EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7053 ISSN 1307-5543 – ejpam.com Published by New York Business Global Weak Forms of µ(σ1, σ2)-Continuity for Multifunctions Monchaya Chiangpradit1, 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 a generalized topological space and a bitopological space, namely upper (lower) weakly µ(σ1, σ2)-continuous multifunctions, has been defined and studied. Moreover, several characterizations and some properties concerning upper weakly µ(σ1, σ2)-continuous multifunctions and lower weakly µ(σ1, σ2)-continuous multifunctions are established. Furthermore, the relationships between almost µ(σ1, σ2)-continuity and weak µ(σ1, σ2)-continuity are considered. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper weakly µ(σ1, σ2)-continuous multifunction, lower weakly µ(σ1, σ2)-continuous multifunction 1. Introduction In 1961, Levine [1] introduced and investigated the notion of weakly continuous func- tions. Husain [2] introduced and studied the concept of almost continuous functions. Janković [3] introduced almost weak continuity as a generalization of both weak conti- nuity and almost continuity. Noiri [4] investigated several characterizations of almost weakly continuous functions. Rose [5] introduced the notion of subweakly continuous functions and investigated the relationships between subweak continuity and weak conti- nuity. Popa and Noiri [6] introduced the concept of weakly (τ,m)-continuous functions as functions from a topological space into a set satisfying some minimal conditions and inves- tigated several characterizations of weakly (τ,m)-continuous functions. In 2002, Császár [7] introduced the concepts of generalized topological spaces and generalized neighbor- hood systems. The classes of topological spaces and neighborhood systems are contained in the classes of generalized topological spaces and generalized neighborhood systems, respectively. Furthermore, Császár [7] introduced two kinds of generalized continuous ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7053 Email addresses: monchaya.c@msu.ac.th (M. Chiangpradit), 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. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7053 2 of 13 functions by utilizing the notions of generalized topological spaces and generalized neigh- borhood systems. In 2009, Kanibir and Reilly [8] extended the concept of generalized continuous functions to multifunctions and defined upper semi generalized continuous multifunctions and lower semi generalized continuous multifunctions. On the other hand, the present authors introduced and investigated four classes of multifunctions defined from a generalized topological space into a generalized topological space, namely up- per α(µX , µY )-continuous multifunctions [9], lower α(µX , µY )-continuous multifunctions [9], upper weakly β(µX , µY )-continuous multifunctions [10] and lower weakly β(µX , µY )- continuous multifunctions [10]. Pue-on et al. [11] introduced and studied new classes of multifunctions between bitopological spaces, namely upper (τ1, τ2)-continuous multifunc- tions and lower (τ1, τ2)-continuous multifunctions. Klanarong et al. [12] introduced and investigated the notions of upper almost (τ1, τ2)-continuous multifunctions and lower al- most (τ1, τ2)-continuous multifunctions. Thongmoon et al. [13] extended the concept of weakly continuous functions to multifunctions and presented two classes of multifunctions defined from a bitopological space into a bitopological space, called upper weakly (τ1, τ2)- continuous multifunctions and lower weakly (τ1, τ2)-continuous multifunctions. Quite re- cently, Pue-on et al. [14] introduced new classes of continuous multifunctions between an ideal topological space and a bitopological space, namely upper almost τ⋆(σ1, σ2)- continuous multifunctions and lower almost τ⋆(σ1, σ2)-continuous multifunctions. More- over, several characterizations and some properties of upper almost τ⋆(σ1, σ2)-continuous multifunctions and lower almost τ⋆(σ1, σ2)-continuous multifunctions were discussed in [14]. In this paper, we introduce new classes of multifunctions between a generalized topological space and a bitopological