EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6571 ISSN 1307-5543 – ejpam.com Published by New York Business Global Almost Quasi Continuity and Weak Quasi Continuity for Multifunctions between an Ideal Topological Space and a Bitopological Space Montri Thongmoon1, Areeyuth Sama-Ae2, Chawalit Boonpok1,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand 2 Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Pattani, 94000, Thailand Abstract. This paper introduces four classes of continuous multifunctions defined from an ideal topological space into a bitopological space, namely upper almost quasi τ⋆(σ1, σ2)-continuous mul- tifunctions, lower almost quasi τ⋆(σ1, σ2)-continuous multifunctions, upper weakly quasi τ⋆(σ1, σ2)- continuous multifunctions and lower weakly quasi τ⋆(σ1, σ2)-continuous multifunctions. Moreover, several characterizations of upper almost quasi τ⋆(σ1, σ2)-continuous multifunctions, lower almost quasi τ⋆(σ1, σ2)-continuous multifunctions, upper weakly quasi τ⋆(σ1, σ2)-continuous multifunc- tions and lower weakly quasi τ⋆(σ1, σ2)-continuous multifunctions are established. Furthermore, the relationships between almost quasi τ⋆(σ1, σ2)-continuity and weak quasi τ⋆(σ1, σ2)-continuity are discussed. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper almost quasi τ⋆(σ1, σ2)-continuous multifunction, lower almost quasi τ⋆(σ1, σ2)-continuous multifunction, upper weakly quasi τ⋆(σ1, σ2)-continuous multifunction, lower weakly quasi τ⋆(σ1, σ2)-continuous multifunction 1. Introduction The concept of quasi continuous functions was introduced by Marcus [1]. Popa [2] introduced and investigated the notion of almost quasi continuous functions. Neubrun- novaá [3] showed that quasi continuity is equivalent to semi-continuity due to Levine [4]. Popa and Stan [5] introduced and studied the notion of weakly quasi continuous functions. Weak quasi continuity is implied by quasi continuity and weak continuity [6] which are independent of each other. It is shown in [7] that weak quasi continuity ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6571 Email addresses: montri.t@msu.ac.th (M. Thongmoon), areeyuth.s@psu.ac.th (A. Sama-Ae), chawalit.b@msu.ac.th (C. Boonpok) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6571 2 of 15 is equivalent to weak semi-continuity due to Arya and Bhamini [8] and Kar and Bhat- tacharyya [9]. Janković and Hamlett [10] introduced the concept of I -open sets in ideal topological spaces. Abd El-Monsef et al. [11] 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 gen- eralizations of continuity in ideal topological spaces. Using these notions many authors introduced and studied various types of generalizations of continuity for functions and multifunctions. Hatir and Noiri [12] introduced and investigated the notions of weakly pre-I -open sets and weakly pre-I -continuous functions. Moreover, Hatir and Noiri [13] investigated further properties of semi-I -open sets and semi-I -continuous functions. On the other hand, the present author introduced and investigated the concepts of almost quasi ⋆-continuous multifunctions [14], weakly quasi ⋆-continuous multifunctions [14], pı- continuous multifunctions [15] and weakly pı-continuous multifunctions [15]. Pue-on et al. [16] introduced and studied the concepts of upper (τ1, τ2)-continuous multifunctions and lower (τ1, τ2)-continuous multifunctions. Klanarong et al. [17] introduced and investigated the notions of upper almost (τ1, τ2)-continuous multifunctions and lower almost (τ1, τ2)- continuous multifunctions. Thongmoon et al. [18] introduced and studied the concepts of upper weakly (τ1, τ2)-continuous multifunctions and lower weakly (τ1, τ2)-continuous multifunctions. Chiangpradit et al. [19] introduced and investigated the notions of upper almost quasi (τ1, τ2)-continuous multifunctions and lower almost quasi (τ1, τ2)-continuous multifunctions. Quite recently, Pue-on et al. [20] introduced and studied the concepts of upper weakly quasi (τ1, τ2)-continuous multifunctions and lower weakly