EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7047 ISSN 1307-5543 – ejpam.com Published by New York Business Global Weakly τ ⋆α(σ1, σ2)-Continuous Multifunctions Nongluk 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 classes of continuous multifunctions defined between an ideal topological space and a bitopological space, called upper weakly τ⋆α(σ1, σ2)-continuous multifunc- tions and lower weakly τ⋆α(σ1, σ2)-continuous multifunctions. Furthermore, several characteri- zations and some properties concerning upper weakly τ⋆α(σ1, σ2)-continuous multifunctions and lower weakly τ⋆α(σ1, σ2)-continuous multifunctions are considered. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper weakly τ⋆α(σ1, σ2)-continuous multifunction, lower weakly τ⋆α(σ1, σ2)-continuous multifunction 1. Introduction The notion of weakly α-continuous functions was first introduced by Noiri [1]. Sen and Bhattacharyya [2] investigated several characterizations of weakly α-continuous func- tions. In 2002, Popa and Noiri [3] extended the concept of α-continuous functions to multifunctions and presented two classes of multifunctions defined between topological spaces, namely upper weakly α-continuous multifunctions and lower weakly α-continuous multifunctions. Furthermore, Popa and Noiri [3] investigated several characterizations and some properties of upper weakly α-continuous multifunctions and lower weakly α- continuous multifunctions. On the other hand, the present author introduced and studied four classes of multifunctions defined from an ideal topological space into an ideal topolog- ical space, called upper weakly ⋆-continuous multifunctions [4], lower weakly ⋆-continuous multifunctions [4], upper weakly α(⋆)-continuous multifunctions [5], lower weakly α(⋆)- continuous multifunctions [5], upper weakly sβ(⋆)-continuous multifunctions [6], lower weakly sβ(⋆)-continuous multifunctions [6], weakly ı⋆-continuous multifunctions [7] and weakly pı-continuous multifunctions [8]. Pue-on et al. [9] introduced and investigated two ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7047 Email addresses: nongluk.h@msu.ac.th (N. 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) N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7047 2 of 11 classes of continuous multifunctions between bitopological spaces, namely upper (τ1, τ2)- continuous multifunctions and lower (τ1, τ2)-continuous multifunctions. Thongmoon et al. [10] introduced and studied the notions of upper weakly (τ1, τ2)-continuous multifunc- tions and lower weakly (τ1, τ2)-continuous multifunctions. In [11], the present authors introduced and investigated the concepts of upper weakly (τ1, τ2)α-continuous multifunc- tions and lower weakly (τ1, τ2)α-continuous multifunctions. Quite recently, Pue-on et al. [12] presented new classes of continuous multifunctions defined from an ideal topological space into a bitopological space, namely upper almost τ⋆(σ1, σ2)-continuous multifunc- tions and lower almost τ⋆(σ1, σ2)-continuous multifunctions. In this paper, we introduce the concepts of continuous multifunctions between an ideal topological space and a bitopo- logical space, called upper weakly τ⋆α(σ1, σ2)-continuous multifunctions and lower weakly τ⋆α(σ1, σ2)-continuous multifunctions. We also investigate several characterizations of up- per weakly τ⋆α(σ1, σ2)-continuous multifunctions and lower weakly τ⋆α(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 [13] 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 [13] 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 [13] of A and is denoted by τ1τ2-Int(A). Lemma 1. [13] 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). A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [11] (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 τ1τ2-δ-open [15] if A is the union N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7047 3 of 11 of (τ1, τ2)r-open sets of X. The complement of a τ1τ2-δ-open set is called τ1τ2-δ-closed [15]. The union of all τ1τ2-δ-open sets of X contained in A is called the τ1τ2-δ-interior [15] 