EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7050 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Upper and Lower Weakly τ ⋆β(σ1, σ2)-Continuous Multifunctions Montri Thongmoon1, Areeyuth Sama-Ae2, 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 Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Pattani, 94000, Thailand Abstract. A new class of continuous multifunctions between an ideal topological space and a bitopological space, called upper (lower) weakly τ⋆β(σ1, σ2)-continuous multifunctions, has been defined and studied. Furthermore, several characterizations and some properties concerning upper weakly τ⋆β(σ1, σ2)-continuous multifunctions and lower weakly τ⋆β(σ1, σ2)-continuous multifunc- tions are discussed. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper weakly τ⋆β(σ1, σ2)-continuous multifunction, lower weakly τ⋆β(σ1, σ2)-continuous multifunction 1. Introduction In 1983, Abd El-Monsef et al. [1] introduced and studied the notion of β-continuous functions. Nasef and Noiri [2] defined almost β-continuous functions by utilizing the no- tion of β-open sets due to Abd El-Monsef et al. [1]. In 1994, Popa and Noiri [3] introduced the concept of weakly β-continuous functions and obtained some characterizations of such functions. The class of almost β-continuity is a generalization of β-continuity and the class of weak β-continuity is a generalization of almost β-continuity. In 1999, Popa and Noiri [4] introduced new classes of multifunctions defined from a topological space into a topological space, namely upper weakly β-continuous multifunctions and lower weakly β-continuous multifunctions. Furthermore, Popa and Noiri [5] investigated several char- acterizations and some properties of upper weakly β-continuous multifunctions and lower weakly β-continuous multifunctions. On the other hand, the present author introduced and investigated four classes of multifunctions defined from an ideal topological space into an ideal topological space, namely upper weakly ⋆-continuous multifunctions [6], ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7050 Email addresses: montri.t@msu.ac.th (M. Thongmoon), areeyuth.s@psu.ac.th (A. Sama-Ae), chawalit.b@msu.ac.th (C. Boonpok) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7050 2 of 14 lower weakly ⋆-continuous multifunctions [6], upper weakly sβ(⋆)-continuous multifunc- tions [7], lower weakly sβ(⋆)-continuous multifunctions [7], upper weakly α-⋆-continuous multifunctions [8], lower weakly α-⋆-continuous multifunctions [8], weakly ı⋆-continuous multifunctions [9] and weakly pı-continuous multifunctions [10]. Pue-on et al. [11] intro- duced and studied two classes of continuous multifunctions between bitopological spaces, namely upper (τ1, τ2)-continuous multifunctions and lower (τ1, τ2)-continuous multifunc- tions. Thongmoon et al. [12] introduced and studied the notions of upper weakly (τ1, τ2)- continuous multifunctions and lower weakly (τ1, τ2)-continuous multifunctions. In this pa- per, we introduce the concepts of continuous multifunctions between an ideal topological space and a bitopological space, called upper weakly τ⋆β(σ1, σ2)-continuous multifunc- tions and lower weakly τ⋆β(σ1, σ2)-continuous multifunctions. We also investigate several characterizations of upper weakly τ⋆β(σ1, σ2)-continuous multifunctions and lower weakly τ⋆β(σ1, σ2)-continuous multifunctions. 