quasi (τ1, τ2)- continuous multifunctions. In this paper, we introduce new classes of continuous multi- functions between an ideal topological space and a bitopological space, namely upper al- most quasi τ⋆(σ1, σ2)-continuous multifunctions, lower almost quasi τ⋆(σ1, σ2)-continuous multifunctions, upper weakly quasi τ⋆(σ1, σ2)-continuous multifunctions and lower weakly quasi τ⋆(σ1, σ2)-continuous multifunctions. We also investigate several characterizations of upper almost quasi τ⋆(σ1, σ2)-continuous multifunctions, lower almost quasi τ⋆(σ1, σ2)- continuous multifunctions, upper weakly quasi τ⋆(σ1, σ2)-continuous multifunctions and lower weakly quasi 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. 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). Lemma 1. [21] Let A and B be subsets of a bitopological space (X, τ1, τ2). For the τ1τ2- M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6571 3 of 15 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 [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 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 [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. Let A be a subset of a bitopological space (X, τ1, τ2). The intersec- tion 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 [25] (resp. (τ1, τ2)s-closure [23], α(τ1, τ2)-closure [26]) 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 [25] (resp. (τ1, τ2)s-interior [23], α(τ1, τ2)-interior [26]) 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 [22] 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 [22] 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 [22] 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 [22] of A and is denoted by (τ1, τ2)θ-Int(A). Lemma 2. [27] For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: (1) (τ1, τ2)-sCl(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∪A [28]; (2) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A. 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}. M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6571 4 of 15 In case there is no chance for confusion, A⋆(I ) is simply written as A⋆. In [29], 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 [10] 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 [30] (resp. semi-I -open [13]) 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 [30] (resp. semi-I -closed [13]). A subset A of an ideal topological space (X, τ,I ) is said to be semi-I ⋆-open [14] if A ⊆ Cl⋆(Int⋆(A)). The complement of a semi-I ⋆-open set is called semi-I ⋆-closed [14]. For a subset A of an ideal topological space (X, τ,I ), the intersection of all semi-I -closed sets containing A is called the semi-I ⋆-closure [14] of A and is denoted by sCl⋆(A) (sClI ⋆(A) [14]). The union of all semi-I -open sets contained in A is called the semi-I ⋆-interior [14] of A and is denoted by sInt⋆(A) (sIntI ⋆(A) [14]). Lemma 3. [14] For a subset A of a an ideal topological space (X, τ,I ), the following properties hold: (1) sCl⋆(A) = A ∪ Int⋆(Cl⋆(A)); (2) sInt⋆(A) = A ∩ Cl⋆(Int⋆(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 quasi τ ⋆(σ1, σ2)-continuous multifunctions In this section, we introduce the concepts of upper almost quasi τ⋆(σ1, σ2)-continuous multifunctions and lower almost quasi τ⋆(σ1, σ2)-continuous multifunctions. Furthermore, several characterizations of upper almost quasi τ⋆(σ1, σ2)-continuous multifunctions and lower almost quasi τ⋆(σ1, σ2)-continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper almost quasi τ⋆(σ1, σ2)-continuous at a point x ∈ X if for every σ1σ2-open set V of Y such that F (x) ⊆ V and each ⋆-open set U of X containing x, there exists a nonempty ⋆-open set G such that G ⊆ U and F (G) ⊆ (σ1, σ2)-sCl(V ). A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper almost quasi τ⋆(σ1, σ2)-continuous if F is upper almost quasi τ⋆(σ1, σ2)- continuous at each point x of X. Theorem 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost quasi τ⋆(σ1, σ2)-continuous at x ∈ X; M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6571 