EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 1244-1253 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and lower almost (τ1, τ2)-continuous multifunctions Chalongchai Klanarong1, Supannee Sompong2, Chawalit Boonpok1,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand 2 Department of Mathematics and Statistics, Faculty of Science and Technology, Sakon Nakhon Rajbhat University, Sakon Nakhon, 47000, Thailand Abstract. This paper is concerned with the concepts of upper and lower almost (τ1, τ2)-continuous multifunctions. Furthermore, some characterizations of upper and lower almost (τ1, τ2)-continuous multifunctions are investigated. 2020 Mathematics Subject Classifications: 54C08, 54C60, 54E55 Key Words and Phrases: Upper almost (τ1, τ2)-continuous multifunction; lower almost (τ1, τ2)- continuous multifunction 1. Introduction It is well-known that the branch of mathematics called topology is related to all ques- tions directly or indirectly concerned with continuity. Semi-open sets, preopen sets, α-open sets, β-open sets and δ-open sets play an important role in the researches of generaliza- tions of continuity in topological spaces. By using these sets many authors introduced and studied various types of weak forms of continuity for functions and multifunctions. Singal and Singal [28] introduced the concept of almost continuous functions as a generaliza- tion of continuity. Munshi and Bassan [16] studied the notion of almost semi-continuous functions. Noiri [18] introduced and investigated the concept of almost α-continuous func- tions. Nasef and Noiri [17] introduced two classes of functions, namely almost precontin- uous functions and almost β-continuous functions by utilizing the notions of preopen sets and β-open sets due to Mashhour et al [15] and Abd El-Monsef et al. [12], respectively. The class of almost precontinuity is a generalization of almost α-continuity. The class of almost β-continuity is a generalization of almost semi-continuity. Keskin and Noiri [13] introduced the concept of almost b-continuous functions by utilizing the notion of b-open ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5192 Email addresses: chalongchai.k@msu.ac.th (C. Klanarong), s−sompong@snru.ac.th (S. Sompong), chawalit.b@msu.ac.th (C. Boonpok) https://www.ejpam.com 1244 © 2024 EJPAM All rights reserved. C. Klanarong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (2) (2024), 1244-1253 1245 sets due to Andrijević [1]. The class of almost b-continuity is a generalization of almost precontinuity and almost semi-continuity. The class of almost β-continuity is a gener- alization of almost b-continuity. Popa [22] introduced the concepts of upper and lower almost continuous multifunctions. Popa and Noiri [23] introduced the notions of upper and lower almost quasi-continuous multifunctions. Several characterizations of upper and lower almost quasi-continuous multifunctions were investigated in [19]. In 1996, Popa and Noiri [24] introduced and investigated the notions of upper and lower almost α-continuous multifunctions. In 1997, Popa et al. [26] introduced the concepts of upper and lower almost precontinuous multifunctions. In particular, several characteriza- tions of upper and lower almost precontinuous multifunctions were presented in [27]. In 1999, Noiri and Popa [20] introduced the concepts of upper and lower almost β-continuous multifunctions. Some characterizations of upper and lower almost β-continuous multi- functions were investigated in [25]. In 2006, Ekici and Park [11] introduced and studied almost γ-continuous multifunctions. Noiri and Popa [21] introduced and investigated the notions of upper and lower almost m-continuous multifunctions as multifunctions from a set satisfying some minimal conditions into a topological space. In [3], the present author introduced and studied the concept of pairwise almost M -continuous functions in bimini- mal structure spaces. Laprom et al. [14] introduced and investigated the notion of almost β(τ1, τ2)-continuous multifunctions. Viriyapong and Boonpok [29] introduced and studied the concept of almost (τ1, τ2)α-continuous multifunctions. Moreover, some characteriza- tions of almost (τ1, τ2)δ-semicontinuous multifunctions, almost weakly (τ1, τ2)-continuous multifunctions, almost (Λ, sp)-continuous multifunctions, almost β(⋆)-continuous multi- functions and almost ⋆-continuous multifunctions were established in [5], [8], [10], [6] and [4] respectively. In this paper, we introduce the concepts of upper and lower almost (τ1, τ2)-continuous multifunctions. Furthermore, several characterizations of upper and lower almost (τ1, τ2)-continuous multifunctions are investigated. 