space, namely upper weakly µ(σ1, σ2)-continuous multifunctions and lower weakly µ(σ1, σ2)-continuous multifunctions. We also investigate several characterizations of upper weakly µ(σ1, σ2)-continuous multifunctions and lower weakly µ(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 [15] 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 [15] 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 [15] of A and is denoted by τ1τ2-Int(A). Lemma 1. [15] 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). M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7053 3 of 13 (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 [16] (resp. (τ1, τ2)s-open [17], (τ1, τ2)p-open [17], (τ1, τ2)β-open [17]) 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 said to be (τ1, τ2)r-closed (resp. (τ1, τ2)s-closed, (τ1, τ2)p-closed, (τ1, τ2)p-closed). For a subset A of a bitopological space (X, τ1, τ2), a point x ∈ X is called a (τ1, τ2)θ-cluster point [16] 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 [16] 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 [16] 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 [16] of A and is denoted by (τ1, τ2)θ-Int(A). Lemma 2. [16] 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. Let X be a nonempty set, and denote P(X) the power set of X. We call a class µ ⊆ P(X) a generalized topology (briefly, GT) if ∅ ∈ µ, and an arbitrary union of elements of µ belongs to µ [7]. A set X with a GT µ on it is said to be a generalized topological space (briefly, GTS) and is denoted by (X,µ). For a GTS (X,µ), the elements of µ are called µ-open sets and the complements of µ-open sets are called µ-closed sets. For A ⊆ X, we denote by cµ(A) the intersection of all µ-closed sets containing A and by iµ(A) the union of all µ-open sets contained in A. Then, we have iµ(iµ(A)) = iµ(A), cµ(cµ(A)) = cµ(A), and iµ(A) = X − cµ(X −A). According to [18], for A ⊆ X and x ∈ X, we have x ∈ cµ(A) if and only if x ∈ M ∈ µ implies M ∩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 µ(σ1, σ2)-continuous multifunctions In this section, we introduce the concepts of upper weakly µ(σ1, σ2)-continuous mul- tifunctions and lower weakly µ(σ1, σ2)-continuous multifunctions. Furthermore, several M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7053 4 of 13 characterizations of upper weakly µ(σ1, σ2)-continuous multifunctions and lower weakly µ(σ1, σ2)-continuous multifunctions are discussed. Definition 1. A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be upper weakly µ(σ1, σ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a µ-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be upper weakly µ(σ1, σ2)-continuous if F is upper weakly µ(σ1, σ2)-continuous at each point x of X. Theorem 1. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly µ(σ1, σ2)-continuous; (2) F+(V ) ⊆ iµ(F +(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (3) cµ(F −(σ1σ2-Int(K))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (4) cµ(F −(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (5) F+(σ1σ2-Int(B)) ⊆ iµ(F +(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ; (6) cµ(F −(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (7) cµ(F −(V )) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (8) cµ(F −(σ1σ2-Int(K))) ⊆ F−(K) for every (σ1, σ2)r-closed set K of Y . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y such that x ∈ F+(V ). Then, we have F (x) ⊆ V and by (1), there exists a µ-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). Thus, U ⊆ F+(σ1σ2-Cl(V )). Since U is µ-open, we have x ∈ iµ(F +(σ1σ2-Cl(V ))) and so F+(V ) ⊆ iµ(F +(σ1σ2-Cl(V ))). (2) ⇒ (3): Let K be any σ1σ2-closed set of Y . Then, Y −K is σ1σ2-open in Y . By (2), X − F−(K) = F+(Y −K) ⊆ iµ(F +(σ1σ2-Cl(Y −K))) = X − cµ(F −(σ1σ2-Int(K))). Thus, cµ(F−(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), cµ(F−(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−(σ1σ2-Cl(B)). (4) ⇒ (5): Let B be any subset of Y . Thus by (4), we have X − iµ(F +(σ1σ2-Cl(σ1σ2-Int(B)))) = cµ(X − F+(σ1σ2-Cl(σ1σ2-Int(B)))) = cµ(F −(σ1σ2-Int(σ1σ2-Cl(Y −B)))) ⊆ F−(σ1σ2-Cl(Y −B)) = X − F+(σ1σ2-Int(B)) and hence F+(σ1σ2-Int(B)) ⊆ iµ(F +(σ1σ2-Cl(σ1σ2-Int(B)))). (5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that F (x) ⊆ V . By (5), x ∈ F+(V ) ⊆ iµ(F +(σ1σ2-Cl(V ))) and there exists a µ-open set U of X containing x M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7053 5 of 13 such that U ⊆ F+(σ1σ2-Cl(V )). Thus, F (U) ⊆ σ1σ2-Cl(V ) and hence F is upper weakly µ(σ1, σ2)-continuous. (4) ⇒ (6) and (6) ⇒ (7): The proofs are obvious. (7) ⇒ (8): Let K be any (σ1, σ2)r-closed set of Y . Thus by (7), cµ(F −(σ1σ2-Int(K))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(K))) = F−(K). (8) ⇒ (3): Let K be any σ1σ2-closed set of Y . Then, σ1σ2-Cl(σ1σ2-Int(K)) is (σ1, σ2)r- closed in Y and σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))) = σ1σ2-Int(σ1σ2-Cl(K)) = σ1σ2-Int(K). By (8), cµ(F −(σ1σ2-Int(K))) = cµ(F −(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ F−(K). Definition 2. A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be lower weakly µ(σ1, σ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that V ∩F (x) ̸= ∅, there exists a µ-open set U of X containing x such that σ1σ2-Cl(V )∩F (z) ̸= ∅ for every z ∈ U . A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be lower weakly µ(σ1, σ2)-continuous if F is lower weakly µ(σ1, σ2)-continuous at each point x of X. Theorem 2. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly µ(σ1, σ2)-continuous; (2) F−(V ) ⊆ iµ(F −(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (3) cµ(F +(σ1σ2-Int(K))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (4) cµ(F +(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (5) F−(σ1σ2-Int(B)) ⊆ iµ(F −(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ; (6) cµ(F +(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (7) cµ(F +(V )) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (8) cµ(F +(σ1σ2-Int(K))) ⊆ F+(K) for every (σ1, σ2)r-closed set K of Y . Proof. The proof is similar to that of Theorem 1. Definition 3. [19] A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be upper almost µ(σ1, σ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a µ-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be upper almost µ(σ1, σ2)-continuous if F is upper almost µ(σ1, σ2)-continuous at each point x of X. M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7053 6 of 13 Definition 4. [19] A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be lower almost µ(σ1, σ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that V ∩F (x) ̸= ∅, there exists a µ-open set U of X containing x such that σ1σ2-Int(σ1σ2-Cl(V ))∩F (z) ̸= ∅ for every z ∈ U . A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be lower almost µ(σ1, σ2)-continuous if F is lower almost µ(σ1, σ2)-continuous at each point x of X. Remark 1. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following implication holds: upper almost µ(σ1, σ2)-continuity ⇒ upper weakly µ(σ1, σ2)-continuity. The converse of the implication is not true in general. We give an example for the implication as follows. Example 1. Let X = {1, 2, 3} with a generalized topology µ = {∅, {2}, {1, 3}, X}. Let Y = {a, b, c} with topologies σ1 = {∅, {a}, {a, b}, Y } and σ2 = {∅, {a}, {b}, {a, b}, Y }. A multifunction F : (X,µ) → (Y, σ1, σ2) is defined as follows: F (1) = {a}, F (2) = {b} and F (3) = {a, c}. Then, F is upper weakly µ(σ1, σ2)-continuous but F is not upper almost µ(σ1, σ2)-continuous. Theorem 3. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly µ(σ1, σ2)-continuous; (2) cµ(F −(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) cµ(F −(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y . Proof. (1) ⇒ (2): This follows from (4) of Theorem 1. (2) ⇒ (3): The proof is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open. (3) ⇒ (1): Since every σ1σ2-open set is (σ1, σ2)s-open, the proof is obvious by (7) of Theorem 1. Theorem 4. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly µ(σ1, σ2)-continuous; (2) cµ(F +(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) cµ(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 3. M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7053 7 of 13 Theorem 5. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly µ(σ1, σ2)-continuous; (2) cµ(F −(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) cµ(F −(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (4) F+(V ) ⊆ iµ(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 σ1σ2-Int(σ1σ2-Cl(V )) is σ1σ2-open, by Theorem 1(7) cµ(F −(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). (2) ⇒ (3): Let V be any (σ1, σ2)p-open set of Y . By (2), we have cµ(F −(V )) ⊆ cµ(F −(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). (3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . Thus by (3), X − iµ(F +(σ1σ2-Cl(V ))) = cµ(X − F+(σ1σ2-Cl(V ))) = cµ(F −(Y − σ1σ2-Cl(V ))) ⊆ F−(σ1σ2-Cl(Y − σ1σ2-Cl(V ))) = X − F+(σ1σ2-Int(σ1σ2-Cl(V ))) ⊆ X − F+(V ) and hence F+(V ) ⊆ iµ(F +(σ1σ2-Cl(V ))). (4) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, V is (σ1, σ2)p-open in Y and by (4), F+(V ) ⊆ iµ(F +(σ1σ2-Cl(V ))). By Theorem 1(2), F is upper weakly µ(σ1, σ2)-continuous. Theorem 6. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly µ(σ1, σ2)-continuous; (2) cµ(F +(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) cµ(F +(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (4) F−(V ) ⊆ iµ(F −(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y . Proof. The proof is similar to that of Theorem 5. M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7053 8 of 13 Lemma 3. If F : (X,µ) → (Y, σ1, σ2) is lower weakly µ(σ1, σ2)-continuous, then for each x ∈ X and each subset B of Y with (σ1, σ2)θ-Int(B)∩ F (x) ̸= ∅, there exists a µ-open set U of X containing x such that U ⊆ F−(B). Proof. Since (σ1, σ2)θ-Int(B) ∩ F (x) ̸= ∅, there exists a nonempty σ1σ2-open set V of Y such that σ1σ2-Cl(V ) ⊆ B and V ∩ F (x) ̸= ∅. Since F is lower weakly µ(σ1, σ2)- continuous, there exists a µ-open set U of X containing x such that σ1σ2-Cl(V )∩F (z) ̸= ∅ for each z ∈ U and hence U ⊆ F−(B). Theorem 7. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly µ(σ1, σ2)-continuous; (2) cµ(F +(B)) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) F (cµ(A)) ⊆ (σ1, σ2)θ-Cl(F (A)) for every subset A of X. Proof. (1) ⇒ (2): Let B be any subset of Y . Suppose that x ̸∈ F+((σ1, σ2)θ-Cl(B)). Then, we have x ∈ F−(Y − (σ1, σ2)θ-Cl(B)) = F−((σ1, σ2)θ-Int(Y − B)). By Lemma 3, there exists a µ-open set U of X containing x such that U ⊆ F−(Y − B) = X − F+(B). Therefore, U ∩ F+(B) = ∅. Thus, x ̸∈ τ1τ2-Cl(F+(B)) and hence cµ(F +(B)) ⊆ F+((σ1, σ2)θ-Cl(B)). (2) ⇒ (3): Let A be any subset of X. By (2), we have cµ(A) ⊆ cµ(F +(F (A))) ⊆ F+((σ1, σ2)θ-Cl(F (A))). Thus, F (cµ(A)) ⊆ (σ1, σ2)θ-Cl(F (A)). (3) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, σ1σ2-Cl(V ) = (σ1, σ2)θ-Cl(V ) and by (3), F (cµ(F +(V ))) ⊆ (σ1, σ2)θ-Cl(F (F+(V ))) ⊆ (σ1, σ2)θ-Cl(V ) = σ1σ2-Cl(V ). Thus, cµ(F +(V )) ⊆ F+(σ1σ2-Cl(V )) and by Theorem 1, F is lower weakly µ(σ1, σ2)-continuous. Theorem 8. For a multifunction F : (X,µ) → (Y, σ1, τ2), the following properties are equivalent: (1) F is upper weakly µ(σ1, σ2)-continuous; (2) cµ(F −(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) cµ(F −(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . Then, (σ1, σ2)θ-Cl(B) is σ1σ2-closed in Y and by Theorem 2, cµ(F−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)). (2) ⇒ (3): The proof is obvious. M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7053 9 of 13 (3) ⇒ (1): Let K be any (σ1, σ2)r-closed set of Y . Then, we have (σ1, σ2)θ-Cl(σ1σ2-Int(K)) = σ1σ2-Cl(σ1σ2-Int(K)) = K and hence cµ(F −(σ1σ2-Int(K))) = cµ(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F−((σ1, σ2)θ-Cl(σ1σ2-Int(K))) = F−(σ1σ2-Cl(σ1σ2-Int(K))) = F−(K). Thus by Theorem 2, F is upper weakly µ(σ1, σ2)-continuous. Theorem 9. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly µ(σ1, σ2)-continuous; (2) cµ(F +(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) cµ(F +(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y . Proof. The proof is similar to that of Theorem 10. Definition 5. [20] A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be upper µ(σ1, σ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a µ-open set U of X containing x such that F (U) ⊆ V . A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be upper µ(σ1, σ2)-continuous if F is upper µ(σ1, σ2)- continuous at each point x of X. Definition 6. [20] A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be