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 [15] 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 [11] 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 [11] 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 [11] 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 [11] of A and is denoted by (τ1, τ2)θ-Int(A). Lemma 2. [11] 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 [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 [17] 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 R-I ⋆-open [4] (resp. I ⋆-preopen [4], semi-I ⋆-open [18], semi-I ⋆-preopen [18]) if A = Int⋆(Cl⋆(A)) (resp. A ⊆ Int⋆(Cl⋆(A)), A ⊆ Cl⋆(Int⋆(A)), A ⊆ Cl⋆(Int⋆(Cl⋆(A)))). The complement of a R-I ⋆-open (resp. I ⋆-preopen, semi-I ⋆-open, semi-I ⋆-preopen) set is said to be R-I ⋆-closed (resp. I ⋆-preclosed, semi-I ⋆-closed, semi-I ⋆-preclosed). 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 [18] of A and is denoted by sCl⋆(A) (sClI ⋆(A) [18]). The union of all semi-I ⋆-open sets contained in A is called the semi-I ⋆-interior [18] of A and is denoted by sInt⋆(A) (sIntI ⋆(A) [18]). Lemma 3. [18] For a subset A of 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)). N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7047 4 of 11 A subset A of an ideal topological space (X, τ,I ) is called τ⋆-α-open [19] (α-I ⋆-open [20]) if A ⊆ Int⋆(Cl⋆(Int⋆(A))). The complement of a τ⋆-α-open set is called τ⋆-α-closed. Lemma 4. [20] For a subset A of an ideal topological space (X, τ,I ), the following properties are equivalent: (1) A is α-I ⋆-open in X. (2) G ⊆ A ⊆ Int⋆(Cl⋆(G)) for some ⋆-open set G. (3) G ⊆ A ⊆ sCl⋆(G) for some ⋆-open set G. (4) A ⊆ sCl⋆(Int⋆(A)). For a subset A of an ideal topological space (X, τ,I ), the intersection of all α-I ⋆- closed sets containing A is called the α-I ⋆-closure [20] of A and is denoted by αCl⋆(A) (αClI ⋆(A) [20]). The α-I ⋆-interior [20] of A is defined by the union of all α-I ⋆-open sets contained in A and is denoted by αInt⋆(A) (αIntI ⋆(A) [20]). Lemma 5. [20] For a subset A of an ideal topological space (X, τ,I ), the following properties hold: (1) A is α-I ⋆-closed in X if and only if sInt⋆(Cl⋆(A)) ⊆ A. (2) sInt⋆(Cl⋆(A)) = Cl⋆(Int⋆(Cl⋆(A))). (3) αCl⋆(A) = A ∪ Cl⋆(Int⋆(Cl⋆(A))). (4) αInt⋆(A) = A ∩ Int⋆(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 weakly τ ⋆α(σ1, σ2)-continuous multifunctions In this section, we introduce the notions of upper weakly τ⋆α(σ1, σ2)-continuous mul- tifunctions and lower weakly τ⋆α(σ1, σ2)-continuous multifunctions. Moreover, several characterizations of upper weakly τ⋆α(σ1, σ2)-continuous multifunctions and lower weakly τ⋆α(σ1, σ2)-continuous multifunctions discussed. Definition 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper weakly τ⋆α(σ1, σ2)-continuous at a point x of X if for each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a τ⋆-α-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⋆α(σ1, σ2)-continuous if F is upper weakly τ⋆α(σ1, σ2)-continuous at each point of X. N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7047 5 of 11 Theorem 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly τ⋆α(σ1, σ2)-continuous at x ∈ X; (2) x ∈ αInt⋆(F+(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y containing F (x); (3) x ∈ Int⋆(Cl⋆(Int⋆(F+(σ1σ2-Cl(V ))))) for every σ1σ2-open set V of Y containing F (x). Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing F (x). Then, there exists a τ⋆-α-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ); hence U ⊆ F+(σ1σ2-Cl(V )). Thus, x ∈ αInt⋆(F+(σ1σ2-Cl(V ))). (2) ⇒ (3): Let V be any σ1σ2-open set of Y containing F (x). Thus by (2), we have x ∈ αInt⋆(F+(σ1σ2-Cl(V ))) and by Lemma 5, x ∈ Int⋆(Cl⋆(Int⋆(F+(σ1σ2-Cl(V ))))). (3) ⇒ (1): Let V be any σ1σ2-open set of Y containing F (x). By (3), we have x ∈ Int⋆(Cl⋆(Int⋆(F+(σ1σ2-Cl(V ))))) and by Lemma 5, x ∈ αInt⋆(F+(σ1σ2-Cl(V ))). Therefore, there exists a τ⋆-α-open set U of X containing x such that U ⊆ F+(σ1σ2-Cl(V )); hence F (U) ⊆ σ1σ2-Cl(V ). This shows that F is upper weakly τ⋆α(σ1, σ2)-continuous at x. Definition 2. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is called lower weakly τ⋆α(σ1, σ2)- continuous at a point x of X if for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a τ⋆-α-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 called lower weakly τ⋆α(σ1, σ2)- continuous if F is lower weakly τ⋆α(σ1, σ2)-continuous at each point 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⋆α(σ1, σ2)-continuous at x ∈ X; (2) x ∈ αInt⋆(F−(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y such that F (x)∩V ̸= ∅; (3) x ∈ Int⋆(Cl⋆(Int⋆(F−(σ1σ2-Cl(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 weakly τ⋆α(σ1, σ2)-continuous; (2) F+(V ) ⊆ Int⋆(Cl⋆(Int⋆(F+(σ1σ2-Cl(V ))))) for every σ1σ2-open set V of Y ; N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7047 6 of 11 (3) Cl⋆(Int⋆(Cl⋆(F−(σ1σ2-Int(K))))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (4) αCl⋆(F−(σ1σ2-Int(K))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (5) αCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (6) F+(σ1σ2-Int(B)) ⊆ αInt⋆(F+(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ; (7) F+(V ) ⊆ αInt⋆(F+(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (8) αCl⋆(F−(σ1σ2-Int(K))) ⊆ F−(K) for every (σ1, σ2)r-closed set K of Y ; (9) αCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (10) αCl⋆(F−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(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 there exists a τ⋆-α-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ); hence U ⊆ F+(σ1σ2-Cl(V )) and so x ∈ U ⊆ Int⋆(Cl⋆(Int⋆(F+(σ1σ2-Cl(V ))))). This shows that F+(V ) ⊆ Int⋆(Cl⋆(Int⋆(F+(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), we have X − F−(K) = F+(Y −K) ⊆ Int⋆(Cl⋆(Int⋆(F+(σ1σ2-Cl(Y −K))))) = Int⋆(Cl⋆(Int⋆(F+(Y − σ1σ2-Int(K))))) = Int⋆(Cl⋆(Int⋆(X − F−(σ1σ2-Int(K))))) = Int⋆(Cl⋆(X − Cl⋆(F−(σ1σ2-Int(K))))) = Int⋆(X − Int⋆(Cl⋆(F−(σ1σ2-Int(K))))) = X − Cl⋆(Int⋆(Cl⋆(F−(σ1σ2-Int(K))))) and hence Cl⋆(Int⋆(Cl⋆(F−(σ1σ2-Int(K))))) ⊆ F−(K). (3) ⇒ (4): Let K be any σ1σ2-closed set of Y . By (3), we have Cl⋆(Int⋆(Cl⋆(F−(σ1σ2-Int(K))))) ⊆ F−(K) and hence αCl⋆(F−(σ1σ2-Int(K))) ⊆ F−(K) by Lemma 5. (4) ⇒ (5): Let B be any subset of Y . Then, σ1σ2-Cl(B) is σ1σ2-closed in Y and by (4), we have αCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−(σ1σ2-Cl(B)). (5) ⇒ (6): Let B be any subset of Y . By (5), F+(σ1σ2-Int(B)) = X − F−(σ1σ2-Cl(Y −B)) ⊆ X − αCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(Y −B)))) = X − αCl⋆(F−(Y − σ1σ2-Cl(σ1σ2-Int(B)))) = X − αCl⋆(X − F+(σ1σ2-Cl(σ1σ2-Int(B)))) N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7047 7 of 11 = αInt⋆(F+(σ1σ2-Cl(σ1σ2-Int(B)))). (6) ⇒ (7): The proof is obvious. (7) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing F (x). It follows from Lemma 5 that x ∈ F+(V ) ⊆ αInt⋆(F+(σ1σ2-Cl(V ))) ⊆ Int⋆(Cl⋆(Int⋆(F+(σ1σ2-Cl(V ))))) and hence F is upper weakly τ⋆α(σ1, σ2)-continuous at x by Theorem 1. This shows that F is upper weakly τ⋆α(σ1, σ2)-continuous. (4) ⇒ (8): The proof is obvious. (8) ⇒ (9): Let V be any σ1σ2-open set of Y . Then, we have σ1σ2-Cl(V ) is (σ1, σ2)r- closed in Y and by (8), αCl⋆(F−(V )) ⊆ αCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). (9) ⇒ (7): Let V be any σ1σ2-open set of Y . Thus by (9), we have X − αInt⋆(F+(σ1σ2-Cl(V ))) = αCl⋆(X − F+(σ1σ2-Cl(V ))) = αCl⋆(F−(Y − σ1σ2-Cl(V ))) ⊆ F−(σ1σ2-Cl(Y − σ1σ2-Cl(V ))) = X − F+(σ1σ2-Int(σ1σ2-Cl(V ))) and hence F+(V ) ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))) ⊆ αInt⋆(F+(σ1σ2-Cl(V ))). (9) ⇒ (10): Let B be any subset of Y . Then, σ1σ2-Int((σ1, σ2)θ-Cl(B)) is σ1σ2-open in Y . By (9) and Lemma 2, αCl⋆(F−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)). (10) ⇒ (8): Let K be any (σ1, σ2)r-closed set of Y . Then by (10) and Lemma 2, we have αCl⋆(F−(σ1σ2-Int(K))) = αCl⋆(F−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))))) = αCl⋆(σ1σ2-Int((σ1, σ2)θ-Cl(σ1σ2-Int(K)))) ⊆ F−((σ1, σ2)θ-Cl(σ1σ2-Int(K))) = F−(σ1σ2-Cl(σ1σ2-Int(K))) = F−(K). Definition 3. [21] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper al- most τ⋆α(σ1, σ2)-continuous at a point x of X if for each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a τ⋆-α-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper almost τ⋆α(σ1, σ2)-continuous if F is upper almost τ⋆α(σ1, σ2)-continuous at each point of X. N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7047 8 of 11 Definition 4. [21] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower almost τ⋆α(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 τ⋆-α-open set U of X containing x such that F (z) ∩ σ1σ2-Int(σ1σ2-Cl(V )) ̸= ∅ for every z ∈ U . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower almost τ⋆α(σ1, σ2)-continuous if F is lower almost τ⋆α(σ1, σ2)-continuous at each point of X. Remark 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following implication holds: upper almost τ⋆α(σ1, σ2)-continuity ⇒ upper weakly τ⋆α(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 = {∅, X} and an ideal I = {∅}. Let Y = {a, b, c} with topologies σ1 = {∅, {a}, {b}, {a, b}, Y } and σ2 = {∅, {a}, {b}, {a, b}, {a, c}, Y }. Define a multifunction F : (X, τ,I ) → (Y, σ1, σ2) as follows: F (1) = {a}, F (2) = {b} and F (3) = {a, c}. Then F is upper weakly τ⋆α(σ1, σ2)-continuous but F is not upper almost τ⋆α(σ1, σ2)-continuous. Theorem 4. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly τ⋆α(σ1, σ2)-continuous; (2) F−(V ) ⊆ Int⋆(Cl⋆(Int⋆(F−(σ1σ2-Cl(V ))))) for every σ1σ2-open set V of Y ; (3) Cl⋆(Int⋆(Cl⋆(F+(σ1σ2-Int(K))))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (4) αCl⋆(F+(σ1σ2-Int(K))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (5) αCl⋆(F+(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (6) F−(σ1σ2-Int(B)) ⊆ αInt⋆(F−(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ; (7) F−(V ) ⊆ αInt⋆(F−(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (8) αCl⋆(F+(σ1σ2-Int(K))) ⊆ F+(K) for every (σ1, σ2)r-closed set K of Y ; (9) αCl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (10) αCl⋆(F+(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y . Proof. The proof is similar to that of Theorem 3. N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7047 9 of 11 Definition 5. A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be weakly τ⋆α(σ1, σ2)- continuous if for each x ∈ X and each σ1σ2-open set V of Y containing f(x), there exists a τ⋆-α-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). Corollary 1. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly τ⋆α(σ1, σ2)-continuous; (2) f−1(V ) ⊆ αInt⋆(f−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (3) αCl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(K) for every (σ1, σ2)r-closed set K of Y ; (4) αCl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (5) αCl⋆(f−1(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (6) Cl⋆(Int⋆(Cl⋆(f−1(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (7) f−1(V ) ⊆ Int⋆(Cl⋆(Int⋆(f−1(σ1σ2-Cl(V ))))) for every σ1σ2-open set V of Y ; (8) f(Cl⋆(Int⋆(Cl⋆(A)))) ⊆ (σ1, σ2)θ-Cl(f(A)) for every subset A of X; (9) Cl⋆(Int⋆(Cl⋆(f−1(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every subset B of Y . Definition 6. [22] A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be τ⋆α(σ1, σ2)- continuous if for each σ1σ2-open set V of Y , f−1(V ) is τ⋆-α-open in X. Definition 7. [21] A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be almost τ⋆α(σ1, σ2)- continuous if f−1(V ) is τ⋆-α-open in X for every (σ1, σ2)r-open set V of Y . Theorem 5. For a function f : (X, τ,I ) → (Y, σ1, σ2) such that αInt⋆(f−1(σ1σ2-Cl(V ))) ⊆ αInt⋆(f−1(V )) for every σ1σ2-open set V of Y , the following properties are equivalent: (1) f is τ⋆α(σ1, σ2)-continuous; (2) f is almost τ⋆α(σ1, σ2)-continuous; (3) f is weakly τ⋆α(σ1, σ2)-continuous. Proof. We prove only the implication (3) ⇒ (1). Suppose that f is weakly τ⋆α(σ1, σ2)- continuous. Let V be any σ1σ2-open set of Y . Since f is weakly τ⋆α(σ1, σ2)-continuous, by Corollary 1, we have f−1(V ) ⊆ αInt⋆(f−1(σ1σ2-Cl(V ))) and hence f−1(V ) ⊆ αInt⋆(f−1(σ1σ2-Cl(V ))) ⊆ αInt⋆(f−1(V )). Thus, f−1(V ) is τ⋆-α-open in X. This shows that f is τ⋆α(σ1, σ2)-continuous. N. Viriyapong, A. 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