EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7045 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Upper and Lower Almost τ ⋆α(σ1, σ2)-Continuous Multifunctions 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 introduces new classes of continuous multifunctions defined between an ideal topological space and a bitopological space, called upper almost τ⋆α(σ1, σ2)-continuous mul- tifunctions and lower almost τ⋆α(σ1, σ2)-continuous multifunctions. Furthermore, several char- acterizations and some properties concerning upper almost τ⋆α(σ1, σ2)-continuous multifunctions and lower almost τ⋆α(σ1, σ2)-continuous multifunctions are investigated. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper almost τ⋆α(σ1, σ2)-continuous multifunction, lower almost τ⋆α(σ1, σ2)-continuous multifunction 1. Introduction In 1988, Noiri [1] introduced a class of functions between topological spaces, called al- most α-continuous functions. Furthermore, Noiri [1] investigated several characterizations and some basic properties of almost α-continuous functions. In 1996, Popa and Noiri [2] extended the concept of almost α-continuous functions to multifunctions and presented classes of multifunctions defined from a topological space into a topological space, namely upper almost α-continuous multifunctions and lower almost α-continuous multifunctions. In particular, several characterizations and some properties concerning upper almost α- continuous multifunctions and lower almost α-continuous multifunctions were established in [2]. On the other hand, the present author introduced and studied four classes of mul- tifunctions defined from an ideal topological space into an ideal topological space, called upper almost ⋆-continuous multifunctions [3], lower almost ⋆-continuous multifunctions [3], upper almost α(⋆)-continuous multifunctions [4], lower almost α(⋆)-continuous multifunc- tions [4], upper almost α-⋆-continuous multifunctions [5], lower almost α-⋆-continuous ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7045 Email addresses: nongluk.h@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 (4) (2025), 7045 2 of 12 multifunctions [5] and almost ı⋆-continuous multifunctions [6]. Pue-on et al. [7] intro- duced and studied two classes of multifunctions between bitopological spaces, namely up- per (τ1, τ2)-continuous multifunctions and lower (τ1, τ2)-continuous multifunctions. More- over, Boonpok and Pue-on [8] introduced and investigated the concepts of upper almost (τ1, τ2)-continuous multifunctions and lower almost (τ1, τ2)-continuous multifunctions. In [9], the present authors introduced and studied the concepts of upper almost (τ1, τ2)α- continuous multifunctions and lower almost (τ1, τ2)α-continuous multifunctions. Quite recently, Viriyapong et al. [10] presented new classes of continuous multifunctions defined from an ideal topological space into a bitopological space, namely upper almost τ⋆(σ1, σ2)- continuous multifunctions and lower almost τ⋆(σ1, σ2)-continuous multifunctions. In this paper, we introduce the concepts of continuous multifunctions between an ideal topologi- cal space and a bitopological space, called upper almost τ⋆α(σ1, σ2)-continuous multifunc- tions and lower almost τ⋆α(σ1, σ2)-continuous multifunctions. We also investigate several characterizations of upper almost τ⋆α(σ1, σ2)-continuous multifunctions and lower almost 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 [11] 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 [11] 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 [11] of A and is denoted by τ1τ2-Int(A). Lemma 1. [11] 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 [9] (resp. (τ1, τ2)s-open [12], (τ1, τ2)p-open [12], (τ1, τ2)β-open [12]) 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). The C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7045 3 of 12 intersection of all (τ1, τ2)s-closed sets of X containing A is called the (τ1, τ2)s-closure [12] of A and is denoted by (τ1, τ2)-sCl(A). The union of all (τ1, τ2)s-open sets of X contained in A is called the (τ1, τ2)s-interior [12] of A and is denoted by (τ1, τ2)-sInt(A). Lemma 2. 