EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6565 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and Lower Continuous Multifunctions Defined Between an Ideal Topological Space and a Bitopological Space Jeeranunt Khampakdee1, 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 from an ideal topological space into a bitopological space, called upper τ⋆(σ1, σ2)-continuous multifunctions and lower τ⋆(σ1, σ2)-continuous multifunctions. Furthermore, several characterizations and some prop- erties concerning upper τ⋆(σ1, σ2)-continuous multifunctions and lower τ⋆(σ1, σ2)-continuous mul- tifunctions are investigated. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper τ⋆(σ1, σ2)-continuous multifunction, lower τ⋆(σ1, σ2)-continuous multifunction 1. Introduction The field of the mathematical science which goes under the name of topology is concerned with all questions directly or indirectly related to continuity. The concept of ideals in topological spaces has been introduced and studied by Kuratowski [1] and Vaidyanathaswamy [2] which is one of the important areas of research in the branch of mathematics. 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 generalizations of continuity. Using these no- tions many authors introduced and studied various types of generalizations of continuity for functions and multifunctions. Hatir and Noiri [3] introduced and investigated the no- tions of weakly pre-I -open sets and weakly pre-I -continuous functions. Moreover, Hatir and Noiri [4] investigated further properties of semi-I -open sets and semi-I -continuous ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6565 Email addresses: jeeranunt.k@msu.ac.th (J. Khampakdee), 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) J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6565 2 of 9 functions. On the other hand, the present author [5] introduced new classes of mul- tifunctions between ideal topological spaces, namely upper ⋆-continuous multifunctions and lower ⋆-continuous multifunctions. Furthermore, several characterizations of upper ⋆- continuous multifunctions, lower ⋆-continuous multifunctions, upper almost ⋆-continuous multifunctions, lower almost ⋆-continuous multifunctions, upper weakly ⋆-continuous mul- tifunctions and lower weakly ⋆-continuous multifunctions were considered in [5]. Quite recently, the present author [6] introduced and investigated the notions of pı-continuous multifunctions and weakly pı-continuous multifunctions. Pue-on et al. [7] introduced and studied the concepts of upper (τ1, τ2)-continuous multifunctions and lower (τ1, τ2)- continuous multifunctions. Klanarong et al. [8] investigated several characterizations of upper (τ1, τ2)-continuous multifunctions and lower (τ1, τ2)-continuous multifunctions by utilizing the notions of (τ1, τ2)θ-closed sets and (τ1, τ2)θ-open sets. Thongmoon et al. [9] studied some characterizations of upper (τ1, τ2)-continuous multifunctions and lower (τ1, τ2)-continuous multifunctions by using τ1τ2-δ-open sets and τ1τ2-δ-closed sets. In this paper, we introduce the concepts of upper τ⋆(σ1, σ2)-continuous multifunctions and lower τ⋆(σ1, σ2)-continuous multifunctions. We also investigate several characterizations of up- per τ⋆(σ1, σ2)-continuous multifunctions and lower 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 [10] 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 [10] 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 [10] of A and is denoted by τ1τ2-Int(A). Lemma 1. [10] 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 [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)))). J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6565 3 of 9 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 [13] 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. 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) [14]. 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) [11]. 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 [1], 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 [15] 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 [16] (resp. semi-I -open [4]) 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 [16] (resp. semi-I -closed [4]). 