5 of 15 (2) for every σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a semi-I ⋆-open set U of X containing x such that F (U) ⊆ (σ1, σ2)-sCl(V ); (3) x ∈ sInt⋆(F+((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y such that F (x) ⊆ V ; (4) x ∈ Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V )))) for every σ1σ2-open set V of Y such that F (x) ⊆ V . Proof. (1) ⇒ (2): Let U (x) be the family of all ⋆-open sets of X containing x. Let V be any σ1σ2-open set of Y such that F (x) ⊆ V . For each H ∈ U (x), there exists a nonempty ⋆-open set GH such that GH ⊆ H and F (GH) ⊆ (σ1, σ2)-sCl(V ). Put W = ∪{GH | H ∈ U (x)}. Then, W is ⋆-open in X, x ∈ Cl⋆(W ) and F (W ) ⊆ (σ1, σ2)-sCl(V ). Put U = W ∪{x}, then W ⊆ U ⊆ Cl⋆(W ). Thus, U is a semi-I ⋆-open set of X containing x such that F (U) ⊆ (σ1, σ2)-sCl(V ). (2) ⇒ (3): Let V be any σ1σ2-open set of Y and F (x) ⊆ V . Then, there exists a semi-I ⋆-open set U of X containing x such that F (U) ⊆ (σ1, σ2)-sCl(V ). Thus, x ∈ U ⊆ F+((σ1, σ2)-sCl(V )) and hence x ∈ U ⊆ sInt⋆(F+((σ1, σ2)-sCl(V ))). (3) ⇒ (4): Let V be any σ1σ2-open set of Y such that F (x) ⊆ V . By (3), we have x ∈ sInt⋆(F+((σ1, σ2)-sCl(V ))). Now, put U = sInt⋆(F+((σ1, σ2)-sCl(V ))). Then, U is semi-I ⋆-open in X and by Lemma 3, x ∈ U ⊆ Cl⋆(Int⋆(U)) ⊆ Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V )))). (4) ⇒ (1): Let U be any ⋆-open set of X containing x and V be any σ1σ2-open set of Y such that F (x) ⊆ V . Thus by (4), x ∈ Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V )))) and hence Int⋆(F+((σ1, σ2)-sCl(V ))) ∩ U ̸= ∅. Put W = Int⋆(F+((σ1, σ2)-sCl(V ))) ∩ U . Then, we have W is a nonempty ⋆-open set of X such that W ⊆ U and F (W ) ⊆ (σ1, σ2)-sCl(V ). This shows that F is upper almost quasi τ⋆(σ1, σ2)-continuous at x. Definition 2. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower almost quasi τ⋆(σ1, σ2)-continuous at a point x ∈ X if for every σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅ and each ⋆-open set U of X containing x, there exists a nonempty ⋆-open set G such that G ⊆ U and (σ1, σ2)-sCl(V ) ∩ F (z) ̸= ∅ for each z ∈ G. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower almost quasi τ⋆(σ1, σ2)-continuous if F is lower almost quasi 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 almost quasi τ⋆(σ1, σ2)-continuous at x ∈ X; (2) for every σ1σ2-open set V of Y such that F (x)∩V ̸= ∅, there exists a semi-I ⋆-open set U of X containing x such that (σ1, σ2)-sCl(V ) ∩ F (z) ̸= ∅ for every z ∈ U ; (3) x ∈ sInt⋆(F−((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅; M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6571 6 of 15 (4) x ∈ Cl⋆(Int⋆(F−((σ1, σ2)-sCl(V )))) for every σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅. 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 quasi τ⋆(σ1, σ2)-continuous; (2) for each x ∈ X and every σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a semi-I ⋆-open set U of X containing x such that F (U) ⊆ (σ1, σ2)-sCl(V ); (3) F+(V ) is semi-I ⋆-open in X for every (σ1, σ2)r-open set V of Y ; (4) F+(V ) ⊆ sInt⋆(F+((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y ; (5) sCl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (6) F+(V ) ⊆ Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V )))) for every σ1σ2-open set V of Y . Proof. (1) ⇒ (2): It follows from Theorem 1. (2) ⇒ (3): Let V be any (σ1, σ2)r-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V and there exists a semi-I ⋆-open set U of X containing x such that F (U) ⊆ V . Thus, x ∈ U ⊆ F+(V ) and hence x ∈ sInt⋆(F+(V )). Therefore, F+(V ) ⊆ sInt⋆(F+(V )). This shows that F+(V ) is semi-I ⋆-open in X. (3) ⇒ (4): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, we have F (x) ⊆ V ⊆ (σ1, σ2)-sCl(V ). Thus, x ∈ F+((σ1, σ2)-sCl(V )). By Lemma 2, we have (σ1, σ2)-sCl(V ) is (σ1, σ2)r-open in Y and by (3), F+((σ1, σ2)-sCl(V )) is semi-I ⋆-open in X and x ∈ sInt⋆(F+((σ1, σ2)-sCl(V ))). Thus, F+(V ) ⊆ sInt⋆(F+((σ1, σ2)-sCl(V ))). (4) ⇒ (5): Let B be any subset of Y . Then, we have Y − σ1σ2-Cl(B) is σ1σ2-open