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 [13] if A = τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is called τ1τ2-open. The intersection of all τ1τ2-closed sets of X containing A is called the τ1τ2-closure [13] 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 [13] of A and is denoted by τ1τ2-Int(A). Lemma 1. [13] 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). (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 subsetA of a bitopological space (X, τ1, τ2) is called (τ1, τ2)r-open [14] (resp. (τ1, τ2)s- open [15], (τ1, τ2)p-open [15], (τ1, τ2)β-open [15]) 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 (resp. (τ1, τ2)s-closed, (τ1, τ2)p-closed, (τ1, τ2)β-closed). The inter- section of all (τ1, τ2)s-closed sets of X containing A is called the (τ1, τ2)s-closure [15] 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 [15] of A and is denoted by (τ1, τ2)-sInt(A). M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7050 3 of 14 Lemma 2. For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: (1) (τ1, τ2)-sCl(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∪A [15]; (2) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A [16]. Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is called a (τ1, τ2)θ-cluster point [14] of A if τ1τ2-Cl(U)∩A ̸= ∅ for every τ1τ2-open set U containing x. The set of all (τ1, τ2)θ-cluster points of A is called the (τ1, τ2)θ-closure [14] of A and is denoted by (τ1, τ2)θ-Cl(A). A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)θ-closed [14] if (τ1, τ2)θ-Cl(A) = A. The complement of a (τ1, τ2)θ-closed set is said to be (τ1, τ2)θ-open. The union of all (τ1, τ2)θ-open sets of X contained in A is called the (τ1, τ2)θ-interior [14] of A and is denoted by (τ1, τ2)θ-Int(A). Lemma 3. [14] For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: (1) If A is τ1τ2-open in X, then τ1τ2-Cl(A) = (τ1, τ2)θ-Cl(A). (2) (τ1, τ2)θ-Cl(A) is τ1τ2-closed in X. An ideal I on a topological space (X, τ) is a nonempty collection of subsets of X satisfying the following properties: (1) A ∈ I and B ⊆ A imply B ∈ I ; (2) A ∈ I and B ∈ I imply A ∪ B ∈ I . A topological space (X, τ) with an ideal I on X is called an ideal topological space and is denoted by (X, τ,I ). For an ideal topological space (X, τ,I ) and a subset A of X, A⋆(I ) is defined as follows: A⋆(I ) = {x ∈ X : U ∩A ̸∈ I for every open neighbourhood U of x}. In case there is no chance for confusion, A⋆(I ) is simply written as A⋆. In [17], A⋆ is called the local function of A with respect to I and τ and Cl⋆(A) = A⋆ ∪ A defines a Kuratowski closure operator for a topology τ⋆(I ) finer than τ . A subset A is said to be ⋆-closed [18] if A⋆ ⊆ A. The interior of a subset A in (X, τ⋆(I )) is denoted by Int⋆(A). A subset A of an ideal topological space (X, τ,I ) is said to be R-I ⋆- open [6] (resp. I ⋆-preopen [6], τ⋆-semi-open [19] (semi-I ⋆-open [20]), τ⋆-β-open [19] (semi-I ⋆-preopen [20])) if A = Int⋆(Cl⋆(A)) (resp. A ⊆ Int⋆(Cl⋆(A)), A ⊆ Cl⋆(Int⋆(A)), A ⊆ Cl⋆(Int⋆(Cl⋆(A)))). The complement of a R-I ⋆-open (resp. I ⋆-preopen, τ⋆-semi- open, τ⋆-β-open) set is said to be R-I ⋆-closed (resp. I ⋆-preclosed, τ⋆-semi-closed, τ⋆- β-closed). For a subset A of an ideal topological space (X, τ,I ), the intersection of all semi-I ⋆-closed sets containing A is called the semi-I ⋆-closure [20] of A and is denoted by sCl⋆(A) (sClI ⋆(A) [20]). The union of all semi-I ⋆-open sets contained in A is called the semi-I ⋆-interior [20] of A and is denoted by sInt⋆(A) (sIntI ⋆(A) [20]). The intersection of all β-I ⋆-closed sets containing A is called the β-I ⋆-closure of A and is denoted by βCl⋆(A). The union of all β-I ⋆-open sets contained in A is called the β-I ⋆-interior of A and is denoted by βInt⋆(A). M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7050 4 of 14 Lemma 4. For a subset A of an ideal topological space (X, τ,I ), the following properties hold: (1) sCl⋆(A) = A ∪ Int⋆(Cl⋆(A)) [20]. (2) sInt⋆(A) = A ∩ Cl⋆(Int⋆(A)) [20]. (3) βCl⋆(A) = A ∪ Int⋆(Cl⋆(Int⋆(A))). (4) βInt⋆(A) = A ∩ Cl⋆(Int⋆(Cl⋆(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 , 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 weakly τ ⋆β(σ1, σ2)-continuous multifunctions In this section, we introduce the notions of upper weakly τ⋆β(σ1, σ2)-continuous mul- tifunctions and lower weakly τ⋆β(σ1, σ2)-continuous multifunctions. Moreover, several characterizations of upper weakly τ⋆β(σ1, σ2)-continuous multifunctions and lower weakly τ⋆β(σ1, σ2)-continuous multifunctions discussed. Definition 