhattacharya and Lahiri [2] generalized the concept of closed sets to semi-generalized closed sets with the help of semi-openness. The notion of α-open sets (originally called α-sets) in topological spaces was introduced by Njåstad [10] in 1965. By using α-open sets, Mashhour et al. [9] defined and studied the notions of α-continuity and α-openness in topological spaces. In 1982, Mashhour et al. [8] introduced and investigated the concepts of preopen sets and precon- tinuous functions in topological spaces. In 1983, Abd El-Monsef et al. [4] introduced a weak form of open sets called β-open sets. The concept of β-open sets is equivalent to that ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4276 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), jeeranunt.k@msu.ac.th (J. Khampakdee) https://www.ejpam.com 572 © 2022 EJPAM All rights reserved. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 573 of semi-preopen sets [1]. In 1996, Andrijević [1] introduced a class of generalized open sets in a topological space, the so-called b-open sets. The class of b-open sets is contained in the class of β-open sets and contains all semi-open sets and preopen sets. The concept of extremally disconnected topological spaces was first introduced by Gillman and Jerison [5]. A topological space is called extremally disconnected if the closure of every open set is open. Sivaraj [13] investigated some characterizations of extremally disconnected spaces by utilizing semi-open sets due to Levine [6]. Noiri [11] obtained several characterizations of extremally disconnected spaces by utilizing preopen sets and semi-preopen sets. In 2004, Noiri and Hatir [12] introduced the notion of Λsp-sets in terms of the concept of β-open sets and investigated the notion of Λsp-closed sets by using Λsp- sets. In [3], the author introduced the concepts of (Λ, sp)-open sets and (Λ, sp)-closed sets which are defined by utilizing the notions of Λsp-sets and β-closed sets. The purpose of the present paper is to investigate some properties of s(Λ, sp)-open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets, β(Λ, sp)-open sets and b(Λ, sp)-open sets. In particular, the relationships between s(Λ, sp)-open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets, β(Λ, sp)- open sets, b(Λ, sp)-open sets and other related sets are explored. Furthermore, some characterizations of Λsp-extremally disconnected spaces are discussed. 2. Preliminaries Throughout the 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 is said to be β-open [4] if A ⊆ Cl(Int(Cl(A))). The complement of a β-open set A is called β-closed. The family of all β-open sets of a topological space (X, τ) is denoted by β(X, τ). A subset Λsp(A) [12] is defined as follows: Λsp(A) = ∩{U | A ⊆ U,U ∈ β(X, τ)}. Lemma 1. [12] For subsets A, B and Aα(α ∈ ∇) of a topological space (X, τ), the following hold: (1) A ⊆ Λsp(A). (2) If A ⊆ B, then Λsp(A) ⊆ Λsp(B). (3) Λsp(Λsp(A)) = Λsp(A). (4) If U ∈ β(X, τ), then Λsp(U) = U . (5) Λsp(∩{Aα|α ∈ ∇}) ⊆ ∩{Λsp(Aα)|α ∈ ∇}. (6) Λsp(∪{Aα|α ∈ ∇}) = ∪{Λsp(Aα)|α ∈ ∇}. A subset A of a topological space (X, τ) is called a Λsp-set [12] if A = Λsp(A). The family of all Λsp-sets of a topological space (X, τ) is denoted by Λsp(X, τ) (or simply Λsp). C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 574 Lemma 2. [12] For subsets A and Aα(α ∈ ∇) of a topological space (X, τ), the following hold: (1) Λsp(A) is a Λsp-set. (2) If A is β-open, then A is a Λsp-set. (3) If Aα is a Λsp-set for each α ∈ ∇, then ∩α∈∇Aα is a Λsp-set. (4) If Aα is a Λsp-set for each α ∈ ∇, then ∪α∈∇Aα is a Λsp-set. A subset A of a topological space (X, τ) is called (Λ, sp)-closed [3] if A = T ∩C, where T is a