lower µ(σ1, σ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a µ-open set U of X containing x such that F (z) ∩ V ̸= ∅ for every z ∈ U . A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be lower µ(σ1, σ2)-continuous if F is lower µ(σ1, σ2)-continuous at each point x of X. Recall that a subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-paracompact [15] if every cover of A by τ1τ2-open sets of X is refined by a cover of A which consists of τ1τ2-open sets of X and is τ1τ2-locally finite in X. A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-regular [15] if for each x ∈ A and each τ1τ2-open set U of X containing x, there exists a τ1τ2-open set V of X such that x ∈ V ⊆ τ1τ2-Cl(V ) ⊆ U . Lemma 4. [15] If A is a τ1τ2-regular τ1τ2-paracompact set of a bitopological space (X, τ1, τ2) and U is a τ1τ2-open neighbourhood of A, then there exists a τ1τ2-open set V of X such that A ⊆ V ⊆ τ1τ2-Cl(V ) ⊆ U . M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7053 10 of 13 Theorem 10. For a multifunction F : (X,µ) → (Y, σ1, σ2) such that F (x) is a σ1σ2- regular σ1σ2-paracompact set of Y for each point x ∈ X, the following properties are equivalent: (1) F is upper µ(σ1, σ2)-continuous; (2) F is upper almost µ(σ1, σ2)-continuous; (3) F is upper weakly µ(σ1, σ2)-continuous. Proof. We show only the implication (3) ⇒ (1) since the others are obvious. Suppose that F is upper weakly µ(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-regular σ1σ2-paracompact, by Lemma 4 there exists a σ1σ2-open set W of Y such that F (x) ⊆ W ⊆ σ1σ2-Cl(W ) ⊆ V . Since F is upper weakly µ(σ1, σ2)-continuous, there exists a µ-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(W ); hence F (U) ⊆ V . This shows that F is upper µ(σ1, σ2)-continuous. Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-compact [15] if every cover of X by τ1τ2-open sets of X has a finite subcover. Definition 7. [21] A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-regular if for each τ1τ2-closed set F and each x ∈ X − F , there exist disjoint τ1τ2-open sets U and V such that x ∈ U and F ⊆ V . Corollary 1. Let F : (X,µ) → (Y, σ1, σ2) be a multifunction such that F (x) is σ1σ2- compact for each point x ∈ X and (Y, σ1, σ2) is (σ1, σ2)-regular. Then, the following properties are equivalent: (1) F is upper µ(σ1, σ2)-continuous; (2) F is upper almost µ(σ1, σ2)-continuous; (3) F is upper weakly µ(σ1, σ2)-continuous. Lemma 5. [22] If A is a τ1τ2-regular set of a bitopological space (X, τ1, τ2), then for each τ1τ2-open set G which intersect A, there exists a τ1τ2-open set W such that A ∩ W ̸= ∅ and τ1τ2-Cl(W ) ⊆ G. Theorem 11. For a multifunction F : (X,µ) → (Y, σ1, σ2) such that F (x) is a σ1σ2- regular set of Y for each point x ∈ X, the following properties are equivalent: (1) F is lower µ(σ1, σ2)-continuous; (2) F is lower almost µ(σ1, σ2)-continuous; (3) F is lower weakly µ(σ1, σ2)-continuous. M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7053 11 of 13 Proof. We show only the implication (3) ⇒ (1) since the others are obvious. Suppose that F is lower weakly µ(σ1, σ2)-continuous. Let x ∈ X and V be any σ1σ2-open set of Y such that V ∩ F (x) ̸= ∅. Since F (x) is σ1σ2-regular, by Lemma 5 there exists a σ1σ2-open set W of Y such that F (x) ∩ W ̸= ∅ and σ1σ2-Cl(W ) ⊆ V . Since F is lower weakly µ(σ1, σ2)-continuous, there exists a µ-open set U of X containing x such that σ1σ2-Cl(W ) ∩ F (z) ̸= ∅; hence F (z) ∩ V ̸= ∅ for each z ∈ U . This shows that F is lower µ(σ1, σ2)-continuous. Definition 8. [23] A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-normal 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 12. If F : (X,µ) → (Y, σ1, σ2) is a multifunction such that F (x) is σ1σ2-closed in Y for each x ∈ X and (Y, σ1, σ2) is a (σ1, σ2)-normal space, then the following properties are equivalent: (1) F is upper µ(σ1, σ2)-continuous; (2) F is upper almost µ(σ1, σ2)-continuous; (3) F is upper weakly µ(σ1, σ2)-continuous. Proof. As in Theorem 10, we prove only the implication (3) ⇒ (1). Suppose that F is upper weakly µ(σ1, σ2)-continuous. Let x ∈ X and G be any σ1σ2-open set of Y containing F (x). 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