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper weakly τ⋆β(σ1, σ2)-continuous at a point x of X if for each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a τ⋆-β-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper weakly τ⋆β(σ1, σ2)-continuous if F is upper weakly τ⋆β(σ1, σ2)-continuous at each point of X. Theorem 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly τ⋆β(σ1, σ2)-continuous at a point x ∈ X; (2) x ∈ Cl⋆(Int(Cl⋆(F+(σ1σ2-Cl(V ))))) for every σ1σ2-open set V of Y containing F (x); (3) x ∈ βInt⋆(F+(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y containing F (x). Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing F (x). Thus by (1), there exists a τ⋆-β-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). Then, x ∈ U ⊆ F+(σ1σ2-Cl(V )). Since U is τ⋆-β-open, we have x ∈ U ⊆ Cl⋆(Int⋆(Cl⋆(U))) ⊆ Cl⋆(Int⋆(Cl⋆(F+(σ1σ2-Cl(V ))))). (2) ⇒ (3): Let V be any σ1σ2-open set of Y containing F (x). Then by (2), we have x ∈ Cl⋆(Int⋆(Cl⋆(F+(σ1σ2-Cl(V ))))). Since x ∈ F+(σ1σ2-Cl(V )) and by Lemma 4, x ∈ F+(σ1σ2-Cl(V )) ∩ Cl⋆(Int⋆(Cl⋆(F+(σ1σ2-Cl(V ))))) = βInt⋆(F+(σ1σ2-Cl(V ))). M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7050 5 of 14 (3) ⇒ (1): Let V be any σ1σ2-open set of Y containing F (x). By (3), we have x ∈ βInt⋆(F+(σ1σ2-Cl(V ))) and so there exists a τ⋆-β-open set U of X containing x such that U ⊆ F+(σ1σ2-Cl(V )); hence F (U) ⊆ σ1σ2-Cl(V ). This shows that F is upper weakly τ⋆β(σ1, σ2)-continuous at x. Definition 2. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is called lower weakly τ⋆β(σ1, σ2)- continuous at a point x of X if for each σ1σ2-open set V of Y such that F (x)∩V ̸= ∅, there exists a τ⋆-β-open set U of X containing x such that F (z) ∩ σ1σ2-Cl(V ) ̸= ∅ for every z ∈ U . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is called lower weakly τ⋆β(σ1, σ2)- continuous if F is lower weakly τ⋆β(σ1, σ2)-continuous at each point of X. Theorem 2. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly τ⋆β(σ1, σ2)-continuous at a point x ∈ X; (2) x ∈ Cl⋆(Int⋆(Cl⋆(F−(σ1σ2-Cl(V ))))) for every σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅; (3) x ∈ βInt⋆(F−(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y such that F (x)∩V ̸= ∅. Proof. The proof is similar to that of Theorem 1. Definition 3. A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be weakly τ⋆β(σ1, σ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing f(x), there exists a τ⋆-β-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be weakly τ⋆β(σ1, σ2)-continuous if f is τ⋆β(σ1, σ2)- continuous at each point of X. Corollary 1. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly τ⋆β(σ1, σ2)-continuous at a point x ∈ X; (2) x ∈ Cl⋆(Int⋆(Cl⋆(f−1(σ1σ2-Cl(V ))))) for every σ1σ2-open set V of Y containing f(x); (3) x ∈ βInt⋆(f−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y containing f(x). Theorem 3. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly τ⋆β(σ1, σ2)-continuous; M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7050 6 of 14 (2) F+(V ) ⊆ Cl⋆(Int⋆(Cl⋆(F+(σ1σ2-Cl(V ))))) for every σ1σ2-open set V of Y ; (3) Int⋆(Cl⋆(Int⋆(F−(V )))) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (4) Int⋆(Cl⋆(Int⋆(F−(σ1σ2-Int(K))))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (5) βCl⋆(F−(σ1σ2-Int(K))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (6) βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (7) F+(σ1σ2-Int(B)) ⊆ βInt⋆(F+(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ; (8) F+(V ) ⊆ βInt⋆(F+(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (9) βCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V and by Theorem 1, x ∈ βInt⋆(F+(σ1σ2-Cl(V ))) and hence F+(V ) ⊆ Cl⋆(Int⋆(Cl⋆(F+(σ1σ2-Cl(V ))))) by Lemma 4. (2) ⇒ (3): Let V be any σ1σ2-open set of Y . Thus by (2), we have X − F−(σ1σ2-Cl(V )) = F+(Y − σ1σ2-Cl(V )) ⊆ Cl⋆(Int⋆(Cl⋆(F+(Cl⋆(Y − σ1σ2-Cl(V )))))) = Cl⋆(Int⋆(Cl⋆(F+(Y − σ1σ2-Int(σ1σ2-Cl(V )))))) ⊆ Cl⋆(Int⋆(Cl⋆(F+(Y − V )))) = Cl⋆(Int⋆(Cl⋆(X − F−(V )))) = X − Int⋆(Cl⋆(Int⋆(F−(V )))) and hence Int⋆(Cl⋆(Int⋆(F−(V )))) ⊆ F−(σ1σ2-Cl(V )). (3) ⇒ (4): Let K be any σ1σ2-closed set of Y . Then, σ1σ2-Int(K) is σ1σ2-open in Y and so Int⋆(Cl⋆(Int⋆(F−(σ1σ2-Int(K))))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ F−(σ1σ2-Cl(K)) = F−(K). (4) ⇒ (5): Let K be any σ1σ2-closed set of Y . Then, we have Int⋆(Cl⋆(Int⋆(F−(σ1σ2-Int(K))))) ⊆ F−(K) and F−(σ1σ2-Int(K)) ⊆ F−(K). Thus by Lemma 4, βCl⋆(F−(σ1σ2-Int(K))) ⊆ F−(K). (5) ⇒ (6): Let B be any subset of Y . Then, σ1σ2-Cl(B) is σ1σ2-closed in Y and by (5), βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−(σ1σ2-Cl(B)). M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7050 7 of 14 (6) ⇒ (7): Let B be any subset of Y . By (6), F+(σ1σ2-Int(B)) = X − F−(σ1σ2-Cl(Y −B)) ⊆ X − βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(Y −B)))) = βInt⋆(F+(σ1σ2-Cl(σ1σ2-Int(B)))). (7) ⇒ (8): The proof is obvious. (8) ⇒ (9): Let V be any σ1σ2-open set of Y . Then by (8), we have βCl⋆(F−(V )) ⊆ βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) = βCl⋆(X − F+(Y − σ1σ2-Int(σ1σ2-Cl(V )))) = X − βInt⋆(F+(Y − σ1σ2-Int(σ1σ2-Cl(V )))) = X − βInt⋆(F+(σ1σ2-Cl(Y − σ1σ2-Cl(V )))) ⊆ X − F+(Y − σ1σ2-Cl(V )) = F−(σ1σ2-Cl(V )). (9) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing F (x). By (9), x ∈ F+(V ) ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))) = X − F−(σ1σ2-Cl(Y − σ1σ2-Cl(V ))) ⊆ X − βCl⋆(F−(Y − σ1σ2-Cl(V ))) = βInt⋆(F+(σ1σ2-Cl(V ))) and hence F is upper weakly τ⋆β(σ1, σ2)-continuous by Theorem 1. Theorem 4. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly τ⋆β(σ1, σ2)-continuous; (2) F−(V ) ⊆ Cl⋆(Int⋆(Cl⋆(F−(σ1σ2-Cl(V ))))) for every σ1σ2-open set V of Y ; (3) Int⋆(Cl⋆(Int⋆(F+(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (4) Int⋆(Cl⋆(Int⋆(F+(σ1σ2-Int(K))))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (5) βCl⋆(F+(σ1σ2-Int(K))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (6) βCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (7) F−(σ1σ2-Int(B)) ⊆ βInt⋆(F−(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ; (8) F−(V ) ⊆ βInt⋆(F−(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (9) βCl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y . M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7050 8 of 14 Proof. The proof is similar to that of Theorem 3. Corollary 2. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly τ⋆β(σ1, σ2)-continuous; (2) f−1(V ) ⊆ Cl⋆(Int⋆(Cl⋆(f−1(σ1σ2-Cl(V ))))) for every σ1σ2-open set V of Y ; (3) Int⋆(Cl⋆(Int⋆(f−1(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (4) Int⋆(Cl⋆(Int⋆(f−1(σ1σ2-Int(K))))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; (5) βCl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; (6) βCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ; (7) f−1(σ1σ2-Int(B)) ⊆ βInt⋆(f−1(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ; (8) f−1(V ) ⊆ βInt⋆(f−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (9) βCl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y . Definition 4. [21] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper al- most τ⋆β(σ1, σ2)-continuous at a point x of X if for each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a τ⋆-β-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper almost τ⋆β(σ1, σ2)-continuous if F is upper almost τ⋆β(σ1, σ2)-continuous at each point of X. Definition 5. [21] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower almost τ⋆β(σ1, σ2)-continuous at a point x of X if for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a τ⋆-β-open set U of X containing x such that F (z) ∩ σ1σ2-Int(σ1σ2-Cl(V )) ̸= ∅ for every z ∈ U . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower almost τ⋆β(σ1, σ2)-continuous if F is lower almost τ⋆β(σ1, σ2)-continuous at each point of X. Remark 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following implication holds: upper almost τ⋆β(σ1, σ2)-continuity ⇒ upper weak τ⋆β(σ1, σ2)-continuity. The converse of the implication is not true in general. We give an example for the implication as follows. M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7050 9 of 14 Example 1. Let X = {1, 2, 3} with a topology τ = {∅, {1}, {2}, {1, 2}, X} and an ideal I = {∅, {1}}. Let Y = {a, b, c} with topologies σ1 = {∅, {a}, {a, b}, Y } and σ2 = {∅, {a}, {b}, {a, b}, Y }. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is defined as follows: F (1) = {c} and F (2) = F (3) = {a, b}. Then, F is upper weakly τ⋆β(σ1, σ2)-continuous but F is not upper almost τ⋆β(σ1, σ2)-continuous. Theorem 5. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly τ⋆β(σ1, σ2)-continuous; (2) βCl⋆(F−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (4) βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (5) βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (6) βCl⋆(F−(σ1σ2-Int(K))) ⊆ F−(K) for every (σ1, σ2)r-closed set K of Y ; (7) βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (8) βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . Thus by Lemma 3, (σ1, σ2)θ-Cl(B) is σ1σ2-closed in Y and by Theorem 3, βCl⋆(F−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)). (2) ⇒ (3): This is obvious since σ1σ2-Cl(B) ⊆ (σ1, σ2)θ-Cl(B) for every subset B of Y . (3) ⇒ (4): This is obvious since σ1σ2-Cl(V ) = (σ1, σ2)θ-Cl(V ) for every σ1σ2-open set V of Y . (4) ⇒ (5): Let V be any (σ1, σ2)p-open set of Y . Then, V ⊆ σ1σ2-Int(σ1σ2-Cl(V )) and so σ1σ2-Cl(V ) = σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))). Now, put G = σ1σ2-Int(σ1σ2-Cl(V )), then G is σ1σ2-open in Y and σ1σ2-Cl(G) = σ1σ2-Cl(V ). Thus by (4), we have βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). (5) ⇒ (6): Let K be any (σ1, σ2)r-closed set of Y . Then, σ1σ2-Int(K) is (σ1, σ2)p-open in Y , by (5) we have βCl⋆(F−(σ1σ2-Int(K))) = βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))))) M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7050 10 of 14 ⊆ F−(σ1σ2-Cl(σ1σ2-Int(K))) = F−(K). (6) ⇒ (7): Let V be any (σ1, σ2)β-open set of Y . Then, we have V ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))). Since σ1σ2-Cl(V ) is (σ1, σ2)r-closed in Y . Thus by (6), βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). (7) ⇒ (8): This is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open. (8) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, since V is (σ1, σ2)s-open set in Y , by (8) we have βCl⋆(F−(V )) ⊆ βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). By Theorem 3, F is upper weakly τ⋆β(σ1, σ2)-continuous. Theorem 6. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly τ⋆β(σ1, σ2)-continuous; (2) βCl⋆(F+(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) βCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (4) βCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (5) βCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (6) βCl⋆(F+(σ1σ2-Int(K))) ⊆ F+(K) for every (σ1, σ2)r-closed set K of Y ; (7) βCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (8) βCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(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, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly τ⋆β(σ1, σ2)-continuous; (2) βCl⋆(f−1(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every subset B of Y ; M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7050 11 of 14 (3) βCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (4) βCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (5) βCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (6) βCl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(K) for every (σ1, σ2)r-closed set K of Y ; (7) βCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (8) βCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y . Theorem 7. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly τ⋆β(σ1, σ2)-continuous; (2) βCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) F+(V ) ⊆ βInt⋆(F+(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y . Proof. (1) ⇒ (2): Let V be any (σ1, σ2)p-open set of Y . Since F is upper weakly τ⋆β(σ1, σ2)-continuous, by Theorem 3 we have βCl⋆(F−(V )) ⊆ βCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). (2) ⇒ (3): Let V be any (σ1, σ2)p-open set of Y . Then, V ⊆ σ1σ2-Int(σ1σ2-Cl(V )) and Y − V ⊇ σ1σ2-Cl(σ1σ2-Int(Y − V )). Thus by (3), X − F+(V ) = F−(Y − V ) ⊇ F−(σ1σ2-Cl(σ1σ2-Int(Y − V ))) ⊇ βCl⋆(F−(σ1σ2-Int(Y − V ))) = βCl⋆(F−(Y − σ1σ2-Cl(V ))) = βCl⋆(X − F+(σ1σ2-Cl(V ))) = X − βInt⋆(F+(σ1σ2-Cl(V ))) and hence F+(V ) ⊆ βInt⋆(F+(σ1σ2-Cl(V ))). (3) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, V is (σ1, σ2)p-open in Y , by (4) we have F+(V ) ⊆ βInt⋆(F+(σ1σ2-Cl(V ))). Thus, F is upper weakly τ⋆β(σ1, σ2)-continuous by Theorem 3. Theorem 8. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7050 12 of 14 (1) F is lower weakly τ⋆β(σ1, σ2)-continuous; (2) βCl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) F−(V ) ⊆ βInt⋆(F−(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y . Proof. The proof is similar to that of Theorem 7. Corollary 4. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly τ⋆β(σ1, σ2)-continuous; (2) βCl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) f−1(V ) ⊆ βInt⋆(f−1(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y . Definition 6. [22] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper τ⋆β(σ1, σ2)-continuous at a point x of X if for each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a τ⋆-β-open set U of X containing x such that F (U) ⊆ V . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper τ⋆β(σ1, σ2)-continuous if F is upper τ⋆β(σ1, σ2)-continuous at each point of X. Definition 7. [22] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower τ⋆β(σ1, σ2)-continuous at a point x of X if for each σ1σ2-open set V of Y such that F (x)∩V ̸= ∅, there exists a τ⋆-β-open set U of X containing x such that F (z)∩V ̸= ∅ for every z ∈ U . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower τ⋆β(σ1, σ2)- continuous if F is lower τ⋆β(σ1, σ2)-continuous at each point of X. Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-normal [23] if for each pair of disjoint τ1τ2-closed sets F and F ′, there exist disjoint τ1τ2-open sets U and V such that F ⊆ U and F ′ ⊆ V . Theorem 9. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2) such that F (x) is σ1σ2-closed in Y for each x ∈ X and (Y, σ1, σ2) is a (σ1, σ2)-normal space, the following properties are equivalent: (1) F is upper τ⋆β(σ1, σ2)-continuous; (2) F is upper almost τ⋆β(σ1, σ2)-continuous; (3) F is upper weakly τ⋆β(σ1, σ2)-continuous. Proof. We show only the implication (3) ⇒ (1) since the others are obvious. Suppose that F is upper weakly τ⋆β(σ1, σ2)-continuous. Let x ∈ X and V be any σ1σ2-open set of Y such that F (x) ⊆ V . Since F (x) is σ1σ2-closed in Y and (Y, σ1, σ2) is (σ1, σ2)-normal, there exists a σ1σ2-open set G of Y such that F (x) ⊆ G ⊆ σ1σ2-Cl(G) ⊆ V . Since F is upper weakly τ⋆β(σ1, σ2)-continuous, there exists a τ⋆-β-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(G); hence F (U) ⊆ V . This shows that F is upper τ⋆β(σ1, σ2)- continuous. M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7050 13 of 14 Theorem 10. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2) such that F (x) is σ1σ2-open in Y for each x ∈ X, the following properties are equivalent: (1) F is lower τ⋆β(σ1, σ2)-continuous; (2) F is lower almost τ⋆β(σ1, σ2)-continuous; (3) F is lower weakly τ⋆β(σ1, σ2)-continuous. Proof. (1) ⇒ (2) and (2) ⇒ (3): The proofs of these implications are obvious. (3) ⇒ (1): Suppose that F is lower weakly τ⋆β(σ1, σ2)-continuous. Let x ∈ X and V be any σ1σ2-open set of Y such that F (x)∩V ̸= ∅. Then, there exists a τ⋆-β-open set U of X containing x such that F (z)∩σ1σ2-Cl(V ) ̸= ∅ for each z ∈ U . Since F (z) is σ1σ2-open, we have F (z) ∩ V ̸= ∅ for each z ∈ U and so F is lower τ⋆β(σ1, σ2)-continuous. 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