Λsp-set and C is a β-closed set. The complement of a (Λ, sp)-closed set is called (Λ, sp)-open. The family of all (Λ, sp)-open (resp. (Λ, sp)-closed) sets of a topological space (X, τ) is denoted by ΛspO(X, τ) (resp. ΛspC(X, τ)). Let A be a subsets of a topological space (X, τ). A point x ∈ X is called a (Λ, sp)-cluster point [3] of A if A∩U 6= ∅ for every (Λ, sp)-open set U of X containing x. The set of all (Λ, sp)-cluster points of A is called the (Λ, sp)-closure of A and is denoted by A(Λ,sp). Lemma 3. [3] Let A and B be subsets of a topological space (X, τ). For the (Λ, sp)-closure, the following properties hold: (1) A ⊆ A(Λ,sp) and [A(Λ,sp)](Λ,sp) = A(Λ,sp). (2) If A ⊆ B, then A(Λ,sp) ⊆ B(Λ,sp). (3) A(Λ,sp) is (Λ, sp)-closed. (4) A is (Λ, sp)-closed if and only if A(Λ,sp) = A. 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 (Λ, sp)-interior [3] of A and is denoted by A(Λ,sp). Lemma 4. [3] For subsets A and B of a topological space (X, τ), the following properties hold: (1) A(Λ,sp) ⊆ A and [A(Λ,sp)](Λ,sp) = A(Λ,sp). (2) If A ⊆ B, then A(Λ,sp) ⊆ B(Λ,sp). (3) A(Λ,sp) is (Λ, sp)-open. (4) A is (Λ, sp)-open if and only if A(Λ,sp) = A. (5) [X −A](Λ,sp) = X −A(Λ,sp). (6) [X −A](Λ,sp) = X −A(Λ,sp). C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 575 3. Generalized (Λ, sp)-open sets In this section, we investigate some properties of s(Λ, sp)-open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets, β(Λ, sp)-open sets and b(Λ, sp)-open sets. Definition 1. [3] A subset A of a topological space (X, τ) is said to be: (i) s(Λ, sp)-open if A ⊆ [A(Λ,sp)] (Λ,sp); (ii) p(Λ, sp)-open if A ⊆ [A(Λ,sp)](Λ,sp); (iii) α(Λ, sp)-open if A ⊆ [[A(Λ,sp)] (Λ,sp)](Λ,sp); (iv) β(Λ, sp)-open if A ⊆ [[A(Λ,sp)](Λ,sp)] (Λ,sp). The family of all s(Λ, sp)-open (resp. p(Λ, sp)-open, α(Λ, sp)-open, β(Λ, sp)-open) sets in a topological space (X, τ) is denoted by sΛspO(X, τ) (resp. pΛspO(X, τ), αΛspO(X, τ), βΛspO(X, τ)). The complement of a s(Λ, sp)-open (resp. p(Λ, sp)-open, α(Λ, sp)-open, β(Λ, sp)-open) set is called s(Λ, sp)-closed (resp. p(Λ, sp)-closed, α(Λ, sp)-closed, β(Λ, sp)- closed). The family of all s(Λ, sp)-closed (resp. p(Λ, sp)-closed, α(Λ, sp)-closed, β(Λ, sp)- closed) sets in a topological space (X, τ) is denoted by sΛspC(X, τ) (resp. pΛspC(X, τ), αΛspC(X, τ), βΛspC(X, τ)). Proposition 1. For a topological space (X, τ), the following properties hold: (1) ΛspO(X, τ) ⊆ αΛspO(X, τ) ⊆ sΛspO(X, τ) ⊆ βΛspO(X, τ). (2) αΛspO(X, τ) ⊆ pΛspO(X, τ) ⊆ βΛspO(X, τ). (3) αΛspO(X, τ) = sΛspO(X, τ) ∩ pΛspO(X, τ). Proof. (1) Let V ∈ ΛspO(X, τ). Then, we have V = V(Λ,sp) ⊆ [[V(Λ,sp)] (Λ,sp)](Λ,sp) ⊆ [V (Λ,sp)](Λ,sp) ⊆ [[V (Λ,sp)](Λ,sp)] (Λ,sp). Thus, ΛspO(X, τ) ⊆ αΛspO(X, τ) ⊆ sΛspO(X, τ) ⊆ βΛspO(X, τ). (2) Let V ∈ αΛspO(X, τ). Then, V ⊆ [V (Λ,sp)](Λ,sp) ⊆ [[V (Λ,sp)](Λ,sp)] (Λ,sp) and hence αΛspO(X, τ) ⊆ pΛspO(X, τ) ⊆ βΛspO(X, τ). (3) By (1) and (2), we have αΛspO(X, τ) ⊆ sΛspO(X, τ) ∩ pΛspO(X, τ). Let V ∈ sΛspO(X, τ) ∩ pΛspO(X, τ). Then, V ∈ sΛspO(X, τ) and V ∈ pΛspO(X, τ). Therefore, V ⊆ [V(Λ,sp)] (Λ,sp) and V ⊆ [V (Λ,sp)](Λ,sp). Thus, V ⊆ [V (Λ,sp)](Λ,sp) ⊆ [[V(Λ,sp)] (Λ,sp)](Λ,sp) and hence V ∈ αΛspO(X, τ). Consequently, we obtain sΛspO(X, τ) ∩ pΛspO(X, τ) ⊆ αΛspO(X, τ). This shows that αΛspO(X, τ) = sΛspO(X, τ) ∩ pΛspO(X, τ). Definition 2. A subset A of a topological space (X, τ) is said to be r(Λ, sp)-open if A = [A(Λ,sp)](Λ,sp). The complement of a r(Λ, sp)-open set is said to be r(Λ, sp)-closed. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 576 The family of all r(Λ, sp)-open (resp. r(Λ, sp)-closed) sets in a topological space (X, τ) is denoted by rΛspO(X, τ) (resp. rΛspC(X, τ)). Proposition 2. Let A be a subset of a topological space (X, τ), the following properties hold: (1) A is r(Λ, sp)-open if and only if A = F(Λ,sp) for some (Λ, sp)-closed set F . (2) A is r(Λ, sp)-closed if and only if A = U (Λ,sp) for some (Λ, sp)-open set U . Proposition 3. Let A be a subset of a topological space (X, τ), the following properties hold: (1) A is s(Λ, sp)-closed if and only if [A(Λ,sp)](Λ,sp) ⊆ A. (2) A is p(Λ, sp)-closed if and only if [A(Λ,sp)] (Λ,sp) ⊆ A. (3) A is α(Λ, sp)-closed if and only if [[A(Λ,sp)](Λ,sp)] (Λ,sp) ⊆ A. (4) A is β(Λ, sp)-closed if and only if [[A(Λ,sp)] (Λ,sp)](Λ,sp) ⊆ A. Proposition 4. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A is r(Λ, sp)-open. (2) A is (Λ, sp)-open and s(Λ, sp)-closed. (3) A is α(Λ, sp)-open and s(Λ, sp)-closed. (4) A is p(Λ, sp)-open and s(Λ, sp)-closed. (5) A is (Λ, sp)-open and β(Λ, sp)-closed. (6) A is α(Λ, sp)-open and β(Λ, sp)-closed. Proof. (1) ⇒ (2) ⇒ (3) ⇒ (4): Obvious. (4) ⇒ (5): Let A be (Λ, sp)-open and s(Λ, sp)-closed. Then, A ⊆ [A(Λ,sp)](Λ,sp) and [A(Λ,sp)](Λ,sp) ⊆ A. This implies that A = [A(Λ,sp)](Λ,sp). Therefore, A is r(Λ, sp)-open and hence A is (Λ, sp)-open. Since every s(Λ, sp)-closed set is β(Λ, sp)-closed. Thus, A is (Λ, sp)-open and β(Λ, sp)-closed. (5) ⇒ (6): The proof is obvious. (6) ⇒ (1): Let A be α(Λ, sp)-open and β(Λ, sp)-closed. Then, A ⊆ [[A(Λ,sp)] (Λ,sp)](Λ,sp) and [[A(Λ,sp)] (Λ,sp)](Λ,sp) ⊆ A. Thus, A = [[A(Λ,sp)] (Λ,sp)](Λ,sp) and hence A(Λ,sp) = [[A(Λ,sp)] (Λ,sp)](Λ,sp) = A. Therefore, [A(Λ,sp)](Λ,sp) = [[A(Λ,sp)] (Λ,sp)](Λ,sp) = A. This shows that A is r(Λ, sp)-open. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 577 Corollary 1. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A is r(Λ, sp)-closed. (2) A is (Λ, sp)-closed and s(Λ, sp)-open. (3) A is α(Λ, sp)-closed and s(Λ, sp)-open. (4) A is p(Λ, sp)-closed and s(Λ, sp)-open. (5) A is (Λ, sp)-closed and β(Λ, sp)-open. (6) A is α(Λ, sp)-closed and β(Λ, sp)-open. Proposition 5. For a subset A of a topological space (X, τ), the following properties hold: (1) [[[A(Λ,sp)] (Λ,sp)](Λ,sp)] (Λ,sp) = [A(Λ,sp)] (Λ,sp). (2) [[[A(Λ,sp)](Λ,sp)] (Λ,sp)](Λ,sp) = [A(Λ,sp)](Λ,sp). Definition 3. A subset A of a topological space (X, τ) is called (Λ, sp)-clopen if A is both (Λ, sp)-open and (Λ, sp)-closed. Proposition 6. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A is (Λ, sp)-clopen. (2) A is r(Λ, sp)-open and r(Λ, sp)-closed. (3) A is (Λ, sp)-open and α(Λ, sp)-closed. (4) A is (Λ, sp)-open and p(Λ, sp)-closed. (5) A is α(Λ, sp)-open and p(Λ, sp)-closed. (6) A is α(Λ, sp)-open and (Λ, sp)-closed. (7) A is p(Λ, sp)-open and (Λ, sp)-closed. (8) A is β(Λ, sp)-open and α(Λ, sp)-closed. Proof. (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5): Obvious. (5) ⇒ (6): Let A be α(Λ, sp)-open and p(Λ, sp)-closed. Then, A ⊆ [[A(Λ,sp)] (Λ,sp)](Λ,sp) and [[A(Λ,sp)] (Λ,sp)](Λ,sp) ⊆ A. Thus, A = [[A(Λ,sp)] (Λ,sp)](Λ,sp) and hence A(Λ,sp) = [[[A(Λ,sp)] (Λ,sp)](Λ,sp)] (Λ,sp). By Proposition 5, A(Λ,sp) = [A(Λ,sp)] (Λ,sp). Since [A(Λ,sp)] (Λ,sp) ⊆ A, we have A(Λ,sp) ⊆ A and hence A(Λ,sp) = A. This shows that A is (Λ, sp)-closed. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 578 (6) ⇒ (7) ⇒ (8): Obvious. (8) ⇒ (1): Let A be β(Λ, sp)-open and α(Λ, sp)-closed. Then, A ⊆ [A(Λ,sp)](Λ,sp) and [[A(Λ,sp)](Λ,sp)] (Λ,sp) ⊆ A. Thus, A(Λ,sp) ⊆ [[A(Λ,sp)](Λ,sp)] (Λ,sp) ⊆ A and hence A(Λ,sp) ⊆ A. Therefore, A is (Λ, sp)-closed. Since [[A(Λ,sp)](Λ,sp)] (Λ,sp) ⊆ A, we have [[[A(Λ,sp)](Λ,sp)] (Λ,sp)](Λ,sp) ⊆ A(Λ,sp), by Proposition 5, A ⊆ [A(Λ,sp)](Λ,sp) ⊆ A(Λ,sp) and hence A ⊆ A(Λ,sp). Thus, A is (Λ, sp)- open. Therefore, A is (Λ, sp)-clopen. Definition 4. A subset A of a topological space (X, τ) is said to be: (i) α(Λ, sp)-regular if A = [[A(Λ,sp)] (Λ,sp)](Λ,sp); (ii) β(Λ, sp)-regular