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 [12]; (2) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A [13]. Lemma 3. [14] Let A be a subset of a bitopological space (X, τ1, τ2). If A is τ1τ2-open in X, then (τ1, τ2)-sCl(A) = τ1τ2-Int(τ1τ2-Cl(A)). A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-δ-open [8] if A is the union of (τ1, τ2)r-open sets of X. The complement of a τ1τ2-δ-open set is called τ1τ2-δ- closed [8]. The union of all τ1τ2-δ-open sets ofX contained in A is called the τ1τ2-δ-interior [8] 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 [8] 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 [9] 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 [9] 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 [9] 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 [9] of A and is denoted by (τ1, τ2)θ-Int(A). Lemma 4. [9] 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 [15], 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 [16] 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 [3] C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7045 4 of 12 (resp. I ⋆-preopen [3], semi-I ⋆-open [17], semi-I ⋆-preopen [17]) 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 [17] of A and is denoted by sCl⋆(A) (sClI ⋆(A) [17]). The union of all semi-I ⋆-open sets contained in A is called the semi-I ⋆-interior [17] of A and is denoted by sInt⋆(A) (sIntI ⋆(A) [17]). Lemma 5. [17] For a subset A of an ideal topological space (X, τ,I ), the following prop- erties hold: (1) sCl⋆(A) = A ∪ Int⋆(Cl⋆(A)). (2) sInt⋆(A) = A ∩ Cl⋆(Int⋆(A)). A subset A of an ideal topological space (X, τ,I ) is said to be τ⋆-α-open [18] (α-I ⋆- open [19]) if A ⊆ Int⋆(Cl⋆(Int⋆(A))). The complement of an τ⋆-α-open set is said to be τ⋆-α-closed. Lemma 6. [19] For a subset A of an ideal topological space (X, τ,I ), the following prop- erties 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 [19] of A and is denoted by αCl⋆(A) (αClI ⋆(A) [19]). The α-I ⋆-interior [19] of A is defined by the union of all α-I ⋆-open sets contained in A and is denoted by αInt⋆(A) (αIntI ⋆(A) [19]). Lemma 7. [19] For a subset A of an ideal topological space (X, τ,I ), the following prop- erties 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). C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7045 5 of 12 3. Upper and lower almost τ ⋆α(σ1, σ2)-continuous multifunctions In this section, we introduce the notions of upper almost τ⋆α(σ1, σ2)-continuous mul- tifunctions and lower almost τ⋆α(σ1, σ2)-continuous multifunctions. Moreover, several characterizations of upper almost τ⋆α(σ1, σ2)-continuous multifunctions and lower almost τ⋆α(σ1, σ2)-continuous multifunctions discussed. Definition 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper almost 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. Theorem 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost τ⋆α(σ1, σ2)-continuous at x ∈ X; (2) for 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) ⊆ (σ1, σ2)-sCl(V ); (3) x ∈ αInt⋆(F+((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y containing F (x); (4) x ∈ Int⋆(Cl⋆(Int⋆(F+((σ1, σ2)-sCl(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 containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )) and by Lemma 3, we have F (U) ⊆ (σ1, σ2)-sCl(V ). (2) ⇒ (3): Let V be any σ1σ2-open set of Y containing F (x). By (2), there ex- ists a τ⋆-α-open set U containing x such that F (U) ⊆ (σ1, σ2)-sCl(V ) and hence U ⊆ F+((σ1, σ2)-sCl(V )). Thus, x ∈ αInt⋆(F+((σ1, σ2)-sCl(V ))). (3) ⇒ (4): Let V be any σ1σ2-open set of Y containing F (x). Then by (3), we have x ∈ αInt⋆(F+((σ1, σ2)-sCl(V ))) and by Lemma 7, x ∈ Int⋆(Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V ))))). (4) ⇒ (1): Let V be any σ1σ2-open set of Y containing F (x). By (4), we have x ∈ Int⋆(Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V ))))) and by Lemma 7, x ∈ αInt⋆(F+((σ1, σ2)-sCl(V ))). Therefore, there exists a τ⋆-α-open set U of X containing x such that U ⊆ F+((σ1, σ2)-sCl(V )); hence F (U) ⊆ (σ1, σ2)-sCl(V ). Since V is σ1σ2-open, by Lemma 3 we have F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). This shows that F is upper almost τ⋆α(σ1, σ2)-continuous at x. C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7045 6 of 12 Definition 2. 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. Theorem 2. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost τ⋆α(σ1, σ2)-continuous at x ∈ X; (2) 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) ∩ (σ1, σ2)-sCl(V ) ̸= ∅; (3) x ∈ αInt⋆(F−((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y such that F (x)∩V ̸= ∅; (4) x ∈ Int⋆(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 τ⋆α(σ1, σ2)-continuous; (2) 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) ⊆ (σ1, σ2)-sCl(V ); (3) for each x ∈ X and each (σ1, σ2)r-open set V of Y containing F (x), there exists a τ⋆-α-open set U of X containing x such that F (U) ⊆ V ; (4) F+(V ) is τ⋆-α-open in X for every (σ1, σ2)r-open set V of Y ; (5) F−(K) is τ⋆-α-closed in X for every (σ1, σ2)r-closed set K of Y ; (6) F+(V ) ⊆ αInt⋆(F+((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y ; (7) αCl⋆(F−((σ1, σ2)-sInt(K))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (8) αCl⋆(F−(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (9) αCl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7045 7 of 12 (10) Cl⋆(Int⋆(Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(K)))))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (11) Cl⋆(Int⋆(Cl⋆(F−((σ1, σ2)-sInt(K))))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (12) F+(V ) ⊆ Int⋆(Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V ))))) for every σ1σ2-open set V of Y . Proof. (1) ⇒ (2): The proof follows from Theorem 1. (2) ⇒ (3): The proof is obvious. (3) ⇒ (4): Let V be any (σ1, σ2)r-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V and by (3), there exists a τ⋆-α-open set Ux of X containing x such that F (Ux) ⊆ V . Thus, x ∈ Ux ⊆ F+(V ) and so F+(V ) = ∪x∈F+(V )Ux is τ⋆-α-open in X. (4) ⇒ (5): This follows from the fact that F+(Y −B) = X − F−(B) for every subset B of Y . (5) ⇒ (6): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, we have F (x) ⊆ V ⊆ (σ1, σ2)-sCl(V ) and so x ∈ F+((σ1, σ2)-sCl(V )) = X−F−(Y −(σ1, σ2)-sCl(V )). Since Y −(σ1, σ2)-sCl(V ) is (σ1, σ2)r-closed in Y and by (5), F−(Y −(σ1, σ2)-sCl(V )) is τ⋆-α-closed inX. This shows that F+((σ1, σ2)-sCl(V )) is τ⋆-α-open in X. Thus, x ∈ αInt⋆(F+((σ1, σ2)-sCl(V ))) and hence F+(V ) ⊆ αInt⋆(F+((σ1, σ2)-sCl(V ))). (6) ⇒ (7): Let K be any σ1σ2-closed set of Y . Then, Y −K is σ1σ2-open and by (6), we have X − F−(K) = F+(Y −K) ⊆ αInt⋆(F+((σ1, σ2)-sCl(Y −K))) = αInt⋆(F+(Y − (σ1, σ2)-sInt(K))) = αInt⋆(X − F−((σ1, σ2)-sInt(K))) = X − αCl⋆(F−((σ1, σ2)-sInt(K))) and hence αCl⋆(F−((σ1, σ2)-sInt(K))) ⊆ F−(K). (7) ⇒ (8): The proof is obvious since (σ1, σ2)-sInt(K) = σ1σ2-Cl(σ1σ2-Int(K)) for every σ1σ2-closed set K of Y . (8) ⇒ (9): The proof is obvious. (9) ⇒ (10): It follows from Lemma 7 that Cl⋆(Int⋆(Cl⋆(B))) ⊆ αCl⋆(B) for every subset B of Y . Thus, for every σ1σ2-closed set K of Y , we have Cl⋆(Int⋆(Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(K)))))) ⊆ αCl⋆(F−(σ1σ2-Cl(σ1σ2-Int(K)))) = αCl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(K))))) ⊆ F−(σ1σ2-Cl(K)) = F−(K). (10) ⇒ (11): The proof is obvious since (σ1, σ2)-sInt(K) = σ1σ2-Cl(σ1σ2-Int(K)) for every σ1σ2-closed set K of Y . C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7045 8 of 12 (11) ⇒ (12): Let V be any σ1σ2-open set of Y . Then, Y − V is σ1σ2-closed in Y and by (11), Cl⋆(Int⋆(Cl⋆(F−((σ1, σ2)sInt(Y −V ))))) ⊆ F−(Y −V ) = X −F+(V ). Moreover, we have Cl⋆(Int⋆(Cl⋆(F−((σ1, σ2)-sInt(Y − V ))))) = Cl⋆(Int⋆(Cl⋆(F−(Y − (σ1, σ2)-sCl(V ))))) = Cl⋆(Int⋆(Cl⋆(X − F+((σ1, σ2)-sCl(V ))))) = X − Int⋆(Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V ))))). Thus, F+(V ) ⊆ Int⋆(Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V ))))). (12) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing F (x). By (12), we have x ∈ F+(V ) ⊆ Int⋆(Cl⋆(Int⋆(F+((σ1, σ2)-sCl(V ))))) and hence F is upper almost τ⋆α(σ1, σ2)-continuous at x by Theorem 1. This shows that F is upper almost τ⋆α(σ1, σ2)-continuous. Definition 3. [20] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper 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) ⊆ V . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper τ⋆α(σ1, σ2)-continuous if F is upper τ⋆α(σ1, σ2)-continuous at each point of X. Definition 4. [20] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower 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 (z)∩V ̸= ∅ for every z ∈ U . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower τ⋆α(σ1, σ2)- continuous if F is lower 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 τ⋆α(σ1, σ2)-continuity ⇒ upper almost 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 = {∅, {1, 2}, X} and an ideal I = {∅, {3}}. Let Y = {a, b, c} with topologies σ1 = {∅, {a, b}, Y } and σ2 = {∅, {c}, {a, b}, Y }. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is defined as follows: F (1) = {c} and F (2) = {a} and F (3) = {a, b}. Then F is upper almost τ⋆α(σ1, σ2)-continuous but F is not upper 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 almost τ⋆α(σ1, σ2)-continuous; (2) for each x ∈ X and 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 U ⊆ F−((σ1, σ2)-sCl(V )); C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7045 9 of 12 (3) for each x ∈ X and each (σ1, σ2)r-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a τ⋆-α-open set U of X containing x such that U ⊆ F−(V ); (4) F−(V ) is τ⋆-α-open in X for every (σ1, σ2)r-open set V of Y ; (5) F+(K) is τ⋆-α-closed in X for every (σ1, σ2)r-closed set K of Y ; (6) F−(V ) ⊆ αInt⋆(F−((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y ; (7) αCl⋆(F+((σ1, σ2)-sInt(K))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (8) αCl⋆(F+(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (9) αCl⋆(F+(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (10) Cl⋆(Int⋆(Cl⋆(F+(σ1σ2-Cl(σ1σ2-Int(K)))))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (11) Cl⋆(Int⋆(Cl⋆(F+((σ1, σ2)-sInt(K))))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (12) F−(V ) ⊆ Int⋆(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 τ⋆α(σ1, σ2)-continuous; (2) αCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) αCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ; (4) F+(V ) ⊆ αInt⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) for every (σ1, σ2)p-open set V of Y . Proof. (1) ⇒ (2): Let V be any (σ1, σ2)β-open set of Y . Then, σ1σ2-Cl(V ) is (σ1, σ2)r- closed in Y . Since F is upper almost τ⋆α(σ1, σ2)-continuous, by Theorem 3 we have F−(σ1σ2-Cl(V )) is τ⋆-α-closed in X and hence αCl⋆(F−(V )) ⊆ αCl⋆(F−(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) ⇒ (1): Let K be any (σ1, σ2)r-closed set of Y . Then, K is (σ1, σ2)s-open in Y and by (3), we have αCl⋆(F−(K)) ⊆ F−(σ1σ2-Cl(K)) = F−(K). Thus, F−(K) is τ⋆-α-closed in X and hence F is upper almost τ⋆α(σ1, σ2)-continuous by Theorem 3. (1) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . Then, we have σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r-open in Y . Since F is upper almost τ⋆α(σ1, σ2)-continuous, by Theorem 3 we have F+(σ1σ2-Int(σ1σ2-Cl(V ))) is τ⋆-α-open in X. Thus, F+(V ) ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))) = αInt⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))). C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7045 10 of 12 (4) ⇒ (1): Let V be any (σ1, σ2)r-open set of Y . Then, V is (σ1, σ2)p-open in Y and by (4), F+(V ) ⊆ αInt⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) = αInt⋆(F+(V )). This shows that F+(V ) is τ⋆-α-open in X. It follows from Theorem 3 that F is upper almost τ⋆α(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 τ⋆α(σ1, σ2)-continuous; (2) αCl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) αCl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ; (4) F−(V ) ⊆ αInt⋆(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. Definition 5. 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 . Corollary 1. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost τ⋆α(σ1, σ2)-continuous; (2) 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) ⊆ (σ1, σ2)-sCl(V ); (3) for each x ∈ X and each (σ1, σ2)r-open set V of Y containing f(x), there exists an α-I ⋆-open set U of X containing x such that f(U) ⊆ V ; (4) 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-Int(σ1σ2-Cl(V )); (5) f−1(K) is τ⋆-α-closed in X for every (σ1, σ2)r-closed set K of Y ; (6) f−1(V ) ⊆ αInt⋆(f−1((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y ; (7) αCl⋆(f−1((σ1, σ2)-sInt(K))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; (8) αCl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; (9) αCl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ; (10) Cl⋆(Int⋆(Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(K)))))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7045 11 of 12 (11) Cl⋆(Int⋆(Cl⋆(f−1((σ1, σ2)-sInt(K))))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; (12) f−1(V ) ⊆ Int⋆(Cl⋆(Int⋆(f−1((σ1, σ2)-sCl(V ))))) for every σ1σ2-open set V of Y . Corollary 2. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost τ⋆α(σ1, σ2)-continuous; (2) αCl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) αCl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ; (4) f−1(V ) ⊆ αInt⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) for every (σ1, σ2)p-open set V of Y . 4. Conclusion In this paper, we have introduced new classes of continuous multifunctions defined from an ideal topological space into a bitopological space, namely upper almost τ⋆α(σ1, σ2)- continuous multifunctions and lower almost τ⋆α(σ1, σ2)-continuous multifunctions. 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