2. Preliminaries Throughout the present paper, spaces (X, τ1, τ2) and (Y, σ1, σ2) (or simply X and Y ) always mean bitopological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a bitopological space (X, τ1, τ2). The closure of A and the interior of A with respect to τi are denoted by τi-Cl(A) and τi-Int(A), respectively, for i = 1, 2. A subset A of a bitopological space (X, τ1, τ2) is called τ1τ2-closed [9] if A = τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is called τ1τ2-open. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection of all τ1τ2-closed sets of X containing A is called the τ1τ2-closure [9] of A and is denoted by τ1τ2-Cl(A). The union of all τ1τ2-open sets of X contained in A is called the τ1τ2-interior [9] of A and is denoted by τ1τ2-Int(A). Lemma 1. [9] Let A and B be subsets of a bitopological space (X, τ1, τ2). For the τ1τ2- closure, the following properties hold: (1) A ⊆ τ1τ2-Cl(A) and τ1τ2-Cl(τ1τ2-Cl(A)) = τ1τ2-Cl(A). C. Klanarong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (2) (2024), 1244-1253 1246 (2) If A ⊆ B, then τ1τ2-Cl(A) ⊆ τ1τ2-Cl(B). (3) τ1τ2-Cl(A) is τ1τ2-closed. (4) A is τ1τ2-closed if and only if A = τ1τ2-Cl(A). (5) τ1τ2-Cl(X −A) = X − τ1τ2-Int(A). A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [29] (resp. (τ1, τ2)s-open [5], (τ1, τ2)p-open [5], (τ1, τ2)β-open [5]) if A = τ1τ2-Int(τ1τ2-Cl(A)) (resp. A ⊆ τ1τ2-Cl(τ1τ2-Int(A)), A ⊆ τ1τ2-Int(τ1τ2-Cl(A)), A ⊆ τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A)))). The complement of a (τ1, τ2)r-open (resp. (τ1, τ2)s-open, (τ1, τ2)p-open, (τ1, τ2)β-open) set is called (τ1, τ2)r-closed, (τ1, τ2)s-closed, (τ1, τ2)p-closed, (τ1, τ2)β-closed. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection of all (τ1, τ2)s-closed sets of X containing A is called the (τ1, τ2)s-closure [5] of A and is denoted by (τ1, τ2)-sCl(A). The union of all (τ1, τ2)s-open sets of X contained in A is called the (τ1, τ2)s-interior [5] of A and is denoted by (τ1, τ2)-sInt(A). A subset A of a bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-open [30] if A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)-open set is called α(τ1, τ2)-closed. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection of all (τ1, τ2)p-closed (resp. α(τ1, τ2)-closed) sets of X containing A is called the (τ1, τ2)p-closure (resp. α(τ1, τ2)-closure) and is denoted by (τ1, τ2)-pCl(A) (resp. (τ1, τ2)-αCl(A)). By a multifunction F : X → Y , we mean a point-to-set correspondence from X into Y , and we always assume that F (x) ̸= ∅ for all x ∈ X. For a multifunction F : X → Y , following [2] we shall denote the upper and lower inverse of a set B of Y by F+(B) and F−(B), respectively, that is, F+(B) = {x ∈ X | F (x) ⊆ B} and F−(B) = {x ∈ X | F (x) ∩B ̸= ∅}. In particular, F−(y) = {x ∈ X | y ∈ F (x)} for each point y ∈ Y . For each A ⊆ X, F (A) = ∪x∈AF (x). 3. Upper and lower almost (τ1, τ2)-continuous multifunctions In this section, we introduce the notions of upper and lower almost (τ1, τ2)-continuous multifunctions. Moreover, several characterizations of upper and lower almost (τ1, τ2)- continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper almost (τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing F (x), there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper almost (τ1, τ2)-continuous if F has this property at each point of X. Lemma 2. For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: C. Klanarong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (2) (2024), 1244-1253 1247 (1) (τ1, τ2)-sCl(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∪A [5]; (2) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A. Lemma 3. Let A be a subset of a bitopological space (X, τ1, τ2). If A is τ1τ2-open in X, then (τ1, τ2)-sCl(A) = τ1τ2-Int(τ1τ2-Cl(A)). Theorem 1. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost (τ1, τ2)-continuous at x ∈ X; (2) x ∈ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y containing F (x); (3) x ∈ τ1τ2-Int(F +((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y containing F (x); (4) x ∈ τ1τ2-Int(F +(V )) for every (σ1, σ2)r-open set V of Y containing F (x); (5) for each (σ1, σ2)r-open set V of Y containing F (x), there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ V . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing F (x). There exists a τ1τ2-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). Thus, x ∈ U ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))) and hence x ∈ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Int(V )))). (2) ⇒ (3): This follows from Lemma 3. (3) ⇒ (4): Let V be any (σ1, σ2)r-open set of Y containing F (x). Then, it follows from Lemma 3 that V = σ1σ2-Int(σ1σ2-Cl(V )) = (σ1, σ2)-sCl(V ). (4) ⇒ (5): Let V be any (σ1, σ2)r-open set of Y containing F (x). Then by (4), x ∈ τ1τ2-Int(F +(V )) and there exists a τ1τ2-open set U of X containing x such that x ∈ U ⊆ F+(V ); hence F (U) ⊆ V . (5) ⇒ (1): Let V be any σ1σ2-open set of Y containing F (x). Since σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r-open, there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). This shows that F is upper almost (τ1, τ2)-continuous at x ∈ X. Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower almost (τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a τ1τ2-open set U of X containing x such that σ1σ2-Int(σ1σ2-Cl(V ))∩F (z) ̸= ∅ for each z ∈ U . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower almost (τ1, τ2)-continuous if F has this property at each point of X. Theorem 2. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost (τ1, τ2)-continuous at x ∈ X; C. Klanarong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (2) (2024), 1244-1253 1248 (2) x ∈ τ1τ2-Int(F −(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅; (3) x ∈ τ1τ2-Int(F −((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅; (4) x ∈ τ1τ2-Int(F −(V )) for every (σ1, σ2)r-open set V of Y such that F (x) ∩ V ̸= ∅; (5) for each (σ1, σ2)r-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a τ1τ2-open set U of X containing x such that U ⊆ F−(V ). Proof. The proof is similar to that of Theorem 1. Definition 3. [7] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost (τ1, τ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing f(x), there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost (τ1, τ2)-continuous if f has this property at each point of X. Corollary 1. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost (τ1, τ2)-continuous at x ∈ X; (2) x ∈ τ1τ2-Int(f −1(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y containing f(x); (3) x ∈ τ1τ2-Int(f −1((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y containing f(x); (4) x ∈ τ1τ2-Int(f −1(V )) for every (σ1, σ2)r-open set V of Y containing f(x); (5) for each (σ1, σ2)r-open set V of Y containing f(x), there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ V . Theorem 3. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost (τ1, τ2)-continuous; (2) F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ; (3) τ1τ2-Cl(F −(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(F −(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (5) F+(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) for every subset B of Y ; C. Klanarong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (2) (2024), 1244-1253 1249 (6) F+(V ) is τ1τ2-open in X for every (σ1, σ2)r-open set V of Y ; (7) F−(K) is τ1τ2-closed in X for every (σ1, σ2)r-closed set K of Y . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V . Thus, by Theorem 1, x ∈ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))) and hence F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))). (2) ⇒ (3): Let K be any σ1σ2-closed set of Y . Then, Y −K is σ1σ2-open in Y and by (2), X − F−(K) = F+(Y −K) ⊆ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(Y −K)))) = τ1τ2-Int(X − F−(σ1σ2-Cl(σ1σ2-Int(K)))) = X − τ1τ2-Cl(F −(σ1σ2-Cl(σ1σ2-Int(K)))). Thus, τ1τ2-Cl(F −(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F−(K). (3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Cl(B) is a σ1σ2-closed set of Y and by (3), τ1τ2-Cl(F −(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F−(σ1σ2-Cl(B)). (4) ⇒ (5): Let B be any subset of Y . Then, we have F+(σ1σ2-Int(B)) = X − F−(σ1σ2-Cl(Y −B)) ⊆ X − τ1τ2-Cl(F −(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(Y −B))))) = X − τ1τ2-Cl(F −(Y − σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) = τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))). (5) ⇒ (6): Let V be any (σ1, σ2)r-open set of Y . By (5), we have F+(V ) ⊆ τ1τ2-Int(F +(V )) and hence F+(V ) is τ1τ2-open in X. (6) ⇒ (7): The proof is obvious. (7) ⇒ (1): Let x ∈ X and V be any (σ1, σ2)r-open set of Y containing F (x). Since Y −V is (σ1, σ2)r-closed and by (7), X −F+(V ) = F−(Y −V ) is τ1τ2-closed in X. Thus, F+(V ) is τ1τ2-open and hence x ∈ τ1τ2-Int(F +(V )). Then, there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ V . It follows from Theorem 1 that F is upper almost (τ1, τ2)-continuous. Theorem 4. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost (τ1, τ2)-continuous; (2) F−(V ) ⊆ τ1τ2-Int(F −(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ; (3) τ1τ2-Cl(F +(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(F +(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; C. Klanarong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (2) (2024), 1244-1253 1250 (5) F−(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F −(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) for every subset B of Y ; (6) F−(V ) is τ1τ2-open in X for every (σ1, σ2)r-open set V of Y ; (7) F+(K) is τ1τ2-closed in X for every (σ1, σ2)r-closed set K of Y . Proof. The proof is similar to that of Theorem 3. Corollary 2. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost (τ1, τ2)-continuous; (2) f−1(V ) ⊆ τ1τ2-Int(f −1(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ; (3) τ1τ2-Cl(f −1(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(f −1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ; (5) f−1(σ1σ2-Int(B)) ⊆ τ1τ2-Int(f −1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) for every subset B of Y ; (6) f−1(V ) is τ1τ2-open in X for every (σ1, σ2)r-open set V of Y ; (7) f−1(K) is τ1τ2-closed in X for every (σ1, σ2)r-closed set K of Y . Theorem 5. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost (τ1, τ2)-continuous; (2) τ1τ2-Cl(F −(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) τ1τ2-Cl(F −(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y . Proof. (1) ⇒ (2): Let V be any (σ1, σ2)β-open set of Y . Then, σ1σ2-Cl(V ) is a (σ1, σ2)r-closed set of Y . Since F is upper almost (τ1, τ2)-continuous and by Theorem 3, F−(σ1σ2-Cl(V )) is τ1τ2-closed in X. Thus, τ1τ2-Cl(F −(V )) ⊆ F−(σ1σ2-Cl(V )). (2) ⇒ (3): The proof is obvious. (3) ⇒ (1): Let K be any (σ1, σ2)r-closed set of Y . Then, K is (σ1, σ2)s-open in Y . Then by (3), τ1τ2-Cl(F −(K)) ⊆ F−(σ1σ2-Cl(K)) = F−(K) and hence F−(K) is τ1τ2-closed in X. By Theorem 3, F is upper almost (τ1, τ2)-continuous. Theorem 6. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost (τ1, τ2)-continuous; C. Klanarong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (2) (2024), 1244-1253 1251 (2) τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y . Proof. The proof is similar to that of Theorem 5. Corollary 3. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost (τ1, τ2)-continuous; (2) τ1τ2-Cl(f −1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) τ1τ2-Cl(f −1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y . Lemma 4. For a bitopological space (X, τ1, τ2), the following properties hold: (1) (τ1, τ2)-αCl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)β-open set V of Y ; (2) (τ1, τ2)-pCl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)s-open set V of Y . Corollary 4. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost (τ1, τ2)-continuous; (2) τ1τ2-Cl(F −(V )) ⊆ F−((σ1, σ2)-αCl(V )) for every (σ1, σ2)β-open set V of Y ; (3) τ1τ2-Cl(F −(V )) ⊆ F−((σ1, σ2)-pCl(V )) for every (σ1, σ2)s-open set V of Y . Corollary 5. 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