if A = [[A(Λ,sp)](Λ,sp)] (Λ,sp). Proposition 7. Let A be a subset of a topological space (X, τ). Then, A is r(Λ, sp)-open if and only if A is α(Λ, sp)-regular. Proof. Suppose that A is a r(Λ, sp)-open set. Then, A = [A(Λ,sp)](Λ,sp). This implies that A is (Λ, sp)-open and so A = [[A(Λ,sp)] (Λ,sp)](Λ,sp). Thus, A is α(Λ, sp)-regular. Conversely, suppose that A is an α(Λ, sp)-regular set. Then, A = [[A(Λ,sp)] (Λ,sp)](Λ,sp). Therefore, A = [[[[A(Λ,sp)] (Λ,sp)](Λ,sp)](Λ,sp)](Λ,sp) = [[A(Λ,sp)] (Λ,sp)](Λ,sp) = A and hence A = [A(Λ,sp)](Λ,sp). Thus, A is (Λ, sp)-open. Proposition 8. Let A be a subset of a topological space (X, τ). Then, A is r(Λ, sp)-closed if and only if A is β(Λ, sp)-regular. Proof. Suppose that A is a r(Λ, sp)-closed set. Then, we have A = [A(Λ,sp)] (Λ,sp) and so A is (Λ, sp)-closed. Therefore, A = [A(Λ,sp)] (Λ,sp) = [[A(Λ,sp)](Λ,sp)] (Λ,sp). This shows that A is β(Λ, sp)-regular. Conversely, suppose that A is a β(Λ, sp)-regular set. Then, A = [[A(Λ,sp)](Λ,sp)] (Λ,sp) = [A(Λ,sp)] (Λ,sp). Thus, A is r(Λ, sp)-closed. Proposition 9. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A is β(Λ, sp)-regular. (2) A is β(Λ, sp)-open and (Λ, sp)-closed. (3) A is β(Λ, sp)-open and α(Λ, sp)-closed. Proposition 10. For a subset A of a topological space (X, τ), the following properties are equivalent: C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 579 (1) A is α(Λ, sp)-regular. (2) A is α(Λ, sp)-open and β(Λ, sp)-closed. Definition 5. A subset A of a topological space (X, τ) is said to be b(Λ, sp)-open if A ⊆ [A(Λ,sp)] (Λ,sp) ∪ [A(Λ,sp)](Λ,sp). The complement of a b(Λ, sp)-open set is said to be b(Λ, sp)-closed. The family of all b(Λ, sp)-open (resp. b(Λ, sp)-closed) sets in a topological space (X, τ) is denoted by bΛspO(X, τ) (resp. bΛspC(X, τ)). Remark 1. It is easy to see that for a topological space (X, τ), sΛspO(X, τ) ∪ pΛspO(X, τ) ⊆ bΛspO(X, τ) ⊆ βΛspO(X, τ). Proposition 11. Let A be a subset of a topological space (X, τ). If A = B ∪ C, where A is s(Λ, sp)-open and C is p(Λ, sp)-open, then A is b(Λ, sp)-open. The following result is an immediate consequence of Proposition 5 and Remark 1. Corollary 2. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A is r(Λ, sp)-open. (2) A is (Λ, sp)-open and b(Λ, sp)-closed. (3) A is α(Λ, sp)-open and b(Λ, sp)-closed. Lemma 5. Let A be a subset of a topological space (X, τ). If A is s(Λ, sp)-closed and β(Λ, sp)-open, then A is s(Λ, sp)-open. Proof. Since A is s(Λ, sp)-closed, it follows from Proposition 3 that [A(Λ,sp)](Λ,sp) ⊆ A. Since A is β(Λ, sp)-open, [A(Λ,sp)](Λ,sp) ⊆ A ⊆ [[A(Λ,sp)](Λ,sp)] (Λ,sp). Thus, [A(Λ,sp)](Λ,sp) ⊆ A(Λ,sp). Therefore, [[A(Λ,sp)](Λ,sp)] (Λ,sp) ⊆ [A(Λ,sp)] (Λ,sp) and hence A is s(Λ, sp)-open. Proposition 12. Let A be a subset of a topological space (X, τ). If A is b(Λ, sp)-open, then A(Λ,sp) is r(Λ, sp)-closed. Proof. Since A is b(Λ, sp)-open, we have A ⊆ [A(Λ,sp)] (Λ,sp) ∪ [A(Λ,sp)](Λ,sp) and hence A(Λ,sp) ⊆ [[A(Λ,sp)] (Λ,sp) ∪ [A(Λ,sp)](Λ,sp)] (Λ,sp) ⊆ [[A(Λ,sp)] (Λ,sp)](Λ,sp) ∪ [[A(Λ,sp)](Λ,sp)] (Λ,sp) = [[A(Λ,sp)](Λ,sp)] (Λ,sp) ⊆ A(Λ,sp). Thus, A(Λ,sp) = [[A(Λ,sp)](Λ,sp)] (Λ,sp). This shows that A(Λ,sp) is r(Λ, sp)-closed. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 580 Corollary 3. For a subset A of a topological space (X, τ), the following properties hold: (1) If A is s(Λ, sp)-open, then A(Λ,sp) is r(Λ, sp)-closed. (2) If A is p(Λ, sp)-open, then A(Λ,sp) is r(Λ, sp)-closed. (3) If A is α(Λ, sp)-open, then A(Λ,sp) is r(Λ, sp)-closed. Proposition 13. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A ∈ βΛspO(X, τ). (2) A(Λ,sp) ∈ rΛspC(X, τ). (3) A(Λ,sp) ∈ βΛspO(X, τ). (4) A(Λ,sp) ∈ sΛspO(X, τ). (5) A(Λ,sp) ∈ bΛspO(X, τ). Proof. (1) ⇒ (2): Let A ∈ βΛspO(X, τ). Then, we have A ⊆ [[A(Λ,sp)](Λ,sp)] (Λ,sp) and hence A(Λ,sp) ⊆ [[A(Λ,sp)](Λ,sp)] (Λ,sp) ⊆ A(Λ,sp). Thus, A(Λ,sp) = [[A(Λ,sp)](Λ,sp)] (Λ,sp). Therefore, A(Λ,sp) ∈ rΛspC(X, τ). (2) ⇒ (3) ⇒ (4) ⇒ (5): Obvious. (5) ⇒ (1): Let A(Λ,sp) ∈ bΛspO(X, τ). Then, A(Λ,sp) ⊆ [[A(Λ,sp)](Λ,sp)](Λ,sp) ∪ [[A(Λ,sp)](Λ,sp)] (Λ,sp) = [A(Λ,sp)](Λ,sp) ∪ [[A(Λ,sp)](Λ,sp)] (Λ,sp) = [[A(Λ,sp)](Λ,sp)] (Λ,sp) and hence A ⊆ [[A(Λ,sp)](Λ,sp)] (Λ,sp). Thus, A ∈ βΛspO(X, τ). Corollary 4. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A ∈ βΛspC(X, τ). (2) A(Λ,sp) ∈ rΛspO(X, τ). (3) A(Λ,sp) ∈ βΛspC(X, τ). (4) A(Λ,sp) ∈ sΛspC(X, τ). (5) A(Λ,sp) ∈ bΛspC(X, τ). Definition 6. A subset A of a topological space (X, τ) is called rs(Λ, sp)-open if there exists a r(Λ, sp)-open set U such that U ⊆ A ⊆ U (Λ,sp). The complement of a rs(Λ, sp)-open set is called rs(Λ, sp)-closed. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 581 The family of all rs(Λ, sp)-open (resp. rs(Λ, sp)-closed) sets in a topological space (X, τ) is denoted by rsΛspO(X, τ) (resp. rsΛspC(X, τ)). Remark 2. It is clear that every r(Λ, sp)-open set is rs(Λ, sp)-open. Proposition 14. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A is rs(Λ, sp)-open. (2) A is s(Λ, sp)-open and s(Λ, sp)-closed. (3) A is b(Λ, sp)-open and s(Λ, sp)-closed. (4) A is β(Λ, sp)-open and s(Λ, sp)-closed. (5) A is s(Λ, sp)-open and β(Λ, sp)-closed. (6) A is s(Λ, sp)-open and β(Λ, sp)-closed. Proof. (1) ⇒ (2): Suppose that A is a rs(Λ, sp)-open set. There exists a r(Λ, sp)- open set U such that U ⊆ A ⊆ U (Λ,sp). Then, U ⊆ A(Λ,sp) and hence A ⊆ U (Λ,sp) ⊆ [A(Λ,sp)](Λ,sp). Therefore, A is s(Λ, sp)-open. On the other hand, since U (Λ,sp) = A(Λ,sp) and U is r(Λ, sp)-open, [A(Λ,sp)](Λ,sp) = [U (Λ,sp)](Λ,sp) = U ⊆ A. Thus, by Proposition 3, A is s(Λ, sp)-closed. (2) ⇒ (3) and (3) ⇒ (4): The proofs are obvious. (4) ⇒ (5): Follows from Lemma 5 and since sΛspO(X, τ) ⊆ bΛspO(X, τ). (5) ⇒ (6): The proof is obvious. (6) ⇒ (1): Since A is s(Λ, sp)-open and β(Λ, sp)-closed, it follows from Lemma 5 that A is s(Λ, sp)-closed. Thus, by Proposition 3, [A(Λ,sp)](Λ,sp) ⊆ A ⊆ [A(Λ,sp)] (Λ,sp) ⊆ [[A(Λ,sp)](Λ,sp)] (Λ,sp). Let U = [A(Λ,sp)](Λ,sp). Then, U is r(Λ, sp)-open and hence U ⊆ A ⊆ U (Λ,sp). Thus, A is rs(Λ, sp)-open. Remark 3. It is clear from Proposition 14 that if A is a rs(Λ, sp)-open set of a topological space (X, τ), then X −A is rs(Λ, sp)-open. Proposition 15. Let (X, τ) be a topological space and x ∈ X. Then, {x} is (Λ, sp)-open if and only if {x} is s(Λ, sp)-open. Proof. The necessity is clear. Suppose that {x} is s(Λ, sp)-open. Then, {x} ⊆ [{x}(Λ,sp)](Λ,sp). Now {x}(Λ,sp) is either {x} or ∅. Since ∅(Λ,sp) = ∅ and {x} ⊆ [{x}(Λ,sp)](Λ,sp), {x}(Λ,sp) 6= ∅. Therefore, {x}(Λ,sp) = {x} and by Lemma 4, {x} is (Λ, sp)-open. Lemma 6. Let (X, τ) be a topological space and A ⊆ X. If U ∈ ΛspO(X, τ) and U∩A = ∅, then U ∩A(Λ,sp) = ∅. Proposition 16. Let (X, τ) be a topological space and x ∈ X. Then, the following properties are equivalent: C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 582 (1) {x} is p(Λ, sp)-open. (2) {x} is b(Λ, sp)-open. (3) {x} is β(Λ, sp)-open. Proof. (1) ⇒ (2) and (2) ⇒ (3) follows from Remark 1. (3) ⇒ (1): Let {x} be β(Λ, sp)-open. Assume that {x} is not p(Λ, sp)-open. Then, {x} * [{x}(Λ,sp)](Λ,sp), that is {x} ∩ [{x}(Λ,sp)](Λ,sp) = ∅. Since [{x}(Λ,sp)](Λ,sp) is (Λ, sp)- open, it follows from Lemma 6 that {x}(Λ,sp)∩[{x}(Λ,sp)](Λ,sp) = ∅. Thus, [{x}(Λ,sp)](Λ,sp) = ∅ and hence [[{x}(Λ,sp)](Λ,sp)](Λ,sp) = ∅(Λ,sp) = ∅. This is a contradiction. Proposition 17. Let (X, τ) be a topological space and x ∈ X. Then, {x} is p(Λ, sp)-open or {x} is α(Λ, sp)-closed. Proof. Assume