in Y . Thus by (4) and Lemma 2, we have X − F−(σ1σ2-Cl(B)) = F+(Y − σ1σ2-Cl(B)) ⊆ sInt⋆(F+((σ1, σ2)-sCl(Y − σ1σ2-Cl(B)))) = sInt⋆(X − F−(σ1σ2-Cl(σ1σ1-Int(σ1σ2-Cl(B))))) = X − sCl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) and hence sCl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F−(σ1σ2-Cl(B)). (5) ⇒ (6): Let V be any σ1σ2-open set of Y . Then, Y − V is σ1σ2-closed in Y . By (5) and Lemma 3, Int⋆(Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(Y −V ))))) ⊆ F−(Y −V ) = X−F+(V ). Moreover, we have Int⋆(Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(Y − V ))))) = Int⋆(Cl⋆(F−(Y − σ1σ2-Int(σ1σ2-Cl(V ))))) M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6571 7 of 15 = Int⋆(Cl⋆(X − F+((σ1, σ2)-sCl(V )))) = X − Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V )))). Thus, F+(V ) ⊆ Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V )))). (6) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that F (x) ⊆ V . By (6), we have x ∈ F+(V ) ⊆ Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V )))) and by Lemma 3, x ∈ F+(V ) ⊆ sInt⋆(F+((σ1, σ2)-sCl(V ))). Put U = sInt⋆(F+((σ1, σ2)-sCl(V ))). Then, U is a semi-I ⋆-open set of X containing x such that F (U) ⊆ (σ1, σ2)-sCl(V ). This shows that F is upper almost quasi τ⋆(σ1, σ2)- continuous. Theorem 4. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost quasi τ⋆(σ1, σ2)-continuous; (2) for each x ∈ X and every σ1σ2-open set V of Y such that F (x)∩V ̸= ∅, there exists a semi-I ⋆-open set U of X containing x such that (σ1, σ2)-sCl(V ) ∩ F (z) ̸= ∅ for every z ∈ U ; (3) F−(V ) is semi-I ⋆-open in X for every (σ1, σ2)r-open set V of Y ; (4) F−(V ) ⊆ sInt⋆(F−((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y ; (5) sCl⋆(F+(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (6) F−(V ) ⊆ Cl⋆(Int⋆(F−((σ1, σ2)-sCl(V )))) for every σ1σ2-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 almost quasi τ⋆(σ1, σ2)-continuous; (2) sCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) sCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ; (4) F+(V ) ⊆ sInt⋆(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 3. Theorem 6. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6571 8 of 15 (1) F is lower almost quasi τ⋆(σ1, σ2)-continuous; (2) sCl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) sCl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ; (4) F−(V ) ⊆ sInt⋆(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 5. 4. Upper and lower weakly quasi τ ⋆(σ1, σ2)-continuous multifunctions In this section, we introduce the concepts of upper weakly quasi τ⋆(σ1, σ2)-continuous multifunctions and lower weakly quasi τ⋆(σ1, σ2)-continuous multifunctions. Moreover, several characterizations of upper weakly quasi τ⋆(σ1, σ2)-continuous multifunctions and lower weakly quasi τ⋆(σ1, σ2)-continuous multifunctions are discussed. Definition 3. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper weakly quasi τ⋆(σ1, σ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ⊆ V and each ⋆-open set U of X containing x, there exists a nonempty ⋆-open set G such that G ⊆ U and F (G) ⊆ σ1σ2-Cl(V ). A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper weakly quasi τ⋆(σ1, σ2)-continuous if F is upper weakly quasi τ⋆(σ1, σ2)-continuous at each point x of X. Remark 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following implication holds: upper almost quasi τ⋆(σ1, σ2)-continuity ⇒ upper weakly quasi 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 = {∅, {1}, {2, 3}, X} and an ideal I = {∅, {1}}. Let Y = {a, b, c} with topologies σ1 = {∅, {a}, {b}, {a, b}, {a, c}, 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) = {b, c}. Then, F is upper weakly quasi τ⋆(σ1, σ2)- continuous but F is not upper almost quasi τ⋆(σ1, σ2)-continuous. Theorem 7. