EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7049 ISSN 1307-5543 – ejpam.com Published by New York Business Global Almost τ ⋆β(σ1, σ2)-Continuity for Multifunctions Napassanan Srisarakham1, 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⋆β(σ1, σ2)-continuous mul- tifunctions and lower almost τ⋆β(σ1, σ2)-continuous multifunctions. Moreover, several character- izations and some properties concerning upper almost τ⋆β(σ1, σ2)-continuous multifunctions and lower almost τ⋆β(σ1, σ2)-continuous multifunctions are established. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper almost τ⋆β(σ1, σ2)-continuous multifunction, lower almost τ⋆β(σ1, σ2)-continuous multifunction 1. Introduction In 1997, Nasef and Noiri [1] introduced and investigated two classes of functions de- fined between topological spaces, namely almost precontinuous functions and almost β- continuous functions by utilizing the notions of preopen sets and β-open sets due to Mash- hour et al. [2] and Abd El-Monsef et al. [3], respectively. In 1998, Noiri and Popa [4] investigated several characterizations and some properties of almost β-continuous func- tions. Noiri [5] introduced the concept of almost α-continuous functions and proved that the notions of almost feeble continuity [6] and almost α-continuity are equivalent. The class of almost precontinuity is a generalization of almost α-continuity and almost feeble continuity. The class of almost β-continuity is a generalization of almost quasi-continuity [7]. In 1999, Noiri and Popa [8] extended the concept of almost β-continuous functions to multifunctions and introduced new classes of multifunctions defined between topological spaces, namely upper almost β-continuous multifunctions and lower almost β-continuous multifunctions. Furthermore, Noiri and Popa [8] investigated several characterizations and some properties concerning upper almost β-continuous multifunctions and lower almost ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7049 Email addresses: napassanan.sri@msu.ac.th (N. Srisarakham), 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. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7049 2 of 11 β-continuous multifunctions. On the other hand, the present author introduced and inves- tigated classes of continuous multifunctions defined from an ideal topological space into an ideal topological space, namely upper almost ⋆-continuous multifunctions [9], lower almost ⋆-continuous multifunctions [9], upper almost α-⋆-continuous multifunctions [10], lower al- most α-⋆-continuous multifunctions [10], upper almost β(⋆)-continuous multifunctions [11], lower almost β(⋆)-continuous multifunctions [11], upper almost sβ(⋆)-continuous multi- functions [12], lower almost sβ(⋆)-continuous multifunctions [12] and almost ı⋆-continuous multifunctions [13]. Pue-on et al. [14] introduced and studied two classes of multifunc- tions between bitopological spaces, called upper (τ1, τ2)-continuous multifunctions and lower (τ1, τ2)-continuous multifunctions. Moreover, Boonpok and Pue-on [15] introduced and investigated the concepts of upper almost (τ1, τ2)-continuous multifunctions and lower almost (τ1, τ2)-continuous multifunctions. Laprom et al. [16] introduced and studied the notions of upper almost β(τ1, τ2)-continuous multifunctions and lower almost β(τ1, τ2)- continuous multifunctions. In this paper, we introduce the concepts of continuous multi- functions between an ideal topological space and a bitopological space, called upper almost τ⋆β(σ1, σ2)-continuous multifunctions and lower almost τ⋆β(σ1, σ2)-continuous multifunc- tions. We also investigate several characterizations of upper almost τ⋆β(σ1, σ2)-continuous multifunctions and lower almost τ⋆β(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 [17] 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 [17] 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 [17] of A and is denoted by τ1τ2-Int(A). Lemma 1. [17] 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 called (τ1, τ2)r-open [18] (resp. (τ1, τ2)s- open [19], (τ1, τ2)p-open [19], (τ1, τ2)β-open [19]) if A = τ1τ2-Int(τ1τ2-Cl(A)) (resp. A ⊆ N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7049 3 of 11 τ1τ2-Cl(τ1τ2-Int(A)), A ⊆ τ1τ2-Int(τ1τ2-Cl(A)), A ⊆ τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A)))). The complement of a (τ1, τ2)r-open (resp. (τ1, τ2)s-open, (τ1, τ2)p-open, (τ1, τ2)β-open) set is called (τ1, τ2)r-closed (resp. (τ1, τ2)s-closed, (τ1, τ2)p-closed, (τ1, τ2)β-closed). The inter- section of all (τ1, τ2)s-closed sets of X containing A is called the (τ1, τ2)s-closure [19] 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 [19] 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 [19]; (2) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A [20]. 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 [21], 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 [22] 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 [9] (resp. I ⋆- preopen [9], τ⋆-semi-open [23] (semi-I ⋆-open [15]), τ⋆-β-open [23] (semi-I ⋆-preopen [15])) 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, τ⋆-semi-open, τ⋆-β-open) set is said to be R-I ⋆-closed (resp. I ⋆-preclosed, τ⋆-semi-closed, τ⋆-β-closed). 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 [15] of A and is denoted by sCl⋆(A) (sClI ⋆(A) [15]). The union of all semi-I ⋆-open sets contained in A is called the semi-I ⋆-interior [15] of A and is denoted by sInt⋆(A) (sIntI ⋆(A) [15]). The intersection of all β-I ⋆-closed sets containing A is called the β-I ⋆-closure of A and is denoted by βCl⋆(A). The union of all β-I ⋆-open sets contained in A is called the β-I ⋆-interior of A and is denoted by βInt⋆(A). Lemma 3. For a subset A of an ideal topological space (X, τ,I ), the following properties hold: (1) sCl⋆(A) = A ∪ Int⋆(Cl⋆(A)) [15]. (2) sInt⋆(A) = A ∩ Cl⋆(Int⋆(A)) [15]. (3) βCl⋆(A) = A ∪ Int⋆(Cl⋆(Int⋆(A))). N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7049 4 of 11 (4) βInt⋆(A) = A ∩ Cl⋆(Int⋆(Cl⋆(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 τ ⋆β(σ1, σ2)-continuous multifunctions In this section, we introduce the notions of upper almost τ⋆β(σ1, σ2)-continuous mul- tifunctions and lower almost τ⋆β(σ1, σ2)-continuous multifunctions. Moreover, several characterizations of upper almost τ⋆β(σ1, σ2)-continuous multifunctions and lower almost τ⋆β(σ1, σ2)-continuous multifunctions discussed. Definition 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper almost τ⋆β(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⋆-β-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⋆β(σ1, σ2)- continuous if F is upper almost τ⋆β(σ1, σ2)-continuous at each point of X. Theorem 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is upper almost τ⋆β(σ1, σ2)- continuous at x ∈ X if and only if x ∈ βInt⋆(F+((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y containing F (x). Proof. Let V be any σ1σ2-open set of Y containing F (x). Then, there exists a τ⋆-β- open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )) = (σ1, σ2)-sCl(V ); hence U ⊆ F+((σ1, σ2)-sCl(V )). Since U is τ⋆-β-open, we have x ∈ U ⊆ Cl⋆(Int⋆(Cl⋆(U))) ⊆ Cl⋆(Int⋆(Cl⋆(F+((σ1, σ2)-sCl(V ))))). Since x ∈ F+(V ) ⊆ F+((σ1, σ2)-sCl(V )) and by Lemma 3, x ∈ F+((σ1, σ2)-sCl(V )) ∩ Cl⋆(Int(Cl⋆((σ1, σ2)-sCl(V )))) = βInt⋆(F+((σ1, σ2)-sCl(V ))). Conversely, let V be any σ1σ2-open set of Y containing F (x). Then, we have x ∈ βInt⋆(F+((σ1, σ2)-sCl(V ))) and so there exists a τ⋆-β-open set U of X containing x such that U ⊆ F+((σ1, σ2)-sCl(V )); hence F (U) ⊆ (σ1, σ2)-sCl(V ) = σ1σ2-Int(σ1σ2-Cl(V )). This shows that F is upper almost τ⋆β(σ1, σ2)-continuous at x. N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7049 5 of 11 Definition 2. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower almost τ⋆β(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⋆-β-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⋆β(σ1, σ2)-continuous if F is lower almost τ⋆β(σ1, σ2)-continuous at each point of X. Theorem 2. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is lower almost τ⋆β(σ1, σ2)- continuous at x ∈ X if and only if x ∈ βInt⋆(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. Definition 3. A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be almost τ⋆β(σ1, σ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing f(x), there exists a τ⋆-β-open set U of X containing x such that f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be almost τ⋆β(σ1, σ2)-continuous if f is τ⋆β(σ1, σ2)-continuous at each point of X. Corollary 1. A function f : (X, τ,I ) → (Y, σ1, σ2) is almost τ⋆β(σ1, σ2)-continuous at x ∈ X if and only if x ∈ βInt⋆(f−1((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y containing f(x). Theorem 3. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost τ⋆β(σ1, σ2)-continuous; (2) for each x ∈ X and each σ1σ2-open set V of Y containing F (x), there exists a τ⋆-β-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⋆-β-open set U of X containing x such that F (U) ⊆ V ; (4) F+(V ) is τ⋆-β-open in X for every (σ1, σ2)r-open set V of Y ; (5) F−(K) is τ⋆-β-closed in X for every (σ1, σ2)r-closed set K of Y ; (6) F+(V ) ⊆ βInt⋆(F+((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y ; (7) βCl⋆(F−((σ1, σ2)-sInt(K))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (8) βCl⋆(F−(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (9) βCl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7049 6 of 11 (10) Int⋆(Cl⋆(Int⋆(F−(σ1σ2-Cl(σ1σ2-Int(K)))))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (11) Int⋆(Cl⋆(Int⋆(F−((σ1, σ2)-sInt(K))))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; (12) F+(V ) ⊆ Cl⋆(Int⋆(Cl⋆(F+((σ1, σ2)-sCl(V ))))) for every σ1σ2-open set V of Y . Proof. (1) ⇒ (2) and (2) ⇒ (3): The proofs are obvious. (3) ⇒ (4): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V and so there exists a τ⋆-β-open set Ux of X containing x such that F (Ux) ⊆ V . Thus, x ∈ Ux ⊆ F+(V ) and hence F+(V ) = ∪x∈F+(V )Ux. This shows that F+(V ) is τ⋆-β-open in X. (4) ⇒ (5): This follows from the fact that F+(Y − B) = Y − F−(B) for every subset B of Y . (5) ⇒ (6): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V ⊆ (σ1, σ2)-sCl(V ) and hence x ∈ F+((σ1, σ2)-sCl(V )) = X − F−(Y − (σ1, σ2)-sCl(V )). Since Y − (σ1, σ2)-sCl(V ) is (σ1, σ2)r-closed, we have F−(Y − (σ1, σ2)-sCl(V )) is τ⋆-β- closed in X. Thus, F+((σ1, σ2)-sCl(V )) is a τ⋆-β-open set of X containing x and so x ∈ βInt⋆(F+((σ1, σ2)-sCl(V ))). This shows that F+(V ) ⊆ βInt⋆(F+((σ1, σ2)-sCl(V ))). (6) ⇒ (7): Let K be any σ1σ2-closed set of Y . Then, since Y −K is σ1σ2-open and by (6), X − F−(K) = F+(Y −K) ⊆ βInt⋆(F+((σ1, σ2)-sCl(Y −K))) = βInt⋆(F+(Y − (σ1, σ2)-sInt(K))) = βInt⋆(X − F−((σ1, σ2)-sInt(K))) = X − βCl⋆(F−((σ1, σ2)-sInt(K))). Thus, βCl⋆(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): By (9) and Lemma 3, Int⋆(Cl⋆(Int⋆(F−(σ1σ2-Cl(σ1σ2-Int(K)))))) ⊆ βCl⋆(F−(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ βCl⋆(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 . (11) ⇒ (12): Let V be any σ1σ2-open set of Y . Then, Y − V is σ1σ2-closed in Y and by (11), Int⋆(Cl⋆(Int⋆(F−((σ1, σ2)-sInt(Y − V ))))) ⊆ F−(Y − V ) = X − F+(V ). N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7049 7 of 11 Moreover, we have Int⋆(Cl⋆(Int⋆(F−((σ1, σ2)-sInt(Y − V ))))) = Int⋆(Cl⋆(Int⋆(F−(Y − (σ1, σ2)-sCl(V ))))) = Int⋆(Cl⋆(Int⋆(X − F+((σ1, σ2)-sCl(V ))))) = X − Cl⋆(Int⋆(Cl⋆(F+((σ1, σ2)-sCl(V ))))). Thus, F+(V ) ⊆ Cl⋆(Int⋆(Cl⋆(F+((σ1, σ2)-sCl(V ))))). (12) ⇒ (1): Let x be any point of X and V be any σ1σ2-open set of Y containing F (x). Then, we have x ∈ F+(V ) ⊆ Cl⋆(Int⋆(Cl⋆(F+((σ1, σ2)-sCl(V ))))) and hence x ∈ βInt⋆(F+((σ1, σ2)-sCl(V ))). Thus, F is upper almost τ⋆β(σ1, σ2)-continuous at x by Theorem 1. Theorem 4. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost τ⋆β(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⋆-β-open set U of X containing x such that U ⊆ F−((σ1, σ2)-sCl(V )); (3) for each x ∈ X and each (σ1, σ2)r-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a τ⋆-β-open set U of X containing x such that U ⊆ F−(V ); (4) F−(V ) is τ⋆-β-open in X for every (σ1, σ2)r-open set V of Y ; (5) F+(K) is τ⋆-β-closed in X for every (σ1, σ2)r-closed set K of Y ; (6) F−(V ) ⊆ βInt⋆(F−((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y ; (7) βCl⋆(F+((σ1, σ2)-sInt(K))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (8) βCl⋆(F+(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (9) βCl⋆(F+(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (10) Int⋆(Cl⋆(Int⋆(F+(σ1σ2-Cl(σ1σ2-Int(K)))))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (11) Int⋆(Cl⋆(Int⋆(F+((σ1, σ2)-sInt(K))))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (12) F−(V ) ⊆ Cl⋆(Int⋆(Cl⋆(F−((σ1, σ2)-sCl(V ))))) for every σ1σ2-open set V of Y . Proof. The proof is similar to that of Theorem 3. Corollary 2. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7049 8 of 11 (1) f is almost τ⋆β(σ1, σ2)-continuous; (2) for each x ∈ X and each σ1σ2-open set V of Y containing f(x), there exists a τ⋆-β-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⋆-β-open set U of X containing x such that f(U) ⊆ V ; (4) f−1(V ) is τ⋆-β-open in X for every (σ1, σ2)r-open set V of Y ; (5) f−1(K) is τ⋆-β-closed in X for every (σ1, σ2)r-closed set K of Y ; (6) f−1(V ) ⊆ βInt⋆(f−1((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y ; (7) βCl⋆(f−1((σ1, σ2)-sInt(K))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; (8) βCl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; (9) βCl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ; (10) Int⋆(Cl⋆(Int⋆(f−1(σ1σ2-Cl(σ1σ2-Int(K)))))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; (11) Int⋆(Cl⋆(Int⋆(f−1((σ1, σ2)-sInt(K))))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; (12) f−1(V ) ⊆ Cl⋆(Int⋆(Cl⋆(f−1((σ1, σ2)-sCl(V ))))) for every σ1σ2-open set V of Y . Definition 4. [24] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper τ⋆β(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⋆-β-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⋆β(σ1, σ2)-continuous if F is upper τ⋆β(σ1, σ2)-continuous at each point of X. Definition 5. [24] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower τ⋆β(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⋆-β-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⋆β(σ1, σ2)- continuous if F is lower τ⋆β(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⋆β(σ1, σ2)-continuity ⇒ upper almost τ⋆β(σ1, σ2)-continuity. The converse of the implication is not true in general. We give an example for the implication as follows. N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7049 9 of 11 Example 1. Let X = {1, 2, 3} with a topology τ = {∅, X} and an ideal I = {∅}. Let Y = {a, b, c} with topologies σ1 = {∅, {b}, Y } and σ2 = {∅, {b}, {a, b}, Y }. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is defined as follows: F (1) = {b} and F (2) = F (3) = {a, c}. Then, F is upper almost τ⋆β(σ1, σ2)-continuous but F is not upper τ⋆β(σ1, σ2)-continuous, since {a, c} is σ1σ2-open in Y but F+({a, c}) is not τ⋆-β-open in X. Theorem 5. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost τ⋆β(σ1, σ2)-continuous; (2) βCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) βCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ; (4) F+(V ) ⊆ βInt⋆(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 . Since σ1σ2-Cl(V ) is (σ1, σ2)r- closed, by Theorem 3 we have F−(σ1σ2-Cl(V )) is τ⋆-β-closed in X and hence βCl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )). (2) ⇒ (3): This is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open. (3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . Then, V ⊆ σ1σ2-Int(σ1σ2-Cl(V )) and Y −V ⊇ σ1σ2-Cl(σ1σ2-Int(Y −V )). Since σ1σ2-Cl(σ1σ2-Int(Y −V )) is (σ1, σ2)s-open in Y and by (3), X − F+(V ) = F−(Y − V ) ⊇ F−(σ1σ2-Cl(σ1σ2-Int(Y − V ))) ⊇ βCl⋆(F−(σ1σ2-Cl(σ1σ2-Int(Y − V )))) = βCl⋆(F−(Y − σ1σ2-Int(σ1σ2-Cl(V )))) = βCl⋆(X − F+(σ1σ2-Int(σ1σ2-Cl(V )))) = X − βInt⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))). Thus, F+(V ) ⊆ βInt⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))). (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 ) ⊆ βInt⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) = βInt⋆(F+(V )) and hence F+(V ) is τ⋆-β-open in X. It follows from Theorem 3 that F is upper almost τ⋆β(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⋆β(σ1, σ2)-continuous; (2) βCl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) βCl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ; N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7049 10 of 11 (4) F−(V ) ⊆ βInt⋆(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. Corollary 3. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost τ⋆β(σ1, σ2)-continuous; (2) βCl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) βCl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ; (4) f−1(V ) ⊆ βInt⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) for every (σ1, σ2)p-open set V of Y . Acknowledgements This research project was financially supported by Mahasarakham University. References [1] A. A. Nasef and T. Noiri. Some weak forms of almost continuity. Acta Mathematica Hungarica, 74(3):211–219, 1997. [2] A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deeb. On precontinuous and weak precontinuous mappings. Proceedings of the Mathematical and Physical Society of Egypt, 53:47–53, 1982. [3] M. E. Abd El-Monsef, S. N. El-Deeb, and R. A. Mahmoud. β-open sets and β- continuous mappings. Bulletin of the Faculty of Science, Assiut University, 12:77–90, 1983. [4] T. Noiri and V. Popa. On almost β-continuous functions. Acta Mathematica Hun- garica, 79(4):329–339, 1998. [5] T. Noiri. Almost α-continuous functions. Kyungpook Mathematical Journal, 28:71–77, 1988. [6] S. N. Maheshwari, G. I. Chae, and P. C. Jain. Almost feebly continuous functions. Ulsan Institute of Science and Technology Report, 13:195–197, 1982. [7] V. Popa. On the decomposition of the quasi-continuity in topological spaces (Rou- manian). Studii şi Cercetǎri de Matematicǎ, 30:31–35, 1978. [8] V. Popa and T. Noiri. On upper and lower almost β-continuous multifunctions. Acta Mathematica Hungarica, 82(1-2):57–73, 1999. [9] C. Boonpok. On continuous multifunctions in ideal topological spaces. Lobachevskii Journal of Mathematics, 40(1):24–35, 2019. [10] C. Boonpok and N. Srisarakham. Almost α-⋆-continuity for multifunctions. Interna- tional Journal of Analysis and Applications, 21:107, 2023. [11] C. Boonpok. Upper and lower β(⋆)-continuity. Heliyon, 7:e05986, 2021. N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7049 11 of 11 [12] C. Boonpok and P. Pue-on. Upper and lower sβ(⋆)-continuous multifunctions. Eu- ropean Journal of Pure and Applied Mathematics, 16(3):1634–1646, 2023. [13] C. Boonpok and P. Pue-on. Continuity for multifunctions in ideal topological spaces. WSEAS Transactions on Mathematics, 19:624–631, 2020. [14] P. Pue-on, S. Sompong, and C. Boonpok. Upper and lower (τ1, τ2)-continuous multi- functions. International Journal of Mathematics and Computer Science, 19(4):1305– 1310, 2024. [15] C. Boonpok and P. Pue-on. Characterizations of almost (τ1, τ2)-continuous multi- functions. International Journal of Analysis and Applications, 22:33, 2024. [16] K. Laprom, C. Boonpok, and C. Viriyapong. β(τ1, τ2)-continuous multifunctions on bitopological spaces. Journal of Mathematics, 2020:4020971, 2020. [17] C. Boonpok, C. Viriyapong, and M. Thongmoon. On upper and lower (τ1, τ2)- precontinuous multifunctions. Journal of Mathematics and Computer Science, 18:282– 293, 2018. [18] C. Viriyapong and C. Boonpok. (τ1, τ2)α-continuity for multifunctions. Journal of Mathematics, 2020:6285763, 2020. [19] C. Boonpok. (τ1, τ2)δ-semicontinuous multifunctions. Heliyon, 6:e05367, 2020. [20] P. Pue-on, S. Sompong, and C. Boonpok. Almost quasi (τ1, τ2)-continuity for multi- functions. International Journal of Analysis and Applications, 22:97, 2024. [21] K. Kuratowski. Topology, Vol. I. Academic Press, New York, 1966. [22] D. Janković and T. R. Hamlett. New topologies from old via ideals. The American Mathematical Monthly, 97:295–310, 1990. [23] T. Noiri and V. Popa. On (mI, nJ)-continuous multifunctions. Romanian Journal of Mathematics and Computer Science, 15(1):1–8, 2025. [24] P. Pue-on, A. Sama-Ae, and C. Boonpok. Upper and lower τ⋆β(σ1, σ2)-continuity. (submitted).