EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6570 ISSN 1307-5543 – ejpam.com Published by New York Business Global Almost Weak Continuity for Multifunctions Defined between an Ideal Topological Space and a Bitopological Space Chokchai Viriyapong1, 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 presents new concepts of continuous multifunctions, called upper almost weakly τ⋆(σ1, σ2)-continuous multifunctions and lower almost weakly τ⋆(σ1, σ2)-continuous multi- functions. Moreover, several characterizations and some properties concerning upper almost weakly τ⋆(σ1, σ2)-continuous multifunctions and lower almost weakly τ⋆(σ1, σ2)-continuous multifunctions are considered. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper almost weakly τ⋆(σ1, σ2)-continuous multifunction, lower al- most weakly τ⋆(σ1, σ2)-continuous multifunction 1. Introduction Topology as a field of mathematics is concerned with all questions directly or indirectly related to continuity. Singal and Singal [1] introduced the concept of almost continuous functions as a generalization of continuity. 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. Levine [5] introduced and in- vestigated the concept of weakly continuous functions. Husain [6] introduced and studied the notion of almost continuous functions. Janković [7] introduced almost weak continuity as a generalization of both weak continuity and almost continuity. Noiri [8] investigated ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6570 Email addresses: chohchai.v@msu.ac.th (C. Viriyapong), 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) C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6570 2 of 14 several characterizations of almost weakly continuous functions. Rose [9] introduced the notion of subweakly continuous functions and investigated the relationships between sub- weak continuity and weak continuity. In 1993, Noiri and Popa [10] extended the concept of almost weakly continuous functions to multifunctions and defined upper almost weakly continuous multifunctions and lower almost weakly continuous multifunctions. Popa and Noiri [11] investigated some characterizations and several properties concerning upper almost weakly continuous multifunctions and lower almost weakly continuous multifunc- tions. Abd El-Monsef et al. [12] introduced and studied the notions of I -closed sets and I -continuous functions. Semi-I -open sets, pre-I -open sets, α-I -open sets, β-I -open sets and δ-I -open sets play an important role in the research of generalizations of conti- nuity in ideal topological spaces. In 2005, Hatir and Noiri [13] introduced and investigated the notions of weakly pre-I -open sets and weakly pre-I -continuous functions. Further- more, Hatir and Noiri [14] investigated further properties of semi-I -open sets and semi- I -continuous functions. On the other hand, the present author introduced and studied new classes of multifunctions between ideal topological spaces, namely upper ⋆-continuous multifunctions [15], lower ⋆-continuous multifunctions [15], upper almost ⋆-continuous mul- tifunctions [15], lower almost ⋆-continuous multifunctions [15], upper weakly ⋆-continuous multifunctions [15], lower weakly ⋆-continuous multifunctions [15], pı-continuous multi- functions [16] and weakly pı-continuous multifunctions [16]. Recently, Pue-on et al. [17] extended the idea of continuous multifunctions to bitopological spaces. Klanarong et al. [18] introduced and investigated the concepts of upper almost (τ1, τ2)-continuous mul- tifunctions and lower almost (τ1, τ2)-continuous multifunctions. Thongmoon et al. [19] introduced and studied the notions of upper weakly (τ1, τ2)-continuous multifunctions and lower weakly (τ1, τ2)-continuous multifunctions. On the other hand, the present authors introduced and investigated the concepts of upper almost weakly (τ1, τ2)-continuous mul- tifunctions and lower almost weakly (τ1, τ2)-continuous multifunctions [20]. In this paper, we introduce the concepts of upper almost weakly τ⋆(σ1, σ2)-continuous multifunctions and lower almost weakly τ⋆(σ1, σ2)-continuous multifunctions. We also investigate several characterizations of upper almost weakly τ⋆(σ1, σ2)-continuous multifunctions and lower almost weakly τ⋆(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 [21] 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 [21] 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 [21] of A and is denoted by τ1τ2-Int(A). A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-clopen [21] if C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6570 3 of 14 A is both τ1τ2-open and τ1τ2-closed. A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [22] (resp. (τ1, τ2)s-open [23], (τ1, τ2)p-open [23], (τ1, τ2)β-open [23]) 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). A subset A of a bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-open [24] 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. For a subset A of a bitopological space (X, τ1, τ2), a point x ∈ X is called a (τ1, τ2)θ- cluster point 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 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 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 of A and is denoted by (τ1, τ2)θ-Int(A) [22]. Lemma 1. [22] 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 [25], 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 [26] 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 semi⋆-I -open [27] (resp. semi-I -open [14]) if A ⊆ Cl(Int⋆(A)) (resp. A ⊆ Cl⋆(Int(A))). The complement of a semi⋆-I -open (resp. semi-I -open) set is said to be semi⋆-I -closed [27] (resp. semi-I - closed [14]). A subset A of an ideal topological space (X, τ,I ) is called I ⋆-preopen [15] if A ⊆ Int⋆(Cl⋆(A)). The complement of a I ⋆-preopen set is called I ⋆-preclosed. For a subset A of an ideal topological space (X, τ,I ), the intersection of all I ⋆-preclosed sets containing A is called the ⋆-preclosure of A and is denoted by pCl⋆(A). The union of all I ⋆-preopen sets contained in A is called the ⋆-preinterior of A and is denoted by pInt⋆(A). C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6570 4 of 14 Lemma 2. For a subset A of an ideal topological space (X, τ,I ), the following properties hold: (1) pCl⋆(A) = A ∪ Cl⋆(Int⋆(A)). (2) pInt⋆(A) = A ∩ Int⋆(Cl⋆(A)). By a multifunction F : X → Y , we mean a point-to-set correspondence from X into Y , and 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 ̸= ∅}. 