EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6566 ISSN 1307-5543 – ejpam.com Published by New York Business Global Almost Continuity for Multifunctions Defined from an Ideal Topological Space into a Bitopological Space 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 presents new concepts of continuous multifunctions defined between an ideal topological space and a bitopological space, namely upper almost τ⋆(σ1, σ2)-continuous multifunc- tions and lower almost τ⋆(σ1, σ2)-continuous multifunctions. Moreover, several characterizations and some properties concerning upper almost τ⋆(σ1, σ2)-continuous multifunctions and lower al- most τ⋆(σ1, σ2)-continuous multifunctions are considered. Furthermore, the relationships between τ⋆(σ1, σ2)-continuity and almost τ⋆(σ1, σ2)-continuity are discussed. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper almost τ⋆(σ1, σ2)-continuous multifunction, lower almost τ⋆(σ1, σ2)-continuous multifunction 1. Introduction It is well-known that the branch of mathematics called topology is related to all ques- tions directly or indirectly concerned with continuity. Singal and Singal [1] introduced the concept of almost continuous functions as a generalization of continuity. Munshi and Bas- san [2] studied the notion of almost semi-continuous functions. Noiri [3] introduced and investigated the concept of almost α-continuous functions. Nasef and Noiri [4] introduced two classes of functions, namely almost precontinuous functions and almost β-continuous functions. The class of almost precontinuity is a generalization of almost α-continuity. The class of almost β-continuity is a generalization of almost semi-continuity. Popa [5] introduced and studied the concepts of upper almost continuous multifunctions and lower almost continuous multifunctions. Furthermore, Popa and Noiri [6] introduced and in- vestigated the notions of upper almost quasi-continuous multifunctions and lower almost quasi-continuous multifunctions. Popa et al. [7] introduced the concepts of upper almost ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6566 Email addresses: chokchai.v@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 (3) (2025), 6566 2 of 14 precontinuous multifunctions and lower almost precontinuous multifunctions. Noiri and Popa [8] introduced the concepts of upper almost β-continuous multifunctions and lower almost β-continuous multifunctions. Moreover, some characterizations of upper almost β-continuous multifunctions and lower almost β-continuous multifunctions were presented in [9]. Popa and Noiri [10] introduced and investigated the notions of upper almost α- continuous multifunctions and lower almost α-continuous multifunctions. Pue-on et al. [11] introduced and studied the concepts of upper (τ1, τ2)-continuous multifunctions and lower (τ1, τ2)-continuous multifunctions. Klanarong et al. [12] introduced and investigated the notions of upper almost (τ1, τ2)-continuous multifunctions and lower almost (τ1, τ2)- continuous multifunctions. The notion of ideal topological spaces was introduced and studied by Kuratowski [13] and Vaidyanathaswamy [14]. Stronger and weaker forms of open sets in ideal topological spaces such as semi-I -open sets, pre-I -open sets, α-I -open sets, β-I -open sets and δ-I -open sets play an important role in the research of general- izations of continuity. Using these notions many authors introduced and studied various types of generalizations of continuity for functions and multifunctions. Hatir and Noiri [15] introduced and investigated the notions of weakly pre-I -open sets and weakly pre- I -continuous functions. Furthermore, Hatir and Noiri [16] investigated further properties of semi-I -open sets and semi-I -continuous functions. On the other hand, the present author [17] introduced the concepts of upper ⋆-continuous multifunctions and lower ⋆- continuous multifunctions. Moreover, several