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 τ ⋆(σ1, σ2)-continuous multifunctions In this section, we introduce the concepts of upper τ⋆(σ1, σ2)-continuous multifunctions and lower τ⋆(σ1, σ2)-continuous multifunctions. Furthermore, several characterizations of upper τ⋆(σ1, σ2)-continuous multifunctions and lower τ⋆(σ1, σ2)-continuous multifunctions are discussed. J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6565 4 of 9 Definition 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is called 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 called upper τ⋆(σ1, σ2)-continuous if F is upper τ⋆(σ1, σ2)- continuous at each point x of X. Theorem 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper τ⋆(σ1, σ2)-continuous; (2) F+(V ) is ⋆-open in X for every σ1σ2-open set V of Y ; (3) F−(K) is ⋆-closed in X for every σ1σ2-closed set K of Y ; (4) Cl⋆(F−(B)) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (5) F+(σ1σ2-Int(B)) ⊆ Int⋆(F+(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 by (1), there exists a ⋆-open set U of X containing x such that F (U) ⊆ V . Thus, x ∈ U ⊆ F+(V ) and hence x ∈ Int⋆(F+(V )). Therefore, F+(V ) ⊆ Int⋆(F+(V )). This shows that F+(V ) is ⋆-open in X. (2) ⇒ (3): This follows from the fact that F+(Y −B) = X − F−(B) for every subset B of Y . (3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Cl(B) is σ1σ2-closed in Y and by (3), Cl⋆(F−(B)) ⊆ Cl⋆(F−(σ1σ2-Cl(B))) = F−(σ1σ2-Cl(B)). (4) ⇒ (5): Let B be any subset of Y . Thus by (4), we have X − Int⋆(F+(B)) = Cl⋆(X − F+(B)) = Cl⋆(F−(Y − B)) ⊆ F−(σ1σ2-Cl(Y − B)) = F−(Y − σ1σ2-Int(B)) = X − F+(σ1σ2-Int(B)) and hence F+(σ1σ2-Int(B)) ⊆ Int⋆(F+(B)). (5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that F (x) ⊆ V . Then, x ∈ F+(V ) = Int⋆(F+(V )). There exists a ⋆-open set U of X containing x such that U ⊆ F+(V ); hence F (U) ⊆ V . This shows that F is upper τ⋆(σ1, σ2)-continuous. Definition 2. 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. Theorem 2. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower τ⋆(σ1, σ2)-continuous; (2) F−(V ) is ⋆-open in X for every σ1σ2-open set V of Y ; J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6565 5 of 9 (3) F+(K) is ⋆-closed in X for every σ1σ2-closed set K of Y ; (4) Cl⋆(F+(B)) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (5) F (Cl⋆(A)) ⊆ σ1σ2-Cl(F (A)) for every subset A of X; (6) F−(σ1σ2-Int(B)) ⊆ Int⋆(F−(B)) for every subset B of Y . Proof. We prove only the implications (4) ⇒ (5) and (5) ⇒ (6) being the proofs of the other similar to those of Theorem 1. (4) ⇒ (5): Let A be any subset ofX. Then by (4), we have Cl⋆(A) ⊆ Cl⋆(F+(F (A))) ⊆ F+(Cl⋆(F (A))) and so F (Cl⋆(A)) ⊆ σ1σ2-Cl(F (A)). (5) ⇒ (6): Let B be any subset of Y . By (5), F (Cl⋆(F+(Y −B))) ⊆ σ1σ2-Cl(F (F+(Y −B))) ⊆ σ1σ2-Cl(Y −B) = Y − σ1σ2-Int(B). Since F (Cl⋆(F+(Y −B))) = F (Cl⋆(X − F−(B))) = F (X − Int⋆(F−(B))), we have X − Int⋆(F−(B)) ⊆ F+(Y − σ1σ2-Int(B)) = X − F−(σ1σ2-Int(B)) and hence F−(σ1σ2-Int(B)) ⊆ Int⋆(F−(B)). Definition 3. A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be τ⋆(σ1, σ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing f(x), there exists a ⋆-open set U of X containing x such that f(U) ⊆ V . A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be τ⋆(σ1, σ2)-continuous if f is τ⋆(σ1, σ2)-continuous at each point x of X. Corollary 1. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is τ⋆(σ1, σ2)-continuous; (2) f−1(V ) is ⋆-open in X for every σ1σ2-open set V of Y ; (3) f−1(K) is ⋆-closed in X for every σ1σ2-closed set K of Y ; (4) Cl⋆(f−1(B)) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ; (5) f(Cl⋆(A)) ⊆ σ1σ2-Cl(f(A)) for every subset A of X; (6) f−1(σ1σ2-Int(B)) ⊆ Int⋆(f−1(B)) for every subset B of Y . Definition 4. [9] 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 2. [9] A bitopological space (X, τ1, τ2) is (τ1, τ2)s-regular if and only if for each x ∈ X and each (τ1, τ2)s-open set U containing x, there exists a (τ1, τ2)s-open set V such that x ∈ V ⊆ (τ1, τ2)-sCl(V ) ⊆ U . J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6565 6 of 9 Lemma 3. [9] 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. Theorem 3. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)s- regular, the following properties are equivalent: (1) F is upper 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 . Proof. (1) ⇒ (2): Let B be any subset of Y . By Lemma 3, σ1σ2-δ-Cl(B) is σ1σ2-closed in Y . Since F is upper τ⋆(σ1, σ2)-continuous, by Theorem 1 F−(σ1σ2-δ-Cl(B)) is ⋆-closed in X. (2) ⇒ (3): Let K be any σ1σ2-δ-closed set of Y . Then, σ1σ2-δ-Cl(K) = K and by (2), we have F−(K) is ⋆-closed in X. (3) ⇒ (4): This follows from the fact that F+(Y −B) = X −F−(B) for any subset B of Y . (4) ⇒ (1): Let V be any σ1σ2-open set of Y . Since (Y, σ1, σ2) is (σ1, σ2)s-regular, we have V is σ1σ2-δ-open in Y and by (4), F+(V ) is ⋆-open in X. Thus, F is upper τ⋆(σ1, σ2)-continuous by Theorem 1. Theorem 4. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)s- regular, 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 . Proof. The proof is similar to that of Theorem 3. Corollary 2. For a function f : (X, τ,I ) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)s- regular, the following properties are equivalent: (1) f is τ⋆(σ1, σ2)-continuous; (2) f−1(σ1σ2-δ-Cl(B)) is ⋆-closed in X for every subset B of Y ; J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6565 7 of 9 (3) f−1(K) is ⋆-closed in X for every σ1σ2-δ-closed set K of Y ; (4) f−1(V ) is ⋆-open in X for every σ1σ2-δ-open set V of Y . Definition 5. [17] A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-regular if for each τ1τ2-closed set F and each x ̸∈ F , there exist disjoint τ1τ2-open sets U and V such that x ∈ U and F ⊆ V . Lemma 4. [17] A bitopological space (X, τ1, τ2) is (τ1, τ2)-regular if and only if for each x ∈ X and each τ1τ2-open set U containing x, there exists a τ1τ2-open set V such that x ∈ V ⊆ τ1τ2-Cl(V ) ⊆ U . Lemma 5. [8] Let (X, τ1, τ2) be a (τ1, τ2)-regular space. Then, the following properties hold: (1) τ1τ2-Cl(A) = (τ1, τ2)θ-Cl(A) for every subset A of X. (2) Every τ1τ2-open set is (τ1, τ2)θ-open. Theorem 5. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)- regular, the following properties are equivalent: (1) F is upper τ⋆(σ1, σ2)-continuous; (2) F−((σ1, σ2)θ-Cl(B)) is ⋆-closed in X for every subset B of Y ; (3) F−(K) is ⋆-closed in X for every (σ1, σ2)θ-closed set K of Y ; (4) F+(V ) is ⋆-open in X for every (σ1, σ2)θ-open set V of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . By Lemma 5, (σ1, σ2)θ-Cl(B) is σ1σ2- closed in Y . Since F is upper τ⋆(σ1, σ2)-continuous, by Theorem 1 F−((σ1, σ2)θ-Cl(B)) is ⋆-closed in X. (2) ⇒ (3): Let K be any (σ1, σ2)θ-closed set of Y . Then, (σ1, σ2)θ-Cl(K) = K and by (2), we have F−(K) is ⋆-closed in X. (3) ⇒ (4): This follows from the fact that F+(Y −B) = X − F−(B) for every subset B of Y . (4) ⇒ (1): Let V be any σ1σ2-open set of Y . Since (Y, σ1, σ2) is (σ1, σ2)-regular, we have V is (σ1, σ2)θ-open in Y and by (4), F+(V ) is ⋆-open in X. Thus, F is upper τ⋆(σ1, σ2)-continuous by Theorem 1. Theorem 6. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)- regular, the following properties are equivalent: (1) F is lower τ⋆(σ1, σ2)-continuous; (2) F+((σ1, σ2)θ-Cl(B)) is ⋆-closed in X for every subset B of Y ; (3) F+(K) is ⋆-closed in X for every (σ1, σ2)θ-closed set K of Y ; J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6565 8 of 9 (4) F−(V ) is ⋆-open in X for every (σ1, σ2)θ-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), where (Y, σ1, σ2) is (σ1, σ2)- regular, the following properties are equivalent: (1) f is τ⋆(σ1, σ2)-continuous; (2) f−1((σ1, σ2)θ-Cl(B)) is ⋆-closed in X for every subset B of Y ; (3) f−1(K) is ⋆-closed in X for every (σ1, σ2)θ-closed set K of Y ; (4) f−1(V ) is ⋆-open in X for every (σ1, σ2)θ-open set V of Y . Acknowledgements This research project was financially supported by Mahasarakham University. References [1] K. Kuratowski. Topology, Vol. I. Academic Press, New York, 1966. [2] R. Vaidyanathaswamy. The localisation theory in set-topology. Proceedings of the Indian Academy of Sciences-Section A, 20:51–61, 1944. [3] E. Hatir and T. Noiri. Weakly pre-I-open sets and decomposition of continuity. Acta Mathematica Hungarica, 106(3):227–238, 2005. [4] E. Hatir and T. Noiri. On decompositions of continuity via idealization. Acta Math- ematica Hungarica, 96:341–349, 2002. [5] C. Boonpok. On continuous multifunctions in ideal topological spaces. Lobachevskii Journal of Mathematics, 40(1):24–35, 2019. [6] C. Boonpok. pı-continuity and weak pı-continuity. Carpathian Mathematical Publi- cations, 17(1):171–186, 2025. [7] 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. [8] C. Klanarong, S. Sompong, and C. Boonpok. (τ1, τ2)-continuity and (τ1, τ2)θ-closed sets. International Journal of Mathematics and Computer Science, 19(4):1299–1304, 2024. [9] 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. [10] 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. J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6565 9 of 9 [11] C. Viriyapong and C. Boonpok. (τ1, τ2)α-continuity for multifunctions. Journal of Mathematics, 2020:6285763, 2020. [12] C. Boonpok. (τ1, τ2)δ-semicontinuous multifunctions. Heliyon, 6:e05367, 2020. [13] 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. [14] C. Boonpok and P. Pue-on. Characterizations of almost (τ1, τ2)-continuous multi- functions. International Journal of Analysis and Applications, 22:33, 2024. [15] D. Janković and T. R. Hamlett. New topologies from old via ideals. The American Mathematical Monthly, 97:295–310, 1990. [16] E. Ekici and T. Noiri. ⋆-extremally disconnected ideal topological spaces. Acta Mathematica Hungarica, 122:81–90, 2009. [17] M. Chiangpradit, S. Sompong, and C. Boonpok. On characterizations of (τ1, τ2)- regular spaces. International Journal of Mathematics and Computer Science, 19(4):1329–1334, 2024.