EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6567 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Weak Forms of Upper and Lower Continuous Multifunctions between an Ideal Topological Space and a Bitopological Space 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 presents new concepts of continuous multifunctions defined from an ideal topological space into a bitopological space, called upper weakly τ⋆(σ1, σ2)-continuous multifunc- tions and lower weakly τ⋆(σ1, σ2)-continuous multifunctions. Furthermore, several characteriza- tions and some properties concerning upper weakly τ⋆(σ1, σ2)-continuous multifunctions and lower weakly τ⋆(σ1, σ2)-continuous multifunctions are investigated. Moreover, the relationships between almost τ⋆(σ1, σ2)-continuity and weak τ⋆(σ1, σ2)-continuity are established. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper weakly τ⋆(σ1, σ2)-continuous multifunction, lower weakly τ⋆(σ1, σ2)-continuous multifunction 1. Introduction The concept of weakly continuous functions was introduced by Levine [1]. Husain [2] introduced and studied the notion of almost continuous functions. Janković [3] introduced almost weak continuity as a generalization of both weak continuity 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 relation- ships between subweak continuity and weak continuity. 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 investigated several characterizations of weakly (τ,m)-continuous functions. Ekici et al. [7] introduced and studied the concept of weakly λ-continuous functions. Popa [8] and Smithson [9] independently introduced the notion of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6567 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 (3) (2025), 6567 2 of 13 weakly continuous multifunctions. Popa and Noiri [10] introduced a class of multifunctions called weakly α-continuous multifunctions. Furthermore, Popa and Noiri [11] investigated some characterizations of upper and lower weakly β-continuous multifunctions. Noiri and Popa [12] introduced and investigated the notion of weakly m-continuous multifunctions as a multifunction from a set satisfying certain minimal condition into a topological space. Pue-on et al. [13] introduced and studied the concepts of upper (τ1, τ2)-continuous mul- tifunctions and lower (τ1, τ2)-continuous multifunctions. Klanarong et al. [14] introduced and investigated the notions of upper almost (τ1, τ2)-continuous multifunctions and lower almost (τ1, τ2)-continuous multifunctions. Thongmoon et al. [15] introduced and studied the concepts of upper weakly (τ1, τ2)-continuous multifunctions and lower weakly (τ1, τ2)- continuous multifunctions. The concept of ideal topological spaces was introduced and studied by Kuratowski [16] and Vaidyanathaswamy [17]. Weaker and stronger forms of open sets in ideal topological spaces such as 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 continuity. Using these notions many authors introduced and studied various types of generalizations of continuity for functions and multifunctions. Hatir and Noiri [18] introduced and investigated the notions of weakly pre-I -open sets and weakly pre-I -continuous functions. Moreover, Hatir and Noiri [19] investigated further prop- erties of semi-I -open sets and semi-I -continuous functions. On the other hand, the present author [20] introduced the notions of upper ⋆-continuous multifunctions and lower ⋆-continuous multifunctions. Furthermore, several characterizations of upper ⋆-continuous multifunctions, lower ⋆-continuous multifunctions, upper almost ⋆-continuous multifunc- tions, lower almost ⋆-continuous multifunctions, upper weakly ⋆-continuous multifunctions and lower weakly ⋆-continuous multifunctions were considered in [20]. Quite recently, the present author [21] introduced and studied the notions of pı-continuous multifunctions and weakly pı-continuous multifunctions. In this paper, we introduce new classes of multifunc- tions between an ideal topological space and a bitopological space, namely upper weakly τ⋆(σ1, σ2)-continuous multifunctions and lower weakly τ⋆(σ1, σ2)-continuous multifunc- tions. We also investigate several characterizations of upper weakly τ⋆(σ1, σ2)-continuous multifunctions and lower 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 [22] 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 [22] 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 [22] of A and is denoted by τ1τ2-Int(A). Lemma 1. [22] Let A and B be subsets of a bitopological space (X, τ1, τ2). For the τ1τ2- P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6567 3 of 13 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 subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [23] (resp. (τ1, τ2)s-open [24], (τ1, τ2)p-open [24], (τ1, τ2)β-open [24]) 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 [25] 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 [26] (resp. (τ1, τ2)s-closure [24], α(τ1, τ2)-closure [27]) 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 [26] (resp. (τ1, τ2)s-interior [24], α(τ1, τ2)-interior [27]) of A and is denoted by (τ1, τ2)-pInt(A) (resp. (τ1, τ2)-sInt(A), α(τ1, τ2)-Int(A)). 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) [23]. 