that {x} is not p(Λ, sp)-open. Then, {x} * [{x}(Λ,sp)](Λ,sp) and hence {x} ∩ [{x}(Λ,sp)](Λ,sp) = ∅. Since [{x}(Λ,sp)](Λ,sp) is (Λ, sp)-open, it follows from Lemma 6 that {x}(Λ,sp) ∩ [{x}(Λ,sp)](Λ,sp) = ∅. Therefore, [{x}(Λ,sp)](Λ,sp) = ∅. This implies that [[{x}(Λ,sp)](Λ,sp)](Λ,sp) = ∅(Λ,sp) = ∅. Thus, by Proposition 3, {x} is α(Λ, sp)-closed. Proposition 18. Let A be a subset of a topological space (X, τ). Then, A is s(Λ, sp)-open if and only if there exists a (Λ, sp)-open set U such that U ⊆ A ⊆ U (Λ,sp). Proof. Suppose that A is s(Λ, sp)-open. Then, A ⊆ [A(Λ,sp)] (Λ,sp). Let U = A(Λ,sp). Thus, we obtain U ⊆ A ⊆ U (Λ,sp). Conversely, assume that there exists a (Λ, sp)-open set U such that U ⊆ A ⊆ U (Λ,sp). Then, U ⊆ A(Λ,sp) and hence U (Λ,sp) ⊆ [A(Λ,sp)] (Λ,sp). Since A ⊆ U (Λ,sp), A ⊆ [A(Λ,sp)] (Λ,sp). Thus, A is s(Λ, sp)-open. Proposition 19. Let A be a subset of a topological space (X, τ). If there exists a p(Λ, sp)- open set U such that U ⊆ A ⊆ U (Λ,sp), then A is β(Λ, sp)-open. Proof. Since U ⊆ A ⊆ U (Λ,sp), we have A(Λ,sp) = U (Λ,sp) and hence [A(Λ,sp)](Λ,sp) = [U (Λ,sp)](Λ,sp). Since U is p(Λ, sp)-open, we have U ⊆ [A(Λ,sp)](Λ,sp). Thus, A ⊆ U (Λ,sp) and hence A ⊆ [[A(Λ,sp)](Λ,sp)] (Λ,sp). This shows that A is β(Λ, sp)-open. A subset D of a topological space (X, τ) is called Λsp-dense [3] if D(Λ,sp) = X. D is called Λsp-codense [3] if X −D is Λsp-dense. Proposition 20. Let (X, τ) be a topological space and D ⊆ X. Then, the following properties are equivalent: (1) D is Λsp-dense. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 583 (2) If F is any (Λ, sp)-closed set and D ⊆ F , then F = X. (3) Each nonempty (Λ, sp)-open set contains an element of D. (4) The complement of D has empty (Λ, sp)-interior. Proof. (1) ⇒ (2): Let F be a (Λ, sp)-closed set such that D ⊆ F . Then, X = D(Λ,sp) ⊆ F (Λ,sp) = F . (2) ⇒ (3): Let U be a nonempty (Λ, sp)-open set such that U ∩ D = ∅; then D ⊆ X − U 6= X, which contradicts (2), since X − U is (Λ, sp)-closed. (3) ⇒ (4): Assume that [X−D](Λ,sp) 6= ∅; since [X−D](Λ,sp) is (Λ, sp)-open, there is a nonempty (Λ, sp)-open set U such that U ⊆ [X−D](Λ,sp), and since [X−D](Λ,sp) ⊆ X−D, U contains no point of D. (3) ⇒ (4): [X −D](Λ,sp) = X −D(Λ,sp) = ∅ so that D(Λ,sp) = X. Remark 4. Let A be a subset of a topological space (X, τ). If A is Λsp-dense, then A is p(Λ, sp)-open. Proposition 21. Let A be a subset of a topological space (X, τ). If A is p(Λ, sp)-open, then A is the intersection of a r(Λ, sp)-open set and a Λsp-dense set. Proof. Suppose that A is a p(Λ, sp)-open set. Then, we have A ⊆ [A(Λ,sp)](Λ,sp) and hence A = [A ∪ [X − A(Λ,sp)]] ∩ [A(Λ,sp)](Λ,sp). Let C = [A(Λ,sp)](Λ,sp) and D = A ∪ [X − A(Λ,sp)]. Then, C is r(Λ, sp)-open, by Proposition 2, A(Λ,sp) ⊆ D(Λ,sp) since A ⊆ D and X −A(Λ,sp) ⊆ D ⊆ D(Λ,sp). Thus, D(Λ,sp) = X. Corollary 5. Let A be a subset of a topological space (X, τ). If A is p(Λ, sp)-closed, then A is the union of a r(Λ, sp)-closed set and a set has empty (Λ, sp)-interior. Proposition 22. Let A be a subset of a topological space (X, τ). If A is s(Λ, sp)-open, then A is the intersection of a r(Λ, sp)-closed set F and a set C such that C(Λ,sp) is Λsp-dense. Proof. Suppose that A is s(Λ, sp)-open. Then, we have A ⊆ [A(Λ,sp)] (Λ,sp) and hence A = [A ∪ [X − [A(Λ,sp)] (Λ,sp)]] ∩ [A(Λ,sp)] (Λ,sp). Let F = [A(Λ,sp)] (Λ,sp) and C = A ∪ [X − [A(Λ,sp)] (Λ,sp)]. Then, F is r(Λ, sp)-closed, by Proposition 2, we have [A(Λ,sp)] (Λ,sp) ⊆ [C(Λ,sp)] (Λ,sp). Since X− [A(Λ,sp)] (Λ,sp) ⊆ C and X− [A(Λ,sp)] (Λ,sp) is (Λ, sp)-open, X− [A(Λ,sp)] (Λ,sp) ⊆ C(Λ,sp) ⊆ [C(Λ,sp)] (Λ,sp). Thus, [C(Λ,sp)] (Λ,sp) = X. Corollary 6. Let A be a subset of a topological space (X, τ). If A is s(Λ, sp)-closed, then A is the union of a r(Λ, sp)-open set and a set whose (Λ, sp)-closure has empty (Λ, sp)-interior. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 584 Proposition 23. Let A be a subset of a topological space (X, τ). If A is β(Λ, sp)-open, then A is the intersection of a r(Λ, sp)-closed set F and a Λsp-dense set D. Proof. Suppose that A is β(Λ, sp)-open. Then, we have A ⊆ [[A(Λ,sp)](Λ,sp)] (Λ,sp) and hence A = [A ∪ [X − A(Λ,sp)]] ∩ [[A(Λ,sp)](Λ,sp)] (Λ,sp). Let F = [[A(Λ,sp)](Λ,sp)] (Λ,sp) and D = A∪ [X −A(Λ,sp)]. Then, F is r(Λ, sp)-closed by Proposition 2, also A(Λ,sp) ⊆ D(Λ,sp). Since X −A(Λ,sp) ⊆ D ⊆ D(Λ,sp), we have D(Λ,sp) = X. Corollary 7. Let A be a subset of a topological space (X, τ). If A is β(Λ, sp)-closed, then A is the union of a r(Λ, sp)-open set and a set has empty (Λ, sp)-interior. Lemma 7. Let A be a subset of a topological space (X, τ). If A is (Λ, sp)-closed and p(Λ, sp)-open, then A is (Λ, sp)-open. Theorem 1. For a topological space (X, τ), the following properties are equivalent: (1) Every s(Λ, sp)-open set of X is α(Λ, sp)-open. (2) Every s(Λ, sp)-open set of X is p(Λ, sp)-open. (3) Every β(Λ, sp)-open set of X is p(Λ, sp)-open. (4) Every b(Λ, sp)-open set of X is p(Λ, sp)-open. (5) Every rs(Λ, sp)-open set of X is p(Λ, sp)-open. (6) Every rs(Λ, sp)-open set of X is r(Λ, sp)-open. (7) Every r(Λ, sp)-closed set of X is p(Λ, sp)-open. (8) Every r(Λ, sp)-closed set of X is (Λ, sp)-open. Proof. (1) ⇒ (2): This is obvious since αΛspO(X, τ) ⊆ sΛspO(X, τ). (2) ⇒ (3): Let A be a β(Λ, sp)-open set. Then, A ⊆ [[A(Λ,sp)](Λ,sp)] (Λ,sp). It follows from Proposition 2 that B = [[A(Λ,sp)](Λ,sp)] (Λ,sp) is r(Λ, sp)-closed and thus s(Λ, sp)-open. By (2), B is p(Λ, sp)-open and hence A ⊆ B ⊆ [B(Λ,sp)](Λ,sp) = B(Λ,sp). Also it is clear that B ⊆ A(Λ,sp) and thus B(Λ,sp) ⊆ [A(Λ,sp)](Λ,sp). Therefore, A ⊆ [A(Λ,sp)](Λ,sp). This shows that A is p(Λ, sp)-open. (3) ⇒ (4): This is obvious since bΛspO(X, τ) ⊆ βΛspO(X, τ). (4) ⇒ (5): It follows from Proposition 14 that rsΛspO(X, τ) ⊆ sΛspO(X, τ). Since sΛspO(X, τ) ⊆ bΛspO(X, τ), rsΛspO(X, τ) ⊆ bΛspO(X, τ). Thus, the result follows from (3). (5) ⇒ (6): Since every rs(Λ, sp)-open set is s(Λ, sp)-closed, it follows from (4) that a rs(Λ, sp)-open set is both s(Λ, sp)-closed and p(Λ, sp)-open. Thus, by Proposition 4, follows. (6) ⇒ (7): Follows from Proposition 4 and Remark 3. (7) ⇒ (8): Follows from Lemma 7. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 585 (8) ⇒ (1): Let A be a s(Λ, sp)-open set. Then, by Corollary 3, A(Λ,sp) is r(Λ, sp)- closed. By (8), A(Λ,sp) is (Λ, sp)-open and hence A(Λ,sp) ⊆ [A(Λ,sp)](Λ,sp). Therefore, A is p(Λ, sp)-open. Since A ∈ sΛspO(X, τ) ∩ pΛspO(X, τ) = αΛspO(X, τ), (1) follows. Corollary 8. For a topological space (X, τ), the following properties are equivalent: (1) αΛspO(X, τ) = sΛspO(X, τ). (2) Every rs(Λ, sp)-open set of X is p(Λ, sp)-closed. (3) Every rs(Λ, sp)-open set of X is r(Λ, sp)-closed. Proof. Follows from Remark 3 and Theorem 1. Definition 7. A subset A of a topological space (X, τ) is called p(Λ, sp)-clopen if A is both p(Λ, sp)-open and p(Λ, sp)-closed. Corollary 9. For a topological space (X, τ), the following properties are equivalent: (1) αΛspO(X, τ) = sΛspO(X, τ). (2) Every rs(Λ, sp)-open set of X is p(Λ, sp)-clopen. (3) Every rs(Λ, sp)-open set of X is (Λ, sp)-clopen. Proof. Follows from Theorem 1 and Corollary 8. Proposition 24. For a topological space (X, τ), the following properties are equivalent: (1) Every p(Λ, sp)-open set of X is α(Λ, sp)-open. (2) Every p(Λ, sp)-open set of X is s(Λ, sp)-open. Proof. Follows from Proposition 1. 