characterizations of upper ⋆-continuous mul- tifunctions, lower ⋆-continuous multifunctions, upper almost ⋆-continuous multifunctions and lower almost ⋆-continuous multifunctions were established in [17]. Quite recently, the present author [18] introduced and studied the notions of pı-continuous multifunc- tions and weakly pı-continuous multifunctions. In this paper, we introduce new classes of multifunctions between an ideal topological space and a bitopological space, namely upper almost τ⋆(σ1, σ2)-continuous multifunctions and lower almost τ⋆(σ1, σ2)-continuous multifunctions. We also investigate several characterizations of upper almost τ⋆(σ1, σ2)- continuous multifunctions and lower almost 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 [19] 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 [19] 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 [19] of A and is denoted by τ1τ2-Int(A). Lemma 1. [19] Let A and B be subsets of a bitopological space (X, τ1, τ2). For the τ1τ2- closure, the following properties hold: C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6566 3 of 14 (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 [20] (resp. (τ1, τ2)s-open [21], (τ1, τ2)p-open [21], (τ1, τ2)β-open [21]) 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 [22] if A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)-open set is said to be α(τ1, τ2)-closed. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection of all (τ1, τ2)p-closed (resp. (τ1, τ2)s-closed, α(τ1, τ2)-closed) sets of X containing A is called the (τ1, τ2)p-closure [23] (resp. (τ1, τ2)s-closure [21], α(τ1, τ2)-closure [24]) 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 [23] (resp. (τ1, τ2)s-interior [21], α(τ1, τ2)-interior [24]) of A and is denoted by (τ1, τ2)-pInt(A) (resp. (τ1, τ2)-sInt(A), α(τ1, τ2)-Int(A)). A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-δ-open 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. The union of all τ1τ2-δ-open sets of X contained in A is called the τ1τ2-δ-interior 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 of A and is denoted by τ1τ2-δ-Cl(A) [25]. For a subset A of a bitopological space (X, τ1, τ2), a point x ∈ X is called a (τ1, τ2)θ-cluster point of A if τ1τ2-Cl(U) ∩A ̸= ∅ for every τ1τ2-open set U containing x. The set of all (τ1, τ2)θ-cluster points of A is called the (τ1, τ2)θ-closure of A and is denoted by (τ1, τ2)θ-Cl(A). A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)θ-closed if (τ1, τ2)θ-Cl(A) = A. The complement of a (τ1, τ2)θ-closed set is said to be (τ1, τ2)θ-open. The union of all (τ1, τ2)θ-open sets of X contained in A is called the (τ1, τ2)θ-interior of A and is denoted by (τ1, τ2)θ-Int(A) [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 [13], A⋆ is called the local function of A with respect to I and τ and Cl⋆(A) = A⋆∪A defines a Kuratowski C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6566 4 of 14 closure operator for a topology τ⋆(I ) finer than τ . A subset A is said to be ⋆-closed [26] 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 [27] (resp. semi-I -open [16]) 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 [27] (resp. semi-I -closed [16]). 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 ⋆(σ1, σ2)-continuous multifunctions In this section, we introduce the concepts of upper almost τ⋆(σ1, σ2)-continuous mul- tifunctions and lower almost τ⋆(σ1, σ2)-continuous multifunctions. Furthermore, several characterizations of upper almost τ⋆(σ1, σ2)-continuous multifunctions and lower almost τ⋆(σ1, σ2)-continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper almost τ⋆(σ1, σ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a ⋆-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper almost τ⋆(σ1, σ2)-continuous if F is upper almost τ⋆(σ1, σ2)-continuous at each point x of X. Lemma 2. [12] 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)). Theorem 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost τ⋆(σ1, σ2)-continuous at x ∈ X; (2) x ∈ Int⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y containing F (x); (3) x ∈ Int⋆(F+((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y containing F (x); (4) x ∈ Int⋆(F+(V )) for every (σ1, σ2)r-open set V of Y containing F (x); (5) for each (σ1, σ2)r-open set V of Y containing F (x), there exists a ⋆-open set U of X containing x such that F (U) ⊆ V . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing F (x). Thus by (1), there exists a ⋆-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). There- fore, x ∈ U ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))) and so x ∈ Int⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))). C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6566 5 of 14 (2) ⇒ (3): This follows from Lemma 2. (3) ⇒ (4): Let V be any σ1σ2-open set of Y containing F (x). It follows from Lemma 2 that V = σ1σ2-Int(σ1σ2-Cl(V )) = (σ1, σ2)-sCl(V ). (4) ⇒ (5): Let V be any (σ1, σ2)r-open set of Y containing F (x). Then by (4), we have x ∈ Int⋆(F+(V )) and there exists a ⋆-open set U of X containing x such that x ∈ U ⊆ F+(V ); hence F (U) ⊆ V . (5) ⇒ (1): Let V be any σ1σ2-open set of Y containing F (x). Since σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r-open in Y and by (5), there exists a ⋆-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). This shows that F is upper almost τ⋆(σ1, σ2)-continuous at x ∈ X. Definition 2. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is called lower almost τ⋆(σ1, σ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that V ∩ F (x) ̸= ∅, there exists a ⋆-open set U of X containing x such that σ1σ2-Int(σ1σ2-Cl(V )) ∩ F (z) ̸= ∅ for every z ∈ U . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is called lower almost τ⋆(σ1, σ2)-continuous if F is lower almost 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 τ⋆(σ1, σ2)-continuous at x ∈ X; (2) x ∈ Int⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y such that V ∩ F (x) ̸= ∅; (3) x ∈ Int⋆(F−((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y such that V ∩ F (x) ̸= ∅; (4) x ∈ Int⋆(F−(V )) for every (σ1, σ2)r-open set V of Y such that V ∩ F (x) ̸= ∅; (5) for each (σ1, σ2)r-open set V of Y such that V ∩ F (x) ̸= ∅, there exists a ⋆-open set U of X containing x such that U ⊆ F−(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⋆(σ1, σ2)-continuous; (2) F+(V ) ⊆ Int⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ; (3) Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F−(K) for every σ1σ2-closed set K of Y ; C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6566 6 of 14 (4) Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (5) F+(σ1σ2-Int(B)) ⊆ Int⋆(F+(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) for every subset B of Y ; (6) F+(V ) is ⋆-open in X for every (σ1, σ2)r-open set V of Y ; (7) F−(K) is ⋆-closed in X for every (σ1, σ2)r-closed set K of Y . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V . Thus by Theorem 1, we have x ∈ Int⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) and hence F+(V ) ⊆ Int⋆(F+(σ1σ2-Int(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), X − F−(K) = F+(Y −K) ⊆ Int⋆(F+(σ1σ2-Int(σ1σ2-Cl(Y −K)))) = Int⋆(X − F−(σ1σ2-Cl(σ1σ2-Int(K)))) = X − Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(K)))). Thus, Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F−(K). (3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Cl(B) is a σ1σ2-closed set of Y and by (3), Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F−(σ1σ2-Cl(B)). (4) ⇒ (5): Let B be any subset of Y . Thus by (4), we have F+(σ1σ2-Int(B)) = X − F−(σ1σ2-Cl(Y −B)) ⊆ X − Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(Y −B))))) = X − Cl⋆(F−(Y − σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) = Int⋆(F+(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))). (5) ⇒ (6): Let V be any (σ1, σ2)r-open set of Y . By (5), we have F+(V ) ⊆ Int⋆(F+(V )) and hence F+(V ) is ⋆-open in X. (6) ⇒ (7): The proof is obvious. (7) ⇒ (1): Let x ∈ X and V be any (σ1, σ2)r-open set of Y containing F (x). Since Y − V is (σ1, σ2)r-closed and by (7), X − F+(V ) = F−(Y − V ) is ⋆-closed in X. Thus, F+(V ) is ⋆-open and hence x ∈ Int⋆(F+(V )). Then, there exists a ⋆-open set U of X containing x such that F (U) ⊆ V . It follows from Theorem 1 that F is upper almost 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 τ⋆(σ1, σ2)-continuous; (2) F−(V ) ⊆ Int⋆(F−(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ; C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6566 7 of 14 (3) Cl⋆(F+(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F+(K) for every σ1σ2-closed set K of Y ; (4) Cl⋆(F+(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (5) F−(σ1σ2-Int(B)) ⊆ Int⋆(F−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) for every subset B of Y ; (6) F−(V ) is ⋆-open in X for every (σ1, σ2)r-open set V of Y ; (7) F+(K) is ⋆-closed in X for every (σ1, σ2)r-closed set K 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⋆(σ1, σ2)-continuous; (2) Cl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) Cl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y . Proof. (1) ⇒ (2): Let V be any (σ1, σ2)β-open set of Y . Then, σ1σ2-Cl(V ) is a (σ1, σ2)r-closed set of Y . Since F is upper almost τ⋆(σ1, σ2)-continuous and by Theorem 3, F−(σ1σ2-Cl(V )) is ⋆-closed in X. Thus, Cl⋆(F−(V )) ⊆ F−(σ1σ2-Cl(V )). (2) ⇒ (3): The proof is obvious. (3) ⇒ (1): Let K be any (σ1, σ2)r-closed set of Y . Then, K is (σ1, σ2)s-open in Y . Then by (3), Cl⋆(F−(K)) ⊆ F−(σ1σ2-Cl(K)) = F−(K) and hence F−(K) is ⋆-closed in X. By Theorem 3, F is upper almost τ⋆(σ1, σ2)-continuous. Theorem 6. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) Cl⋆(F+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y . Proof. The proof is similar to that of Theorem 5. Lemma 3. [28] For a bitopological space (X, τ1, τ2), the following properties hold: (1) α(τ1, τ2)-Cl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)β-open set V of X; (2) (τ1, τ2)-pCl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)s-open set V of X. C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6566 8 of 14 Corollary 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F−(V )) ⊆ F−(α(σ1, σ2)-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) Cl⋆(F−(V )) ⊆ F−((σ1, σ2)-pCl(V )) for every (σ1, σ2)s-open set V of Y . Corollary 2. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F+(V )) ⊆ F+(α(σ1, σ2)-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) Cl⋆(F+(V )) ⊆ F+((σ1, σ2)-pCl(V )) for every (σ1, σ2)s-open set V of Y . Theorem 7. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (4) F+(V ) ⊆ Int⋆(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)p-open set of Y . Then, σ1σ2-Cl(V ) is σ1σ2- closed in Y and by Theorem 3, we have Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))))) ⊆ F−(σ1σ2-Cl(V )). (2) ⇒ (3): Let V be any (σ1, σ2)p-open set of Y . By (2), Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(V )))) ⊆ Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))))) ⊆ F−(σ1σ2-Cl(V )). (3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . Thus by (3), we have X − Int⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) = Cl⋆(X − F+(σ1σ2-Int(σ1σ2-Cl(V )))) = Cl⋆(F−(Y − σ1σ2-Int(σ1σ2-Cl(V )))) = Cl⋆(F−(σ1σ2-Cl(Y − σ1σ2-Cl(V )))) = Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(Y − σ1σ2-Cl(V ))))) ⊆ F−(σ1σ2-Cl(Y − σ1σ2-Cl(V ))) C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6566 9 of 14 = F−(Y − σ1σ2-Int(σ1σ2-Cl(V ))) ⊆ F−(Y − V ) = X − F+(V ) and hence F+(V ) ⊆ Int⋆(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 ) ⊆ Int⋆(F+(σ1σ2-Int(σ1σ2-Cl(V )))) = Int⋆(F+(V )). Thus, F+(V ) is ⋆-open in X. It follows from Theorem 3 that F is upper almost τ⋆(σ1, σ2)-continuous. Theorem 8. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F+(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) Cl⋆(F+(σ1σ2-Cl(σ1σ2-Int(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (4) F−(V ) ⊆ Int⋆(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 7. Lemma 4. [29] Let A be a subset of a bitopological space (X, τ1, τ2). Then, the following properties hold: (1) If A is τ1τ2-open in X, then τ1τ2-Cl(A) = τ1τ2-δ-Cl(A). (2) τ1τ2-δ-Cl(A) is τ1τ2-closed. Theorem 9. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-δ-Cl(B))))) ⊆ F−(σ1σ2-δ-Cl(B)) for every subset B of Y ; (3) Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F−(σ1σ2-δ-Cl(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . By Lemma 4, σ1σ2-δ-Cl(B) is σ1σ2-closed in Y and by Theorem 3, Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-δ-Cl(B))))) ⊆ F−(σ1σ2-δ-Cl(B)). (2) ⇒ (3): This is obvious since σ1σ2-Cl(B) ⊆ σ1σ2-δ-Cl(B) for every subset B of Y . (3) ⇒ (1): Let K be any (σ1, σ2)r-closed set of Y . Then by (3), we have Cl⋆(F−(K)) = Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(K)))) C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6566 10 of 14 = Cl⋆(F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(K))))) ⊆ F−(σ1σ2-δ-Cl(K)) = F−(K) and hence F−(K) is ⋆-closed inX. By Theorem 3, F is upper almost 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 almost τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F+(σ1σ2-Cl(σ1σ2-Int(σ1σ2-δ-Cl(B))))) ⊆ F+(σ1σ2-δ-Cl(B)) for every subset B of Y ; (3) Cl⋆(F+(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F+(σ1σ2-δ-Cl(B)) for every subset B of Y . Proof. The proof is similar to that of Theorem 9. Lemma 5. If F : (X, τ,I ) → (Y, σ1, σ2) is lower almost τ⋆(σ1, σ2)-continuous, then for each x ∈ X and each subset B of Y with σ1σ2-δ-Int(B) ∩ F (x) ̸= ∅, there exists a ⋆-open set U of X containing x such that U ⊆ F−(B). Proof. Let x ∈ X and B be a subset of Y with σ1σ2-δ-Int(B) ∩ F (x) ̸= ∅. Since σ1σ2-δ-Int(B) ∩ F (x) ̸= ∅, there exists a nonempty (σ1, σ2)r-open set V of Y such that V ⊆ B and V ∩ F (x) ̸= ∅. Since F is lower almost τ⋆(σ1, σ2)-continuous, there exists a ⋆-open set U of X containing x such that V ∩F (z) ̸= ∅ for each z ∈ U ; hence U ⊆ F−(B). Theorem 11. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost τ⋆(σ1, σ2)-continuous; (2) Cl⋆(F+(B)) ⊆ F+(σ1σ2-δ-Cl(B)) for every subset B of Y ; (3) F (Cl⋆(A)) ⊆ σ1σ2-δ-Cl(F (A)) for every subset A of X; (4) F+(K) is ⋆-closed in X for every σ1σ2-δ-closed set K of Y ; (5) F−(V ) is ⋆-open in X for every σ1σ2-δ-open set V of Y ; (6) F−(σ1σ2-δ-Int(B)) ⊆ Int⋆(F−(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . Suppose that x ̸∈ F+(σ1σ2-δ-Cl(B)). Then, we have x ∈ F−(Y −σ1σ2-δ-Cl(B)) = F−(σ1σ2-δ-Int(Y −B)). There exists a ⋆-open set U of X containing x such that U ⊆ F−(Y −B) = X − F+(B). Thus, U ∩ F+(B) = ∅ and hence x ∈ X − Cl⋆(F+(B)). This shows that Cl⋆(F+(B)) ⊆ F+(σ1σ2-δ-Cl(B)). C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6566 11 of 14 (2) ⇒ (3): Let A be any subset of X. By (2), we have Cl⋆(A) ⊆ τ1τ2-Cl(F +(F (A))) ⊆ F+(σ1σ2-δ-Cl(F (A))) and hence F (Cl⋆(A)) ⊆ σ1σ2-δ-Cl(F (A)). (3) ⇒ (1): Let B be any subset of Y . Then, by the hypothesis and Lemma 4, F (Cl⋆(F+(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B)))))) ⊆ τ1τ2-δ-Cl(F (F+(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B)))))) ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))) ⊆ σ1σ2-Cl(B) and hence Cl⋆(F+(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F+(σ1σ2-Cl(B)). By Theorem 4, F is lower almost τ⋆(σ1, σ2)-continuous. (2) ⇒ (4): Let K be any σ1σ2-δ-closed set of Y . Then, σ1σ2-δ-Cl(K) = K. By (2), we have Cl⋆(F+(K)) ⊆ F+(σ1σ2-δ-Cl(K)) = F+(K) and hence F+(K) is ⋆-closed in X. (4) ⇒ (5): The proof is obvious. (5) ⇒ (6): Let B be any subset of Y . Then by (5), we have F−(σ1σ2-δ-Int(B)) = Int⋆(F−(σ1σ2-δ-Int(B))) ⊆ Int⋆(F−(B)). (6) ⇒ (1): Let V be any (σ1, σ2)r-open set of Y . Then, we have V is σ1σ2-δ-open and σ1σ2-δ-Int(V ) = V . Thus by (6), F−(V ) ⊆ Int⋆(F−(V )) and hence F−(V ) is ⋆-open in X. By Theorem 4, F is lower almost τ⋆(σ1, σ2)-continuous. Definition 3. [30] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper τ⋆(σ1, σ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a ⋆-open set U of X containing x such that F (U) ⊆ V . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper τ⋆(σ1, σ2)-continuous if F is upper τ⋆(σ1, σ2)-continuous at each point x of X. Definition 4. [30] A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is called lower τ⋆(σ1, σ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a ⋆-open set U of X containing x such that F (z) ∩ V ̸= ∅ for every z ∈ U . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is called lower τ⋆(σ1, σ2)-continuous if F is lower 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 τ⋆(σ1, σ2)-continuity ⇒ upper almost τ⋆(σ1, σ2)-continuity. The converse of the implication is not true in general. We give an example for the implication as follows. C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6566 12 of 14 Example 1. Let X = {1, 2, 3} with a topology τ = {∅, {1}, {2}, {1, 2}, X} and an ideal I = {∅, {1}}. Let Y = {a, b, c} with topologies σ1 = {∅, {a}, {a, b}, Y } and σ2 = {∅, {a}, {b}, {a, b}, Y }. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is defined as follows: F (1) = {c} and F (2) = F (3) = {a, b}. Then, F is upper almost τ⋆(σ1, σ2)-continuous but F is not upper τ⋆(σ1, σ2)-continuous. Definition 5. [31] A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)s-regular if for each (τ1, τ2)s-closed set F and each x ̸∈ F , there exist disjoint (τ1, τ2)s-open sets U and V such that x ∈ U and F ⊆ V . Lemma 6. [31] Let (X, τ1, τ2) be a (τ1, τ2)s-regular space. Then, the following properties hold: (1) τ1τ2-Cl(A) = τ1τ2-δ-Cl(A) for every subset A of X. (2) Every τ1τ2-open set is τ1τ2-δ-open. Lemma 7. [31] For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), where (Y, σ1, σ2) is a (σ1, σ2)s-regular space, the following properties are equivalent: (1) F is lower τ⋆(σ1, σ2)-continuous; (2) F+(σ1σ2-δ-Cl(B)) is ⋆-closed in X for every subset B of Y ; (3) F+(K) is ⋆-closed in X for every σ1σ2-δ-closed set K of Y ; (4) F−(V ) is ⋆-open in X for every σ1σ2-δ-open set V of Y . Theorem 12. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), where (Y, σ1, σ2) is a (σ1, σ2)s-regular space, the following properties are equivalent: (1) F is lower τ⋆(σ1, σ2)-continuous; (2) F+(σ1σ2-δ-Cl(B)) is ⋆-closed in X for every subset B of Y ; (3) F+(K) is ⋆-closed in X for every σ1σ2-δ-closed set K of Y ; (4) F−(V ) is ⋆-open in X for every σ1σ2-δ-open set V of Y ; (5) F is lower almost τ⋆(σ1, σ2)-continuous. Proof. The proofs of the implications (1) ⇒ (2) ⇒ (3) ⇒ (4) are similar as in Lemma 7. (4) ⇒ (5): Let V be any (σ1, σ2)r-open set of Y . Then, V is σ1σ2-open in Y and by Lemma 6, V is σ1σ2-δ-open in Y . By (4), we have F−(V ) is ⋆-open in X. Thus by Theorem 4, F is lower almost τ⋆(σ1, σ2)-continuous. C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6566 13 of 14 (5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that V ∩F (x) ̸= ∅. Since (Y, σ1, σ2) is (σ1, σ2)s-regular, there exists a (σ1, σ2)r-open set W such that W ∩F (x) ̸= ∅ and W ⊆ V . Since F is lower almost τ⋆(σ1, σ2)-continuous, there exists a ⋆-open set U of X containing x such that W ∩ F (z) ̸= ∅ for every z ∈ U . Thus, F (z) ∩ V ̸= ∅ for every z ∈ U . This shows that F is lower τ⋆(σ1, σ2)-continuous. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] M. K. Singal and A. R. Singal. Almost continuous mappings. Yokohama Mathematical Journal, 16:63–73, 1968. [2] B. M. Munshi and D. S. Bassan. Almost semi-continuous mappings. The Mathematics Student, 49:239–248, 1981. [3] T. Noiri. Almost α-continuous functions. Kyungpook Mathematical Journal, 28:71–77, 1988. [4] A. A. Nasef and T. Noiri. Some weak forms of almost continuity. Acta Mathematica Hungarica, 74(3):211–219, 1997. [5] V. Popa. Almost continuous multifunctions. Matematički Vesnik, 6(9)(34):75–84, 1982. [6] V. Popa and T. Noiri. On upper and lower almost quasi-continuous multifunctions. Bulletin of the Institute of Mathematics, Academia Sinica, 21:337–349, 1993. [7] V. Popa, T. Noiri, and M. Ganster. On upper and lower almost precontinuous multi- functions. Far East Journal of Mathematical Sciences, Special Volume(Part I):49–68, 1997. [8] T. Noiri and V. Popa. On upper and lower almost β-continuous multifunctions. Acta Mathematica Hungarica, 82:57–73, 1999. [9] V. Popa and T. Noiri. On upper and lower weakly β-continuous multifunctions. Annales Universitatis Scientiarum Budapestinensis, 43:25–48, 2000. [10] V. Popa and T. Noiri. On upper and lower almost α-continuous multifunctions. Demonstratio Mathematica, 29:381–396, 1996. [11] 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. [12] C. Klanarong, S. Sompong, and C. Boonpok. Upper and lower almost (τ1, τ2)- continuous multifunctions. European Journal of Pure and Applied Mathematics, 17(2):1244–1253, 2024. [13] K. Kuratowski. Topology, Vol. I. Academic Press, New York, 1966. [14] R. Vaidyanathaswamy. The localisation theory in set-topology. Proceedings of the Indian Academy of Sciences-Section A, 20:51–61, 1944. C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6566 14 of 14 [15] E. Hatir and T. Noiri. Weakly pre-I-open sets and decomposition of continuity. Acta Mathematica Hungarica, 106(3):227–238, 2005. [16] E. Hatir and T. Noiri. On decompositions of continuity via idealization. Acta Math- ematica Hungarica, 96:341–349, 2002. [17] C. Boonpok. On continuous multifunctions in ideal topological spaces. Lobachevskii Journal of Mathematics, 40(1):24–35, 2019. [18] C. Boonpok. pı-continuity and weak pı-continuity. Carpathian Mathematical Publi- cations, 17(1):171–186, 2025. [19] 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. [20] C. Viriyapong and C. Boonpok. (τ1, τ2)α-continuity for multifunctions. Journal of Mathematics, 2020:6285763, 2020. [21] C. Boonpok. (τ1, τ2)δ-semicontinuous multifunctions. Heliyon, 6:e05367, 2020. [22] N. Viriyapong, S. Sompong, and C. Boonpok. (τ1, τ2)-extremal disconnectedness in bitopological spaces. International Journal of Mathematics and Computer Science, 19(3):855–860, 2024. [23] N. Viriyapong, S. Sompong, and C. Boonpok. Upper and lower s-(τ1, τ2)p-continuous multifunctions. European Journal of Pure and Applied Mathematics, 17(3):2210–2220, 2024. [24] C. Viriyapong, S. Sompong, and C. Boonpok. Upper and lower slight α(τ1, τ2)- continuity. European Journal of Pure and Applied Mathematics, 17(3):2142–2154, 2024. [25] C. Boonpok and P. Pue-on. Characterizations of almost (τ1, τ2)-continuous multi- functions. International Journal of Analysis and Applications, 22:33, 2024. [26] D. Janković and T. R. Hamlett. New topologies from old via ideals. The American Mathematical Monthly, 97:295–310, 1990. [27] E. Ekici and T. Noiri. ⋆-extremally disconnected ideal topological spaces. Acta Mathematica Hungarica, 122:81–90, 2009. [28] P. Pue-on, S. Sompong, and C. Boonpok. Almost contra-(τ1, τ2)p-continuity for func- tions. European Journal of Pure and Applied Mathematics, 18(2):6038, 2025. [29] M. Thongmoon, S. Sompong, and C. Boonpok. Almost (τ1, τ2)-continuity and τ1τ2- δ-open sets. International Journal of Analysis and Applications, 22:109, 2024. [30] J. Khampakdee, A. Sama-Ae, and C. Boonpok. Upper and lower continuous multi- functions defined between an ideal topological space and a bitopological space. (sub- mitted). [31] M. Thongmoon, S. Sompong, and C. Boonpok. (τ1, τ2)-continuous multifunctions and τ1τ2-δ-open sets. International Journal of Mathematics and Computer Science, 19(4):1369–1375, 2024.