EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7052 ISSN 1307-5543 – ejpam.com Published by New York Business Global Almost µ(σ1, σ2)-Continuous Multifunctions Prapart Pue-on1, 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. This paper introduces new concepts of continuous multifunctions defined between a generalized topological space and a bitopological space, namely upper almost µ(σ1, σ2)-continuous multifunctions and lower almost µ(σ1, σ2)-continuous multifunctions. Moreover, several charac- terizations and some properties concerning upper almost µ(σ1, σ2)-continuous multifunctions and lower almost µ(σ1, σ2)-continuous multifunctions are investigated. Furthermore, the relationships between µ(σ1, σ2)-continuity and almost µ(σ1, σ2)-continuity are considered. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper almost µ(σ1, σ2)-continuous multifunction, lower almost µ(σ1, σ2)- continuous multifunction 1. Introduction The concept of almost continuous functions was introduced by Singal and Singal [1]. Munshi and Bassan [2] studied the notion of almost semi-continuous functions. Noiri [3] introduced and investigated the concept of almost α-continuous functions. Nasef and Noiri [4] introduced two classes of functions, namely almost precontinuous functions and almost β-continuous functions. 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. The concepts of generalized topological spaces and generalized neighborhood systems were introduced by Császár [5]. The classes of topological spaces and neighbor- hood systems are contained in the classes of generalized topological spaces and generalized neighborhood systems, respectively. Moreover, Császár [5] introduced two kinds of gen- eralized continuous functions by utilizing the concepts of generalized topological spaces and generalized neighborhood systems. Kanibir and Reilly [6] extended the concept of generalized continuous functions to multifunctions and introduced generalized continuous ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7052 Email addresses: prapart.p@msu.ac.th (P. Pue-on), 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) P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7052 2 of 14 multifunctions between generalized topological spaces. On the other hand, the present au- thors introduced and investigated four classes of multifunctions defined from a generalized topological space into a generalized topological space, namely upper almost β(µX , µY )- continuous multifunctions [7], lower almost β(µX , µY )-continuous multifunctions [7], upper α(µX , µY )-continuous multifunctions [8] and lower α(µX , µY )-continuous multifunctions [8]. Pue-on et al. [9] introduced and studied the concepts of upper (τ1, τ2)-continuous multifunctions and lower (τ1, τ2)-continuous multifunctions. Klanarong et al. [10] intro- duced and investigated the notions of upper almost (τ1, τ2)-continuous multifunctions and lower almost (τ1, τ2)-continuous multifunctions. Quite recently, Viriyapong et al. [11] pre- sented 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. Furthermore, several characterizations and some properties of upper almost τ⋆(σ1, σ2)-continuous multifunctions and lower almost τ⋆(σ1, σ2)-continuous multifunctions were discussed in [11]. In this paper, we introduce new classes of multifunctions between a generalized topological space and a bitopolog- ical space, namely upper almost µ(σ1, σ2)-continuous multifunctions and lower almost µ(σ1, σ2)-continuous multifunctions. We also investigate several characterizations of upper almost µ(σ1, σ2)-continuous multifunctions and lower almost µ(σ1, σ2)-continuous multi- functions. 