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 [16], 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 [3] 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 [28] (resp. semi-I -open P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6567 4 of 13 [19]) 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 [28] (resp. semi-I -closed [19]). 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 ⋆(σ1, σ2)-continuous multifunctions In this section, we introduce the concepts of upper weakly τ⋆(σ1, σ2)-continuous mul- tifunctions and lower weakly τ⋆(σ1, σ2)-continuous multifunctions. Furthermore, several characterizations of upper weakly τ⋆(σ1, σ2)-continuous multifunctions and lower weakly τ⋆(σ1, σ2)-continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper weakly τ⋆(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 ⋆-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⋆(σ1, σ2)-continuous if F is upper weakly τ⋆(σ1, σ2)-continuous at each point x of X. Definition 2. [29] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be: (i) upper almost τ⋆(σ1, σ2)-continuous if for each point x ∈ X and each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a ⋆-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )); (ii) lower almost τ⋆(σ1, σ2)-continuous if for each point x ∈ X and each σ1σ2-open set V of Y such that V ∩ F (x) ̸= ∅, there exists a ⋆-open set U of X containing x such that σ1σ2-Int(σ1σ2-Cl(V )) ∩ F (z) ̸= ∅ for every z ∈ U . Remark 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following implication holds: upper almost τ⋆(σ1, σ2)-continuity ⇒ upper weakly τ⋆(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 topology τ = {∅, {2}, {1, 3}, X} and an ideal I = {∅, {2}}. 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) = {a}, F (2) = {b} and F (3) = {a, c}. Then, F is upper weakly τ⋆(σ1, σ2)-continuous but F is not upper almost τ⋆(σ1, σ2)-continuous. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6567 5 of 13 Theorem 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly τ⋆(σ1, σ2)-continuous; (2) F+(V ) ⊆ Int⋆(F+(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (3) Cl⋆(F−(σ1σ2-Int(K))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (4) Cl⋆(F−(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (5) F+(σ1σ2-Int(B)) ⊆ Int⋆(F+(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ; (6) Cl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (7) Cl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (8) Cl⋆(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 ⋆-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). Thus, U ⊆ F+(σ1σ2-Cl(V )). Since U is ⋆-open, we have x ∈ Int⋆(F+(σ1σ2-Cl(V ))) and so F+(V ) ⊆ Int⋆(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) ⊆ Int⋆(F+(σ1σ2-Cl(Y − K))) = X − Cl⋆(F−(σ1σ2-Int(K))). Thus, Cl⋆(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), Cl⋆(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 − Int⋆(F+(σ1σ2-Cl(σ1σ2-Int(B)))) = Cl⋆(X − F+(σ1σ2-Cl(σ1σ2-Int(B)))) = Cl⋆(F−(σ1σ2-Int(σ1σ2-Cl(Y −B)))) ⊆ F−(σ1σ2-Cl(Y −B)) = X − F+(σ1σ2-Int(B)) and hence F+(σ1σ2-Int(B)) ⊆ Int⋆(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 ) ⊆ Int⋆(F+(σ1σ2-Cl(V ))) and there exists a ⋆-open set U of X containing x such that U ⊆ F+(σ1σ2-Cl(V )). Thus, F (U) ⊆ σ1σ2-Cl(V ) and hence F is upper weakly τ⋆(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), Cl⋆(F−(σ1σ2-Int(K))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(K))) = F−(K). P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6567 6 of 13 (8) ⇒ (3): LetK 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), Cl⋆(F−(σ1σ2-Int(K))) = Cl⋆(F−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ F−(K). Definition 3. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower weakly τ⋆(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 ⋆-open set U of X containing x such that σ1σ2-Cl(V ) ∩ F (z) ̸= ∅ for every z ∈ U . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower weakly τ⋆(σ1, σ2)- continuous if F is lower weakly τ⋆(σ1, σ2)-continuous at each point x 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⋆(σ1, σ2)-continuous; (2) F−(V ) ⊆ Int⋆(F−(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (3) Cl⋆(F+(σ1σ2-Int(K))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (4) Cl⋆(F+(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (5) F−(σ1σ2-Int(B)) ⊆ Int⋆(F−(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ; (6) Cl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (7) Cl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (8) Cl⋆(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. Theorem 3. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) Cl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y . P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6567 7 of 13 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, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) Cl⋆(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. Theorem 5. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) Cl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (4) F+(V ) ⊆ Int⋆(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) Cl⋆(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 Cl⋆(F−(V )) ⊆ Cl⋆(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 − Int⋆(F+(Cl⋆(V ))) = Cl⋆(X − F+(Cl⋆(V ))) = Cl⋆(F−(Y − Cl⋆(V ))) ⊆ F−(σ1σ2-Cl(Y − σ1σ2-Cl(V ))) = X − F+(σ1σ2-Int(σ1σ2-Cl(V ))) P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6567 8 of 13 ⊆ X − F+(V ) and hence F+(V ) ⊆ Int⋆(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 ) ⊆ Int⋆(F+(σ1σ2-Cl(V ))). By Theorem 1(2), F is upper 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 weakly τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) Cl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (4) F−(V ) ⊆ Int⋆(F−(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y . Proof. The proof is similar to that of Theorem 5. Lemma 2. If F : (X, τ,I ) → (Y, σ1, σ2) is lower weakly τ⋆(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 ⋆-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 τ⋆(σ1, σ2)- continuous, there exists a ⋆-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, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F+(B)) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) F (Cl⋆(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 ⋆-open set U of X containing x such that U ⊆ F−(Y − B) = X − F+(B). Thus, U ∩ F+(B) = ∅ and hence x ̸∈ Cl⋆(F+(B)). This shows that Cl⋆(F+(B)) ⊆ F+((σ1, σ2)θ-Cl(B)). P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6567 9 of 13 (2) ⇒ (3): Let A be any subset of X. By (2), we have Cl⋆(A) ⊆ Cl⋆(F+(F (A))) ⊆ F+((σ1, σ2)θ-Cl(F (A))). Thus, F (Cl⋆(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 (Cl⋆(F+(V ))) ⊆ (σ1, σ2)θ-Cl(F (F+(V ))) ⊆ (σ1, σ2)θ-Cl(V ) = σ1σ2-Cl(V ). Thus, Cl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) and by Theorem 1, F is lower weakly τ⋆(σ1, σ2)- continuous. Theorem 8. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) Cl⋆(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, Cl⋆(F−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)). (2) ⇒ (3): The proof is obvious. (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 Cl⋆(F−(σ1σ2-Int(K))) = Cl⋆(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 τ⋆(σ1, σ2)-continuous. Theorem 9. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F+(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) Cl⋆(F+(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y . P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6567 10 of 13 Proof. The proof is similar to that of Theorem 10. Recall that a subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-paracompact [22] 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 [22] 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. [22] 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 . Theorem 10. For a multifunction F : (X, τ,I ) → (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 τ⋆(σ1, σ2)-continuous; (2) F is upper almost τ⋆(σ1, σ2)-continuous; (3) F is upper weakly τ⋆(σ1, σ2)-continuous. Proof. We show only the implication (3) ⇒ (1) since the others are obvious. Suppose that F is upper weakly τ⋆(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 3 there exists a σ1σ2-open set W of Y such that F (x) ⊆ W ⊆ σ1σ2-Cl(W ) ⊆ V . Since F is upper weakly τ⋆(σ1, σ2)-continuous, there exists a ⋆-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(W ); hence F (U) ⊆ V . This shows that F is upper τ⋆(σ1, σ2)-continuous. Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-compact [22] if every cover of X by τ1τ2-open sets of X has a finite subcover. Definition 4. [30] 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, τ,I ) → (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 τ⋆(σ1, σ2)-continuous; (2) F is upper almost τ⋆(σ1, σ2)-continuous; (3) F is upper weakly τ⋆(σ1, σ2)-continuous. Lemma 4. [31] 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. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6567 11 of 13 Theorem 11. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2) such that F (x) is a σ1σ2- regular set of Y for each x ∈ X, the following properties are equivalent: (1) F is lower τ⋆(σ1, σ2)-continuous; (2) F is lower almost τ⋆(σ1, σ2)-continuous; (3) F is lower weakly τ⋆(σ1, σ2)-continuous. Proof. We show only the implication (3) ⇒ (1) since the others are obvious. Suppose that F is lower weakly τ⋆(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 4 there exists a σ1σ2-open set W of Y such that F (x) ∩ W ̸= ∅ and σ1σ2-Cl(W ) ⊆ V . Since F is lower weakly τ⋆(σ1, σ2)-continuous, there exists a ⋆-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 τ⋆(σ1, σ2)-continuous. Definition 5. [32] 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, τ,I ) → (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 τ⋆(σ1, σ2)-continuous; (2) F is upper almost τ⋆(σ1, σ2)-continuous; (3) F is upper weakly τ⋆(σ1, σ2)-continuous. Proof. As in Theorem 10, we prove only the implication (3) ⇒ (1). Suppose that F is upper weakly τ⋆(σ1, σ2)-continuous. Let x ∈ X and G be any σ1σ2-open set of Y containing F (x). Since F (x) is σ1σ2-closed in Y , by the (σ1, σ2)-normality of Y there exists a σ1σ2-open set V of Y such that F (x) ⊆ V ⊆ σ1σ2-Cl(V ) ⊆ G. Since F is upper weakly τ⋆(σ1, σ2)-continuous, there exists a ⋆-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ) ⊆ G. This shows that F is upper τ⋆(σ1, σ2)-continuous. Acknowledgements This research project was financially supported by Mahasarakham University. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. 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