4. Some characterizations of Λsp-extremally disconnected spaces In this section, we investigate some characterizations of Λsp-extremally disconnected spaces. Definition 8. [3] A topological space (X, τ) is called Λsp-extremally disconnected if U (Λ,sp) is (Λ, sp)-open in X for every (Λ, sp)-open set U of X. Theorem 2. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is Λsp-extremally disconnected. (2) For each V ∈ βΛspO(X, τ), V (Λ,sp) ∈ rΛspO(X, τ). C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 586 (3) For each V ∈ bΛspO(X, τ), V (Λ,sp) ∈ rΛspO(X, τ). (4) For each V ∈ sΛspO(X, τ), V (Λ,sp) ∈ rΛspO(X, τ). (5) For each V ∈ αΛspO(X, τ), V (Λ,sp) ∈ rΛspO(X, τ). (6) For each V ∈ ΛspO(X, τ), V (Λ,sp) ∈ rΛspO(X, τ). (7) For each V ∈ rΛspO(X, τ), V (Λ,sp) ∈ rΛspO(X, τ). (8) For each V ∈ pΛspO(X, τ), V (Λ,sp) ∈ rΛspO(X, τ). Proof. (1) ⇒ (2): Let V ∈ βΛspO(X, τ). By (1) and Proposition 13, we have V (Λ,sp) = [[V (Λ,sp)](Λ,sp)] (Λ,sp) = [[[V (Λ,sp)](Λ,sp)] (Λ,sp)](Λ,sp) = [V (Λ,sp)](Λ,sp) and hence V (Λ,sp) = [V (Λ,sp)](Λ,sp) = [[V (Λ,sp)](Λ,sp)](Λ,sp). Thus, V (Λ,sp) ∈ rΛspO(X, τ). (2) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (6) ⇒ (7): Obvious. (7) ⇒ (8): Let V ∈ pΛspO(X, τ). Then, we have [V (Λ,sp)](Λ,sp) is r(Λ, sp)-open, by (7), [[V (Λ,sp)](Λ,sp)] (Λ,sp) is r(Λ, sp)-open. Therefore, V (Λ,sp) = [[V (Λ,sp)](Λ,sp)] (Λ,sp) = [[[[V (Λ,sp)](Λ,sp)] (Λ,sp)](Λ,sp)](Λ,sp) = [V (Λ,sp)](Λ,sp) = [[V (Λ,sp)](Λ,sp)](Λ,sp). Thus, V (Λ,sp) ∈ rΛspO(X, τ). (8) ⇒ (1): The proof is obvious. Theorem 3. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is Λsp-extremally disconnected. (2) rΛspC(X, τ) ⊆ ΛspO(X, τ). (3) rΛspC(X, τ) ⊆ αΛspO(X, τ). (4) rΛspC(X, τ) ⊆ pΛspO(X, τ). (5) sΛspO(X, τ) ⊆ αΛspO(X, τ). (6) sΛspC(X, τ) ⊆ αΛspC(X, τ). (7) sΛspC(X, τ) ⊆ pΛspC(X, τ). (8) sΛspO(X, τ) ⊆ pΛspO(X, τ). (9) βΛspO(X, τ) ⊆ pΛspO(X, τ). C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 572-588 587 (10) βΛspC(X, τ) ⊆ pΛspC(X, τ). (11) bΛspC(X, τ) ⊆ pΛspC(X, τ). (12) bΛspO(X, τ) ⊆ pΛspO(X, τ). (13) rΛspO(X, τ) ⊆ pΛspC(X, τ). (14) rΛspO(X, τ) ⊆ ΛspC(X, τ). (15) rΛspO(X, τ) ⊆ αΛspC(X, τ). Proof. (1) ⇒ (2): Let V ∈ rΛspC(X, τ). Then, we have V = [V(Λ,sp)] (Λ,sp). Since (X, τ) is Λsp-extremally disconnected, V(Λ,sp) = [[V(Λ,sp)] (Λ,sp)](Λ,sp) = [V(Λ,sp)] (Λ,sp) = V and hence V ∈ ΛspO(X, τ). Consequently, we obtain rΛspC(X, τ) ⊆ ΛspO(X, τ). (2) ⇒ (3) ⇒ (4): Obvious. (4) ⇒ (5): Let V ∈ sΛspO(X, τ). Then, we have V ⊆ [V(Λ,sp)] (Λ,sp). Since [V(Λ,sp)] (Λ,sp) is r(Λ, sp)-closed, by (4), [V(Λ,sp)] (Λ,sp) is p(Λ, sp)-open and hence [V(Λ,sp)] (Λ,sp) ⊆ [[[V(Λ,sp)] (Λ,sp)](Λ,sp)](Λ,sp) = [[V(Λ,sp)] (Λ,sp)](Λ,sp). This implies that V ⊆ [[V(Λ,sp)] (Λ,sp)](Λ,sp) and hence V ∈ αΛspO(X, τ). Thus, sΛspO(X, τ) ⊆ αΛspO(X, τ). (5) ⇒ (6) ⇒ (7) ⇒ (8): Obvious. (8) ⇒ (9): Let V ∈ βΛspO(X, τ). By Proposition 13, V (Λ,sp) is s(Λ, sp)-open, by (8), V (Λ,sp) is p(Λ, sp)-open. Thus, V (Λ,sp) ⊆ [[V (Λ,sp)](Λ,sp)](Λ,sp) = [V (Λ,sp)](Λ,sp) and hence V ⊆ [V (Λ,sp)](Λ,sp). Therefore, V ∈ pΛspO(X, τ). This shows that βΛspO(X, τ) ⊆ pΛspO(X, τ). (9) ⇒ (10) ⇒ (11) ⇒ (12): Obvious. (12) ⇒ (13): Let V ∈ rΛspO(X, τ). Then, V is β(Λ, sp)-open, by Proposition 13, V (Λ,sp) is b(Λ, sp)-open and by (12), V (Λ,sp) is p(Λ, sp)-open. Thus, V (Λ,sp) ⊆ [[V (Λ,sp)](Λ,sp)](Λ,sp) = [V (Λ,sp)](Λ,sp) = V. Therefore, V is p(Λ, sp)-closed and hence rΛspO(X, τ) ⊆ pΛspC(X, τ). (13) ⇒ (14): Let V ∈ rΛspO(X, τ). By (13), we have V is p(Λ, sp)-closed and hence [V(Λ,sp)] (Λ,sp) ⊆ V . Since V is (Λ, sp)-open, V (Λ,sp) ⊆ V . Thus, V ∈ ΛspC(X, τ). Conse- quently, we obtain rΛspO(X, τ) ⊆ ΛspC(X, τ). (14) ⇒ (15): The proof is obvious. (15) ⇒ (1): Let V be a (Λ, sp)-open set. Then, [V (Λ,sp)](Λ,sp) is r(Λ, sp)-open, by (15), [V (Λ,sp)](Λ,sp) is α(Λ, sp)-closed. 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