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly quasi τ⋆(σ1, σ2)-continuous; (2) for each x ∈ X and every σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a semi-I ⋆-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ); (3) Int⋆(Cl⋆(F−(σ1σ2-Int(K)))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6571 9 of 15 (4) F+(V ) ⊆ sInt⋆(F+(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (5) sCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y . Proof. (1) ⇒ (2): Let U (x) be the family of all ⋆-open sets of X containing x. Let V be any σ1σ2-open set of Y such that F (x) ⊆ V . For each H ∈ U (x), there exists a nonempty ⋆-open set GH such that GH ⊆ H and F (GH) ⊆ σ1σ2-Cl(V ). Let W = ∪{GH | H ∈ U (x)}. Then, W is ⋆-open in X, x ∈ Cl⋆(W ) and F (W ) ⊆ σ1σ2-Cl(V ). Put U = W ∪{x}, then W ⊆ U ⊆ Cl⋆(W ). Thus, U is a semi-I ⋆-open set of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). (2) ⇒ (4): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V and there exists a semi-I ⋆-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). Thus, x ∈ U ⊆ sInt⋆(F+(σ1σ2-Cl(V ))) and so F+(V ) ⊆ sInt⋆(F+(σ1σ2-Cl(V ))). (4) ⇒ (5): Let V be any σ1σ2-open set of Y . Then by (4), we have X − F−(σ1σ2-Cl(V )) = F+(Y − σ1σ2-Cl(V )) ⊆ sInt⋆(F+(σ1σ2-Cl(Y − σ1σ2-Cl(V )))) = sInt⋆(F+(Y − σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ sInt⋆(F+(Y − V )) = sInt⋆(X − F−(V )) = X − sCl⋆(F−(V )) and hence sCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )). (5) ⇒ (3): Let K be any σ1σ2-closed set of Y . By (5) and Lemma 3, we have Int⋆(Cl⋆(F−(σ1σ2-Int(K)))) ⊆ sCl⋆(F−(σ1σ2-Int(K))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ F−(σ1σ2-Cl(K)) = F−(K). (3) ⇒ (4): Let V be any σ1σ2-open set of Y . By (3) and Lemma 3, X − sInt⋆(F+(σ1σ2-Cl(V ))) = sCl⋆(F−(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 so F+(V ) ⊆ sInt⋆(F+(σ1σ2-Cl(V ))). (4) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that F (x) ⊆ V . Thus by (4), we have F+(V ) ⊆ sInt⋆(F+(σ1σ2-Cl(V ))). Put U = sInt⋆(F+(σ1σ2-Cl(V ))), then U is a semi-I ⋆-open set of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). This shows that F is upper weakly quasi τ⋆(σ1, σ2)-continuous. M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6571 10 of 15 Definition 4. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower weakly quasi τ⋆(σ1, σ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅ and each ⋆-open set U of X containing x, there exists a nonempty ⋆-open set G such that G ⊆ U and σ1σ2-Cl(V ) ∩ F (z) ̸= ∅ for each z ∈ G. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower weakly quasi τ⋆(σ1, σ2)-continuous if F is lower weakly quasi τ⋆(σ1, σ2)-continuous at each point x of X. Theorem 8. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly quasi τ⋆(σ1, σ2)-continuous; (2) for each x ∈ X and every σ1σ2-open set V of Y such that F (x)∩V ̸= ∅, there exists a semi-I ⋆-open set U of X containing x such that σ1σ2-Cl(V )∩F (z) ̸= ∅ for every z ∈ U ; (3) Int⋆(Cl⋆(F+(σ1σ2-Int(K)))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (4) F−(V ) ⊆ sInt⋆(F−(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (5) sCl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y . Proof. The proof is similar to that of Theorem 7. Theorem 9. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly quasi τ⋆(σ1, σ2)-continuous; (2) sCl⋆(F−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) sCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (4) sCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (5) sCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (6) sCl⋆(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 . Since (σ1, σ2)θ-Cl(B) is σ1σ2-closed in Y , by Theorem 7 we have Int⋆(Cl⋆(F−(σ1σ2-Int((σ1, σ2)θ-Cl(B))))) ⊆ F−((σ1, σ2)θ-Cl(B)) and by Lemma 3, sCl⋆(F−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)). (2) ⇒ (3): This is obvious since σ1σ2-Cl(B) ⊆ (σ1, σ2)θ-Cl(B) for every subset B of Y . (3) ⇒ (4): This is obvious since σ1σ2-Cl(V ) = (σ1, σ2)θ-Cl(V ) for every σ1σ2-open set V of Y . M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6571 11 of 15 (4) ⇒ (5): Let V be any (σ1, σ2)p-open set of Y . Then, V ⊆ σ1σ2-Int(σ1σ2-Cl(V )) and σ1σ2-Cl(V ) = σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))). Now, put G = σ1σ2-Int(σ1σ2-Cl(V )), then G is σ1σ2-open in Y and σ1σ2-Cl(G) = σ1σ2-Cl(V ). Thus by (4), sCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). (5) ⇒ (6): Let K be any (σ1, σ2)r-closed set of Y . Since σ1σ2-Int(K) is (σ1, σ2)p-open in Y , by (5) we have sCl⋆(F−(σ1σ2-Int(K))) = sCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(K))) = F−(K). (6) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, σ1σ2-Cl(V ) is (σ1, σ2)r-closed in Y and by (6), sCl⋆(F−(V )) ⊆ sCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). It follows from Theorem 7 that F is upper weakly quasi τ⋆(σ1, σ2)-continuous. Theorem 10. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly quasi τ⋆(σ1, σ2)-continuous; (2) sCl⋆(F+(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) sCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (4) sCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (5) sCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (6) sCl⋆(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 9. Theorem 11. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly quasi τ⋆(σ1, σ2)-continuous; (2) sCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) sCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ; (4) sCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y . M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6571 12 of 15 Proof. (1) ⇒ (2): Let V be any (σ1, σ2)β-open set of Y . Then, we have V ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))) and hence σ1σ2-Cl(V ) = σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))). Since σ1σ2-Cl(V ) is (σ1, σ2)r- closed in Y and by Theorem 9, sCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). (2) ⇒ (3): This is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open. (3) ⇒ (4): For any (σ1, σ2)p-open set V of Y , σ1σ2-Cl(V ) is (σ1, σ2)r-closed and σ1σ2-Cl(V ) is (σ1, σ2)s-open in Y . (4) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, V is (σ1, σ2)p-open in Y . By (4), we have sCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). It follows from Theorem 9 that F is upper weakly quasi τ⋆(σ1, σ2)-continuous. Theorem 12. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly quasi τ⋆(σ1, σ2)-continuous; (2) sCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) sCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ; (4) sCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y . Proof. The proof is similar to that of Theorem 11. Theorem 13. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly quasi τ⋆(σ1, σ2)-continuous; (2) Int⋆(Cl⋆(F−(V ))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) sCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (4) F+(V ) ⊆ sInt⋆(F+(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y . Proof. (1) ⇒ (2): Let V be any (σ1, σ2)p-open set of Y . Since F is upper weakly quasi τ⋆(σ1, σ2)-continuous, by Lemma 3 and Theorem 9, Int⋆(Cl⋆(F−(V ))) ⊆ Int⋆(Cl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V ))))) ⊆ F−(σ1σ2-Cl(V )). (2) ⇒ (3): Let V be any (σ1, σ2)p-open set of Y . By (2) and Lemma 3, we have sCl⋆(F−(V )) = F−(V ) ∪ Int⋆(Cl⋆(F−(V ))) ⊆ F−(σ1σ2-Cl(V )). M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6571 13 of 15 (3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . Then by (3), we have X − sInt⋆(F+(σ1σ2-Cl(V ))) = sCl⋆(X − F+(σ1σ2-Cl(V ))) = sCl⋆(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 ) ⊆ sInt⋆(F+(σ1σ2-Cl(V ))). (4) ⇒ (1): Since every σ1σ2-open set is (σ1, σ2)p-open, this follows from Theorem 7. Theorem 14. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly quasi τ⋆(σ1, σ2)-continuous; (2) Int⋆(Cl⋆(F+(V ))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) sCl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (4) F−(V ) ⊆ sInt⋆(F−(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y . Proof. The proof is similar to that of Theorem 13. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] S. Marcus. Sur les fonctions quasicontinues au sense de S. Kempisty. Colloquium Mathematicum, 8:47–53, 1961. [2] V. Popa. 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