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 [12] 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 [12] 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 [12] of A and is denoted by τ1τ2-Int(A). Lemma 1. [12] 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). P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7052 3 of 14 A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [13] (resp. (τ1, τ2)s-open [14], (τ1, τ2)p-open [14], (τ1, τ2)β-open [14]) 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)β-closed). A subset A of a bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-open [15] if A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)-open set is said to be 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. (τ1, τ2)s-closed, α(τ1, τ2)-closed) sets of X containing A is called the (τ1, τ2)p-closure [16] (resp. (τ1, τ2)s-closure [14], α(τ1, τ2)-closure [17]) of A and is denoted by (τ1, τ2)-pCl(A) (resp. (τ1, τ2)-sCl(A), α(τ1, τ2)-Cl(A)). The union of all (τ1, τ2)p-open (resp. (τ1, τ2)s-open, α(τ1, τ2)-open) sets of X contained in A is called the (τ1, τ2)p-interior [16] (resp. (τ1, τ2)s-interior [14], α(τ1, τ2)-interior [17]) of A and is denoted by (τ1, τ2)-pInt(A) (resp. (τ1, τ2)-sInt(A), α(τ1, τ2)-Int(A)). Lemma 2. [10] 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)). A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-δ-open [18] if A is the union of (τ1, τ2)r-open sets of X. The complement of a τ1τ2-δ-open set is called τ1τ2-δ-closed [18]. The union of all τ1τ2-δ-open sets of X contained in A is called the τ1τ2-δ-interior [18] of A and is denoted by τ1τ2-δ-Int(A). The intersection of all τ1τ2- δ-closed sets of X containing A is called the τ1τ2-δ-closure [18] of A and is denoted by τ1τ2-δ-Cl(A). Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is called a (τ1, τ2)θ-cluster point [13] 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 [13] 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 [13] 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 [13] of A and is denoted by (τ1, τ2)θ-Int(A). Lemma 3. [13] 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 [5]. 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), P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7052 4 of 14 and iµ(A) = X − cµ(X −A). According to [19], 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 almost µ(σ1, σ2)-continuous multifunctions In this section, we introduce the concepts of upper almost µ(σ1, σ2)-continuous mul- tifunctions and lower almost µ(σ1, σ2)-continuous multifunctions. Furthermore, several characterizations of upper almost µ(σ1, σ2)-continuous multifunctions and lower almost µ(σ1, σ2)-continuous multifunctions are discussed. Definition 1. 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. Theorem 1. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost µ(σ1, σ2)-continuous at x ∈ X; (2) x ∈ iµ(F +(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y containing F (x); (3) x ∈ iµ(F +((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y containing F (x); (4) x ∈ iµ(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 µ-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). Thus by (1), there exists a µ-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). Therefore, x ∈ U ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))) and so x ∈ iµ(F +(σ1σ2-Int(σ1σ2-Cl(V )))). (2) ⇒ (3): This follows from Lemma 2. (3) ⇒ (4): Let V be any σ1σ2-open set of Y containing F (x). It follows from Lemma 2 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), we have x ∈ iµ(F +(V )) and there exists a µ-open set U of X containing x such that x ∈ U ⊆ F+(V ); hence F (U) ⊆ V . P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7052 5 of 14 (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 µ-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). This shows that F is upper almost µ(σ1, σ2)-continuous at x ∈ X. Definition 2. A multifunction F : (X,µ) → (Y, σ1, σ2) is called 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 called lower almost µ(σ1, σ2)- continuous if F is lower almost 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 almost µ(σ1, σ2)-continuous at x ∈ X; (2) x ∈ iµ(F −(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y such that V ∩ F (x) ̸= ∅; (3) x ∈ iµ(F −((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y such that V ∩F (x) ̸= ∅; (4) x ∈ iµ(F −(V )) for every (σ1, σ2)r-open set V of Y such that V ∩ F (x) ̸= ∅; (5) for each (σ1, σ2)r-open set V of Y such that V ∩F (x) ̸= ∅, there exists a µ-open set U of X containing x such that U ⊆ F−(V ). Proof. The proof is similar to that of Theorem 1. Theorem 3. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost µ(σ1, σ2)-continuous; (2) F+(V ) ⊆ iµ(F +(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ; (3) cµ(F −(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (4) cµ(F −(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (5) F+(σ1σ2-Int(B)) ⊆ iµ(F +(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) for every subset B of Y ; (6) F+(V ) is µ-open in X for every (σ1, σ2)r-open set V of Y ; (7) F−(K) is µ-closed in X for every (σ1, σ2)r-closed set K of Y . P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7052 6 of 14 Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V . Thus by Theorem 1, we have x ∈ iµ(F +(σ1σ2-Int(σ1σ2-Cl(V )))) and hence F+(V ) ⊆ iµ(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) ⊆ iµ(F +(σ1σ2-Int(σ1σ2-Cl(Y −K)))) = iµ(X − F−(σ1σ2-Cl(σ1σ2-Int(K)))) = X − cµ(F −(σ1σ2-Cl(σ1σ2-Int(K)))). Thus, cµ(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), cµ(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 − cµ(F −(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(Y −B))))) = X − cµ(F −(Y − σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) = iµ(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 ) ⊆ iµ(F +(V )) and hence F+(V ) is µ-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 µ-closed in X. Thus, F+(V ) is µ-open and hence x ∈ iµ(F +(V )). Then, there exists a µ-open set U of X containing x such that F (U) ⊆ V . It follows from Theorem 1 that F is upper almost µ(σ1, σ2)-continuous. Theorem 4. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost µ(σ1, σ2)-continuous; (2) F−(V ) ⊆ iµ(F −(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ; (3) cµ(F +(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (4) cµ(F +(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (5) F−(σ1σ2-Int(B)) ⊆ iµ(F −(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) for every subset B of Y ; (6) F−(V ) is µ-open in X for every (σ1, σ2)r-open set V of Y ; P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7052 7 of 14 (7) F+(K) is µ-closed in X for every (σ1, σ2)r-closed set K of Y . Proof. The proof is similar to that of Theorem 3. Theorem 5. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost µ(σ1, σ2)-continuous; (2) cµ(F −(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) cµ(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 µ(σ1, σ2)-continuous and by Theorem 3, F−(σ1σ2-Cl(V )) is µ-closed in X. Thus, cµ(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), cµ(F−(K)) ⊆ F−(σ1σ2-Cl(K)) = F−(K) and hence F−(K) is µ-closed in X. By Theorem 3, F is upper almost µ(σ1, σ2)-continuous. Theorem 6. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost µ(σ1, σ2)-continuous; (2) cµ(F +(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) cµ(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. Lemma 4. [20] For a bitopological space (X, τ1, τ2), the following properties hold: (1) α(τ1, τ2)-Cl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)β-open set V of X; (2) (τ1, τ2)-pCl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)s-open set V of X. Corollary 1. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost µ(σ1, σ2)-continuous; (2) cµ(F −(V )) ⊆ F−(α(σ1, σ2)-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) cµ(F −(V )) ⊆ F−((σ1, σ2)-pCl(V )) for every (σ1, σ2)s-open set V of Y . Corollary 2. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7052 8 of 14 (1) F is lower almost µ(σ1, σ2)-continuous; (2) cµ(F +(V )) ⊆ F+(α(σ1, σ2)-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) cµ(F +(V )) ⊆ F+((σ1, σ2)-pCl(V )) for every (σ1, σ2)s-open set V of Y . Theorem 7. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost µ(σ1, σ2)-continuous; (2) cµ(F −(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) cµ(F −(σ1σ2-Cl(σ1σ2-Int(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (4) F+(V ) ⊆ iµ(F +(σ1σ2-Int(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 . Then, σ1σ2-Cl(V ) is σ1σ2- closed in Y and by Theorem 3, we have cµ(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), cµ(F −(σ1σ2-Cl(σ1σ2-Int(V )))) ⊆ cµ(F −(σ1σ2-Cl(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), we have X − iµ(F +(σ1σ2-Int(σ1σ2-Cl(V )))) = cµ(X − F+(σ1σ2-Int(σ1σ2-Cl(V )))) = cµ(F −(Y − σ1σ2-Int(σ1σ2-Cl(V )))) = cµ(F −(σ1σ2-Cl(Y − σ1σ2-Cl(V )))) = cµ(F −(σ1σ2-Cl(σ1σ2-Int(Y − σ1σ2-Cl(V ))))) ⊆ F−(σ1σ2-Cl(Y − σ1σ2-Cl(V ))) = F−(Y − σ1σ2-Int(σ1σ2-Cl(V ))) ⊆ F−(Y − V ) = X − F+(V ) and hence F+(V ) ⊆ iµ(F +(σ1σ2-Int(σ1σ2-Cl(V )))). (4) ⇒ (1): Let V be any (σ1, σ2)r-open set of Y . Then, V is (σ1, σ2)p-open in Y and by (4), F+(V ) ⊆ iµ(F +(σ1σ2-Int(σ1σ2-Cl(V )))) = iµ(F +(V )). Thus, F+(V ) is µ-open in X. It follows from Theorem 3 that F is upper almost µ(σ1, σ2)-continuous. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7052 9 of 14 Theorem 8. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost µ(σ1, σ2)-continuous; (2) cµ(F +(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) cµ(F +(σ1σ2-Cl(σ1σ2-Int(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (4) F−(V ) ⊆ iµ(F −(σ1σ2-Int(σ1σ2-Cl(V )))) for every (σ1, σ2)p-open set V of Y . Proof. The proof is similar to that of Theorem 7. Lemma 5. [21] Let A be a subset of a bitopological space (X, τ1, τ2). Then, the following properties hold: (1) If A is τ1τ2-open in X, then τ1τ2-Cl(A) = τ1τ2-δ-Cl(A). (2) τ1τ2-δ-Cl(A) is τ1τ2-closed. Theorem 9. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost µ(σ1, σ2)-continuous; (2) cµ(F −(σ1σ2-Cl(σ1σ2-Int(σ1σ2-δ-Cl(B))))) ⊆ F−(σ1σ2-δ-Cl(B)) for every subset B of Y ; (3) cµ(F −(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F−(σ1σ2-δ-Cl(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . By Lemma 5, σ1σ2-δ-Cl(B) is σ1σ2-closed in Y and by Theorem 3, cµ(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-δ-Cl(B))))) ⊆ F−(σ1σ2-δ-Cl(B)). (2) ⇒ (3): This is obvious since σ1σ2-Cl(B) ⊆ σ1σ2-δ-Cl(B). (3) ⇒ (1): Let K be any (σ1, σ2)r-closed set of Y . Then by (3), we have cµ(F −(K)) = cµ(F −(σ1σ2-Cl(σ1σ2-Int(K)))) = cµ(F −(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(K))))) ⊆ F−(σ1σ2-δ-Cl(K)) = F−(K) and hence F−(K) is µ-closed in X. By Theorem 3, F is upper almost µ(σ1, σ2)-continuous. Theorem 10. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7052 10 of 14 (1) F is lower almost µ(σ1, σ2)-continuous; (2) cµ(F +(σ1σ2-Cl(σ1σ2-Int(σ1σ2-δ-Cl(B))))) ⊆ F+(σ1σ2-δ-Cl(B)) for every subset B of Y ; (3) cµ(F +(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F+(σ1σ2-δ-Cl(B)) for every subset B of Y . Proof. The proof is similar to that of Theorem 9. Lemma 6. If F : (X,µ) → (Y, σ1, σ2) is lower almost µ(σ1, σ2)-continuous, then for each x ∈ X and each subset B of Y with σ1σ2-δ-Int(B)∩F (x) ̸= ∅, there exists a µ-open set U of X containing x such that U ⊆ F−(B). Proof. Let x ∈ X and B be a subset of Y with σ1σ2-δ-Int(B) ∩ F (x) ̸= ∅. Since σ1σ2-δ-Int(B) ∩ F (x) ̸= ∅, there exists a nonempty (σ1, σ2)r-open set V of Y such that V ⊆ B and V ∩ F (x) ̸= ∅. Since F is lower almost µ(σ1, σ2)-continuous, there exists a µ-open set U of X containing x such that V ∩F (z) ̸= ∅ for each z ∈ U ; hence U ⊆ F−(B). Theorem 11. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost µ(σ1, σ2)-continuous; (2) cµ(F +(B)) ⊆ F+(σ1σ2-δ-Cl(B)) for every subset B of Y ; (3) F (cµ(A)) ⊆ σ1σ2-δ-Cl(F (A)) for every subset A of X; (4) F+(K) is µ-closed in X for every σ1σ2-δ-closed set K of Y ; (5) F−(V ) is µ-open in X for every σ1σ2-δ-open set V of Y ; (6) F−(σ1σ2-δ-Int(B)) ⊆ iµ(F −(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . Suppose that x ̸∈ F+(σ1σ2-δ-Cl(B)). Then, we have x ∈ F−(Y −σ1σ2-δ-Cl(B)) = F−(σ1σ2-δ-Int(Y −B)). There exists a µ-open set U of X containing x such that U ⊆ F−(Y −B) = X − F+(B). Thus, U ∩ F+(B) = ∅ and hence x ∈ X − cµ(F +(B)). This shows that cµ(F +(B)) ⊆ F+(σ1σ2-δ-Cl(B)). (2) ⇒ (3): Let A be any subset of X. By (2), we have cµ(A) ⊆ cµ(F +(F (A))) ⊆ F+(σ1σ2-δ-Cl(F (A))) and hence F (cµ(A)) ⊆ σ1σ2-δ-Cl(F (A)). (3) ⇒ (1): Let B be any subset of Y . Then, by the hypothesis and Lemma 5, F (cµ(F +(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B)))))) ⊆ τ1τ2-δ-Cl(F (F+(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B)))))) P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7052 11 of 14 ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))) ⊆ σ1σ2-Cl(B) and hence cµ(F +(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F+(σ1σ2-Cl(B)). By Theorem 4, F is lower almost µ(σ1, σ2)-continuous. (2) ⇒ (4): Let K be any σ1σ2-δ-closed set of Y . Then, σ1σ2-δ-Cl(K) = K. By (2), we have cµ(F +(K)) ⊆ F+(σ1σ2-δ-Cl(K)) = F+(K) and so F+(K) is µ-closed in X. (4) ⇒ (5): The proof is obvious. (5) ⇒ (6): Let B be any subset of Y . Then by (5), we have F−(σ1σ2-δ-Int(B)) = iµ(F −(σ1σ2-δ-Int(B))) ⊆ iµ(F −(B)). (6) ⇒ (1): Let V be any (σ1, σ2)r-open set of Y . Then, we have V is σ1σ2-δ-open and σ1σ2-δ-Int(V ) = V . Thus by (6), F−(V ) ⊆ iµ(F −(V )) and hence F−(V ) is µ-open in X. By Theorem 4, F is lower almost µ(σ1, σ2)-continuous. Definition 3. [22] 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 4. [22] 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. Remark 1. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following implication holds: upper µ(σ1, σ2)-continuity ⇒ upper almost 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 = {∅, {1}, {2}, {1, 2}, X}. Let Y = {p, q, r} with topologies σ1 = {∅, {p}, {p, q}, Y } and σ2 = {∅, {p}, {q}, {p, q}, Y }. A multifunction F : (X,µ) → (Y, σ1, σ2) is defined as follows: F (1) = {r} and F (2) = F (3) = {p, q}. Then, F is upper almost µ(σ1, σ2)-continuous but F is not upper µ(σ1, σ2)- continuous. Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)s-regular [23] if for each (τ1, τ2)s-closed set F and each x ̸∈ F , there exist disjoint (τ1, τ2)s-open sets U and V such that x ∈ U and F ⊆ V . Lemma 7. [23] Let (X, τ1, τ2) be a (τ1, τ2)s-regular space. Then, the following properties hold: P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7052 12 of 14 (1) τ1τ2-Cl(A) = τ1τ2-δ-Cl(A) for every subset A of X. (2) Every τ1τ2-open set is τ1τ2-δ-open. Lemma 8. [22] For a multifunction F : (X,µ) → (Y, σ1, σ2), where (Y, σ1, σ2) is a (σ1, σ2)s-regular space, the following properties are equivalent: (1) F is lower µ(σ1, σ2)-continuous; (2) F+(σ1σ2-δ-Cl(B)) is µ-closed in X for every subset B of Y ; (3) F+(K) is µ-closed in X for every σ1σ2-δ-closed set K of Y ; (4) F−(V ) is µ-open in X for every σ1σ2-δ-open set V of Y . Theorem 12. For a multifunction F : (X,µ) → (Y, σ1, σ2), where (Y, σ1, σ2) is a (σ1, σ2)s- regular space, the following properties are equivalent: (1) F is lower µ(σ1, σ2)-continuous; (2) F+(σ1σ2-δ-Cl(B)) is µ-closed in X for every subset B of Y ; (3) F+(K) is µ-closed in X for every σ1σ2-δ-closed set K of Y ; (4) F−(V ) is µ-open in X for every σ1σ2-δ-open set V of Y ; (5) F is lower almost µ(σ1, σ2)-continuous. Proof. The proofs of the implications (1) ⇒ (2) ⇒ (3) ⇒ (4) are similar as in Lemma 8. (4) ⇒ (5): Let V be any (σ1, σ2)r-open set of Y . Then, V is σ1σ2-open in Y and by Lemma 7, V is σ1σ2-δ-open in Y . By (4), we have F−(V ) is µ-open in X. Thus by Theorem 4, F is lower almost µ(σ1, σ2)-continuous. (5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that V ∩F (x) ̸= ∅. Since (Y, σ1, σ2) is (σ1, σ2)s-regular, there exists a (σ1, σ2)r-open set W such that W ∩F (x) ̸= ∅ and W ⊆ V . 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