3. Upper and lower almost weakly τ ⋆(σ1, σ2)-continuous multifunctions In this section, we introduce the notions of upper almost weakly τ⋆(σ1, σ2)-continuous multifunctions and lower almost weakly τ⋆(σ1, σ2)-continuous multifunctions. Moreover, several characterizations of upper almost weakly τ⋆(σ1, σ2)-continuous multifunctions and lower almost weakly τ⋆(σ1, σ2)-continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper almost weakly τ⋆(σ1, σ2)-continuous if for each x ∈ X and each σ1σ2-open set V of Y such that F (x) ⊆ V , x ∈ Int⋆(Cl⋆(F+(σ1σ2-Cl(V )))). Theorem 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost weakly τ⋆(σ1, σ2)-continuous; (2) F+(V ) ⊆ Int⋆(Cl⋆(F+(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ; (3) Cl⋆(Int⋆(F−(V ))) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (4) pCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (5) F+(V ) ⊆ pInt⋆(F+(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (6) for each x ∈ X and each σ1σ2-open set V of Y containing F (x), there exists an I ⋆-preopen set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V and by (1), we have x ∈ Int⋆(Cl⋆(F+(σ1σ2-Cl(V )))). Therefore, F+(V ) ⊆ Int⋆(Cl⋆(F+(σ1σ2-Cl(V )))). (2) ⇒ (3): Let V be any σ1σ2-open set of Y . Since Y − σ1σ2-Cl(V ) is σ1σ2-open and by (2), X − F−(σ1σ2-Cl(V )) = F+(Y − σ1σ2-Cl(V )) C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6570 5 of 14 ⊆ Int⋆(Cl⋆(F+(σ1σ2-Cl(Y − σ1σ2-Cl(V ))))) ⊆ Int⋆(Cl⋆(F+(Y − V ))) = Int⋆(Cl⋆(X − F−(V ))) = X − Cl⋆(Int⋆(F−(V ))). Thus, Cl⋆(Int⋆(F−(V ))) ⊆ F−(σ1σ2-Cl(V )). (3) ⇒ (4): Let V be any σ1σ2-open set of Y . By (3) and Lemma 2, pCl⋆(F−(V )) = Cl⋆(Int⋆(F−(V ))) ∪ F−(V ) ⊆ F−(σ1σ2-Cl(V )). (4) ⇒ (5): Let V be any σ1σ2-open set of Y . Then, Y − σ1σ2-Cl(V ) is σ1σ2-open in Y . Thus by (4), X − pInt⋆(F+(σ1σ2-Cl(V ))) = pCl⋆(X − F+(σ1σ2-Cl(V ))) = pCl⋆(F−(Y − σ1σ2-Cl(V ))) ⊆ F−(σ1σ2-Cl(Y − σ1σ2-Cl(V ))) ⊆ F−(Y − V ) = X − F+(V ) and hence F+(V ) ⊆ pInt⋆(F+(σ1σ2-Cl(V ))). (5) ⇒ (6): Let x ∈ X and V be any σ1σ2-open set of Y containing F (x). By (5), x ∈ F+(V ) ⊆ pInt⋆(F+(σ1σ2-Cl(V ))) and there exists a I ⋆-preopen set U ofX containing x such that F (U) ⊆ σ1σ2-Cl(V ). (6) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing F (x). By (6), there exists an I ⋆-preopen set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ); hence U ⊆ F+(σ1σ2-Cl(V )). Thus, x ∈ U ⊆ Int⋆(Cl⋆(U)) ⊆ Int⋆(Cl⋆(F+(σ1σ2-Cl(V )))). This shows that F is upper almost weakly τ⋆(σ1, σ2)-continuous. Definition 2. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower almost weakly τ⋆(σ1, σ2)-continuous if for each x ∈ X and each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, x ∈ Int⋆(Cl⋆(F−(σ1σ2-Cl(V )))). Theorem 2. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost weakly τ⋆(σ1, σ2)-continuous; (2) F−(V ) ⊆ Int⋆(Cl⋆(F−(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ; (3) Cl⋆(Int⋆(F+(V ))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (4) pCl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (5) F−(V ) ⊆ pInt⋆(F−(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (6) for each x ∈ X and each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there exists an I ⋆-preopen set U of X containing x such that F (z) ∩ σ1σ2-Cl(V ) ̸= ∅ for each z ∈ U . C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6570 6 of 14 Proof. The proof is similar to that of Theorem 1. Theorem 3. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost weakly τ⋆(σ1, σ2)-continuous; (2) Cl⋆(Int⋆(F−(σ1σ2-Int(K)))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (3) pCl⋆(F−(σ1σ2-Int(K))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (4) pCl⋆(F−(σ1σ2-Int(σ1σ2Cl(B)))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (5) F+(σ1σ2-Int(B)) ⊆ pInt⋆(F+(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y . Proof. (1) ⇒ (2): Let K be any σ1σ2-closed set of Y . Then, Y −K is σ1σ2-open in Y , by Theorem 1, we have X − F−(K) = F+(Y −K) ⊆ Int⋆(Cl⋆(F+(σ1σ2-Cl(Y −K)))) = Int⋆(Cl⋆(F+(Y − σ1σ2-Int(K)))) = Int⋆(Cl⋆(X − F−(σ1σ2-Int(K)))) = X − Cl⋆(Int⋆(F−(σ1σ2-Int(K)))) and so Cl⋆(Int⋆(F−(σ1σ2-Int(K)))) ⊆ F−(K). (2) ⇒ (3): Let K be any σ1σ2-closed set of Y . By Lemma 2, we have pCl⋆(F−(σ1σ2-Int(K))) = F−(σ1σ2-Int(K)) ∪ Cl⋆(Int⋆(F−(σ1σ2-Int(K)))) ⊆ F−(K). (3) ⇒ (4): The proof is obvious. (4) ⇒ (5): Let B be any subset of Y . By (4), we have X − pInt⋆(F+(σ1σ2-Cl(σ1σ2-Int(B)))) = pCl⋆(X − F+(σ1σ2-Cl(σ1σ2-Int(B)))) = pCl⋆(F−(Y − σ1σ2-Cl(σ1σ2-Int(B)))) = pCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(Y −B)))) ⊆ F−(σ1σ2-Cl(Y −B)) = X − F+(σ1σ2-Int(B)). Thus, F+(σ1σ2-Int(B)) ⊆ pInt⋆(F+(σ1σ2-Cl(σ1σ2-Int(B)))). (5) ⇒ (1): Let V be any σ1σ2-open set of Y . Then by (5), we have F+(V ) ⊆ pInt⋆(F+(σ1σ2-Cl(V ))) and hence F is upper almost weakly τ⋆(σ1, σ2)-continuous by Theorem 1. Theorem 4. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6570 7 of 14 (1) F is lower almost weakly τ⋆(σ1, σ2)-continuous; (2) Cl⋆(Int⋆(F+(σ1σ2-Int(K)))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (3) pCl⋆(F+(σ1σ2-Int(K))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (4) pCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (5) F−(σ1σ2-Int(B)) ⊆ pInt⋆(F−(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y . Proof. The proof is similar to that of Theorem 3. Theorem 5. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost weakly τ⋆(σ1, σ2)-continuous; (2) pCl⋆(F−(σ1σ2-Int((σ1, σ2)-θCl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) pCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (4) pCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (5) pCl⋆(F−(σ1σ2-Int(K))) ⊆ F−(K) for every (σ1, σ2)r-closed set K of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . Let x ∈ X −F−((σ1, σ2)θ-Cl(B)). Then, x ∈ F+(Y − (σ1, σ2)θ-Cl(B)) and (σ1, σ2)θ-Cl(B) is σ1σ2-closed in Y . By Theorem 1, there exists an I ⋆-preopen set U of X containing x such that U ⊆ F+(σ1σ2-Cl(Y − (σ1, σ2)θ-Cl(B))) = F+(Y − σ1σ2-Int((σ1, σ2)θ-Cl(B))) = X − F−(σ1σ2-Int((σ1, σ2)θ-Cl(B))). Thus, U ∩ F−(σ1σ2-Int((σ1, σ2)θ-Cl(B))) = ∅ and hence x ∈ X − pCl⋆(F−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))). Therefore, pCl⋆(F−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)). (2) ⇒ (3): The proof is obvious since (σ1, σ2)θ-Cl(V ) = σ1σ2-Cl(V ) for every σ1σ2- open set V of Y . (3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . Then, V ⊆ σ1σ2-Int(σ1σ2-Cl(V )) and by (3), we have pCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) = pCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V )))))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Int(V )))) = F−(σ1σ2-Cl(V )). C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6570 8 of 14 (4) ⇒ (5): Let K be any (σ1, σ2)r-closed set of Y . Then, σ1σ2-Int(K) is (σ1, σ2)p-open in Y and by (4), pCl⋆(F−(σ1σ2-Int(K))) = pCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(K))) = F−(K). (5) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, σ1σ2-Cl(V ) is (σ1, σ2)r-closed in Y and by (5), pCl⋆(F−(V )) ⊆ pCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). It follows from Theorem 1 that F is upper almost weakly τ⋆(σ1, σ2)-continuous. Theorem 6. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost weakly τ⋆(σ1, σ2)-continuous; (2) pCl⋆(F+(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) pCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (4) pCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (5) pCl⋆(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 5. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), by ClFı : (X, τ,I ) → (Y, σ1, σ2) (resp. pClFı : (X, τ,I ) → (Y, σ1, σ2)) we denote a multifunction defined as follows: ClFı(x) = σ1σ2-Cl(F (x)) (resp. pClFı(x) = (σ1, σ2)-pCl(F (x))) for each x ∈ X. Definition 3. [21] A subset A of a bitopological space (X, τ1, τ2) is said to be: (1) τ1τ2-paracompact 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; (2) τ1τ2-regular 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 3. [21] 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 . Lemma 4. If F : (X, τ,I ) → (Y, σ1, σ2) is a multifunction such that F (x) is σ1σ2- regular and σ1σ2-paracompact for each x ∈ X, then ClF+ ı (V ) = pClF+ ı (V ) = F+(V ) for each σ1σ2-open set V of Y . C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6570 9 of 14 Proof. It follows from Lemma 5 of [28]. Theorem 7. Let F : (X, τ,I ) → (Y, σ1, σ2) be a multifunction such that F (x) is σ1σ2- paracompact and σ1σ2-regular for each x ∈ X. Then, the following properties are equiva- lent: (1) F is upper almost weakly τ⋆(σ1, σ2)-continuous; (2) pClFı is upper almost weakly τ⋆(σ1, σ2)-continuous; (3) ClFı is upper almost weakly τ⋆(σ1, σ2)-continuous. Proof. We put G = ClFı or pClFı in the sequel. Suppose that F is upper almost weakly τ⋆(σ1, σ2)-continuous. Let x ∈ X and V be any σ1σ2-open set of Y containing G(x). By Lemma 4, we have x ∈ G+(V ) = F+(V ) and hence there exists an I ⋆- preopen set U containing x such that F (U) ⊆ σ1σ2-Cl(V ). Since F (z) is σ1σ2-paracompact and σ1σ2-regular for each z ∈ U , by Lemma 3 there exists a σ1σ2-open set W such that F (z) ⊆ W ⊆ σ1σ2-Cl(W ) ⊆ V ; hence G(z) ⊆ σ1σ2-Cl(W ) ⊆ σ1σ2-Cl(V ) for each z ∈ U . Thus, G(U) ⊆ σ1σ2-Cl(V ). This shows that G is upper almost weakly τ⋆(σ1, σ2)- continuous. Conversely, suppose that G is upper almost weakly τ⋆(σ1, σ2)-continuous. Let x ∈ X and V be any σ1σ2-open set of Y containing G(x). By Lemma 4, we have x ∈ F+(V ) = G+(V ) and hence G(x) ⊆ V . Then, there exists an I ⋆-preopen set U containing x such that F (U) ⊆ σ1σ2-Cl(V ). Thus, U ⊆ G+(V ) = F+(V ) and hence F (U) ⊆ σ1σ2-Cl(V ). This shows that F is upper almost weakly τ⋆(σ1, σ2)-continuous. Lemma 5. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), ClF− ı (V ) = pClF− ı (V ) = F−(V ) for each σ1σ2-open set V of Y . Proof. It follows from Lemma 3 of [28]. Theorem 8. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost weakly τ⋆(σ1, σ2)-continuous; (2) pClFı is lower almost weakly τ⋆(σ1, σ2)-continuous; (3) ClFı is lower almost weakly τ⋆(σ1, σ2)-continuous. Proof. By using Lemma 5 this can be shown similarly to that of Theorem 7. The ⋆-prefrontier of a subset A of an ideal topological space (X, τ,I ), denoted by pfr⋆(A), is defined by pfr⋆(A) = pCl⋆(A) ∩ pCl⋆(X −A) = pCl⋆(A)− pInt⋆(A). C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6570 10 of 14 Theorem 9. The set of all points x of X at which a multifunction F : (X, τ,I ) → (Y, σ1, σ2) is not upper almost weakly τ⋆(σ1, σ2)-continuous is identical with the union of the ⋆- prefrontier of the upper inverse images of the σ1σ2-closure of σ1σ2-open sets containing F (x). Proof. Let x ∈ X at which F is not upper almost weakly τ⋆(σ1, σ2)-continuous. There exists a σ1σ2-open set V of Y containing F (x) such that U ∩ (X − F+(V )) ̸= ∅ for every I ⋆-preopen set U of X containing x. Therefore, we have x ∈ pCl⋆(X − F+(σ1σ2-Cl(V ))) = X − pInt⋆(F+(σ1σ2-Cl(V ))). Since x ∈ F+(V ), we have x ∈ pCl⋆(F+(σ1σ2-Cl(V ))) and so x ∈ pfr⋆(F+(σ1σ2-Cl(V ))). Conversely, if F is upper almost weakly τ⋆(σ1, σ2)-continuous, then for any σ1σ2-open set V of Y containing F (x) there exists an I ⋆-preopen set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ); hence U ⊆ F+(σ1σ2-Cl(V )). Therefore, x ∈ pInt⋆(F+(σ1σ2-Cl(V ))). This contradicts with the fact that x ∈ pfr⋆(F+(σ1σ2-Cl(V ))). Thus, F is not upper almost weakly τ⋆(σ1, σ2)-continuous at x. Theorem 10. The set of all points x of X at which a multifunction F : (X, τ,I ) → (Y, σ1, σ2) is not lower almost weakly τ⋆(σ1, σ2)-continuous is identical with the union of the ⋆- prefrontier of the lower inverse images of σ1σ2-closure of σ1σ2-open sets meeting F (x). Proof. The proof is similar to that of Theorem 9. Definition 4. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper τ⋆(σ1, σ2)- precontinuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists an I ⋆-preopen 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⋆(σ1, σ2)-precontinuous if F is upper τ⋆(σ1, σ2)-precontinuous at each point x of X. Theorem 11. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper τ⋆(σ1, σ2)-precontinuous; (2) F+(V ) is I ⋆-preopen in X for every σ1σ2-open set V of Y ; (3) F−(K) is I ⋆-preclosed in X for every σ1σ2-closed set K of Y ; (4) pCl⋆(F−(B)) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6570 11 of 14 (5) F+(σ1σ2-Int(B)) ⊆ pInt⋆(F+(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V and by (1), there exists an I ⋆-preopen set U of X containing x such that F (U) ⊆ V . Thus, x ∈ U ⊆ F+(V ) and hence x ∈ pInt⋆(F+(V )). Therefore, F+(V ) ⊆ pInt⋆(F+(V )). This shows that F+(V ) is I ⋆-preopen in X. (2) ⇒ (3): This follows from the fact that F+(Y −B) = X − F−(B) for every subset B of Y . (3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Cl(B) is σ1σ2-closed in Y and by (3), pCl⋆(F−(B)) ⊆ pCl⋆(F−(σ1σ2-Cl(B))) = F−(σ1σ2-Cl(B)). (4) ⇒ (5): Let B be any subset of Y . By (4), X−pInt⋆(F+(B)) = pCl⋆(X−F+(B)) = pCl⋆(F−(Y −B)) ⊆ F−(σ1σ2-Cl(Y −B)) = F−(Y −σ1σ2-Int(B)) = X−F+(σ1σ2-Int(B)) and hence F+(σ1σ2-Int(B)) ⊆ pInt⋆(F+(B)). (5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that F (x) ⊆ V . Then, x ∈ F+(V ) = pInt⋆(F+(V )). There exists an I ⋆-preopen set U of X containing x such that U ⊆ F+(V ); hence F (U) ⊆ V . This shows that F is upper τ⋆(σ1, σ2)-precontinuous. Definition 5. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower τ⋆(σ1, σ2)- precontinuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there exists an I ⋆-preopen 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 called lower τ⋆(σ1, σ2)-precontinuous if F is lower τ⋆(σ1, σ2)-precontinuous at each point x of X. Theorem 12. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower τ⋆(σ1, σ2)-precontinuous; (2) F−(V ) is I ⋆-preopen in X for every σ1σ2-open set V of Y ; (3) F+(K) is I ⋆-preclosed in X for every σ1σ2-closed set K of Y ; (4) pCl⋆(F+(B)) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (5) F (pCl⋆(A)) ⊆ σ1σ2-Cl(F (A)) for every subset A of X; (6) F−(σ1σ2-Int(B)) ⊆ pInt⋆(F−(B)) for every subset B of Y . Proof. We prove only the implications (4) ⇒ (5) and (5) ⇒ (6) being the proofs of the other similar to those of Theorem 11. (4) ⇒ (5): Let A be any subset of X. Thus by (4), pCl⋆(A) ⊆ pCl⋆(F+(F (A))) ⊆ F+(σ1σ2-Cl(F (A))) and so F (pCl⋆(A)) ⊆ σ1σ2-Cl(F (A)). (5) ⇒ (6): Let B be any subset of Y . By (5), F (pCl⋆(F+(Y −B))) ⊆ σ1σ2-Cl(F (F+(Y −B))) ⊆ σ1σ2-Cl(Y −B) = Y − σ1σ2-Int(B). C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6570 12 of 14 Since F (pCl⋆(F+(Y −B))) = F (pCl⋆(X − F−(B))) = F (X − pInt⋆(F−(B))), we have X − pInt⋆(F−(B)) ⊆ F+(Y − σ1σ2-Int(B)) = X − F−(σ1σ2-Int(B)) and hence F−(σ1σ2-Int(B)) ⊆ pInt⋆(F−(B)). Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-regular [29] if for each τ1τ2-closed set F and each x ̸∈ F , there exist disjoint τ1τ2-open sets U and V such that x ∈ U and F ⊆ V . Lemma 6. [30] Let (X, τ1, τ2) be a (τ1, τ2)-regular space. Then, the following properties hold: (1) τ1τ2-Cl(A) = (τ1, τ2)θ-Cl(A) for every subset A of X. (2) Every τ1τ2-open set is (τ1, τ2)θ-open. Theorem 13. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)-regular, the following properties are equivalent: (1) F is upper τ⋆(σ1, σ2)-precontinuous; (2) F−((σ1, σ2)θ-Cl(B)) is I ⋆-preclosed in X for every subset B of Y ; (3) F−(K) is I ⋆-preclosed in X for every (σ1, σ2)θ-closed set K of Y ; (4) F+(V ) is I ⋆-preopen in X for every (σ1, σ2)θ-open set V 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 11, F−((σ1, σ2)θ-Cl(B)) is I ⋆-preclosed in X. (2) ⇒ (3): The proof is obvious. (3) ⇒ (4): Let V be any (σ1, σ2)θ-open set of Y . By (3), F−(Y − V ) is I ⋆-preclosed in X and F−(Y − V ) = X − F+(V ). Thus, F+(V ) is I ⋆-preopen in X. (4) ⇒ (1): Let V be any σ1σ2-open set of Y . Since (Y, σ1, σ2) is (σ1, σ2)-regular, by Lemma 6 we have V is (σ1, σ2)θ-open in Y and by (4), F+(V ) is I ⋆-preopen in X. Thus by Theorem 11, F is upper τ⋆(σ1, σ2)-precontinuous. Theorem 14. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)-regular, the following properties are equivalent: (1) F is lower τ⋆(σ1, σ2)-precontinuous; (2) F+((σ1, σ2)θ-Cl(B)) is I ⋆-preclosed in X for every subset B of Y ; (3) F+(K) is I ⋆-preclosed in X for every (σ1, σ2)θ-closed set K of Y ; (4) F−(V ) is I ⋆-preopen in X for every (σ1, σ2)θ-open set V of Y ; (5) F is lower almost weakly τ⋆(σ1, σ2)-continuous. C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6570 13 of 14 Proof. We prove only the implication (5) ⇒ (1), the proof of the other being similar to that of Theprem 13. The proof of the implication (4) ⇒ (5) is obvious. (5) ⇒ (1): Let V be any σ1σ2-open set of Y and x ∈ F−(V ). Then, F (x)∩V ̸= ∅. Since (Y, σ1, σ2) is (σ1, σ2)-regular, there exists a σ1σ2-open set W of Y such that F (x)∩W ̸= ∅ and σ1σ2-Cl(W ) ⊆ V . 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