EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7048 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and Lower τ ⋆β(σ1, σ2)-Continuity Prapart Pue-on1, 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. A new class of continuous multifunctions between an ideal topological space and a bitopological space, called upper (lower) τ⋆β(σ1, σ2)-continuous multifunctions, has been de- fined and studied. Furthermore, several characterizations and some properties concerning upper τ⋆β(σ1, σ2)-continuous multifunctions and lower τ⋆β(σ1, σ2)-continuous multifunctions are dis- cussed. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper τ⋆β(σ1, σ2)-continuous multifunction, lower τ⋆β(σ1, σ2)-continuous multifunction 1. Introduction The notion of β-continuous functions was introduced by Abd El-Monsef et al. [1]. Borsík and Doboš [2] introduced the concept of almost quasicontinuity which is weaker than that of quasicontinuity [3]. Popa and Noiri [4] investigated several characterizations of β-continuity and shown that almost quasi-continuity is equivalent to β-continuity. The equivalence of almost quasicontinuity and β-continuity is also shown by Borsík [5] and Ew- ert [6]. In 1996-1997, Popa and Noiri [7] extended the concept of β-continuous functions to multifunctions and presented new classes of multifunctions defined from a topologi- cal space into a topological space, namely upper β-continuous multifunctions and lower β-continuous multifunctions. Moreover, Popa and Noiri [7] investigated several charac- terizations and some properties concerning upper β-continuous multifunctions and lower β-continuous multifunctions. On the other hand, the present author introduced and in- vestigated four classes of multifunctions defined from an ideal topological space into an ideal topological space, namely upper ⋆-continuous multifunctions [8], lower ⋆-continuous ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7048 Email addresses: prapart.p@msu.ac.th (P. Pue-on), 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) P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7048 2 of 9 multifunctions [8], upper β(⋆)-continuous multifunctions [9], lower β(⋆)-continuous mul- tifunctions [9], upper sβ(⋆)-continuous multifunctions [10], lower sβ(⋆)-continuous multi- functions [10], upper α-⋆-continuous multifunctions [11], lower α-⋆-continuous multifunc- tions [11], ı⋆-continuous multifunctions [12] and pı-continuous multifunctions [13]. Pue- on et al. [14] introduced and studied two classes of multifunctions between bitopological spaces, namely upper (τ1, τ2)-continuous multifunctions and lower (τ1, τ2)-continuous mul- tifunctions. Klanarong et al. [15] investigated several characterizations of upper (τ1, τ2)- continuous multifunctions and lower (τ1, τ2)-continuous multifunctions by utilizing the no- tions of (τ1, τ2)θ-closed sets and (τ1, τ2)θ-open sets. Thongmoon et al. [16] 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. Laprom et al. [17] in- troduced and investigated the notions of upper β(τ1, τ2)-continuous multifunctions and lower β(τ1, τ2)-continuous multifunctions. In this paper, we introduce the concepts of continuous multifunctions between an ideal topological space and a bitopological space, called upper τ⋆β(σ1, σ2)-continuous multifunctions and lower τ⋆β(σ1, σ2)-continuous mul- tifunctions. We also investigate several characterizations of upper τ⋆β(σ1, σ2)-continuous multifunctions and lower 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 [18] 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 [18] 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 [18] of A and is denoted by τ1τ2-Int(A). Lemma 1. [18] 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 [19] (resp. (τ1, τ2)s-open [20], (τ1, τ2)p-open [20], (τ1, τ2)β-open [20]) 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)))). P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7048 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 τ1τ2-δ-open [21] 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 [21]. The union of all τ1τ2-δ-open sets of X contained in A is called the τ1τ2-δ-interior [21] 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 [21] 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 [19] 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 [19] 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 [19] 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 [19] of A and is denoted by (τ1, τ2)θ-Int(A). A subset A of a bitopological space (X, τ1, τ2) is called τ1τ2-nowhere dense if τ1τ2-Int(τ1τ2-Int(A)) = ∅. 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 [22], 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 [23] 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 [8] (resp. I ⋆- preopen [8], τ⋆-semi-open [24] (semi-I ⋆-open [25]), τ⋆-β-open [24] (semi-I ⋆-preopen [25])) 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, 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 [25] of A and is denoted by sCl⋆(A) (sClI ⋆(A) [25]). The union of all semi-I ⋆-open sets contained in A is called the semi-I ⋆-interior [25] of A and is denoted by sInt⋆(A) (sIntI ⋆(A) [25]). 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 2. For a subset A of an ideal topological space (X, τ,I ), the following properties hold: (1) sCl⋆(A) = A ∪ Int⋆(Cl⋆(A)) [25]. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7048 4 of 9 (2) sInt⋆(A) = A ∩ Cl⋆(Int⋆(A)) [25]. (3) βCl⋆(A) = A ∪ Int⋆(Cl⋆(Int⋆(A))). (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 τ ⋆β(σ1, σ2)-continuous multifunctions In this section, we introduce the notions of upper τ⋆β(σ1, σ2)-continuous multifunctions and lower τ⋆β(σ1, σ2)-continuous multifunctions. Moreover, several characterizations of upper τ⋆β(σ1, σ2)-continuous multifunctions and lower τ⋆β(σ1, σ2)-continuous multifunc- tions discussed. Definition 1. 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. Theorem 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is upper τ⋆β(σ1, σ2)-continuous at x ∈ X if and only if x ∈ βInt⋆(F+(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) ⊆ V . Then, U ⊆ F+(V ). Since U is τ⋆- β-open, we have x ∈ U ⊆ Cl⋆(Int⋆(Cl⋆(U))) ⊆ Cl⋆(Int⋆(Cl⋆(F+(V )))). Since x ∈ F+(V ) and by Lemma 2, x ∈ F+(V ) ∩ Cl⋆(Int⋆(Cl⋆(F+(V )))) = βInt⋆(F+(V )). Conversely, let V be any σ1σ2-open set of Y containing F (x). By (2), x ∈ sβInt⋆(F+(V )) and so there exists a τ⋆-β-open set U of X containing x such that U ⊆ F+(V ); hence F (U) ⊆ V . This shows that F is upper τ⋆β(σ1, σ2)-continuous at x. Definition 2. 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. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7048 5 of 9 Theorem 2. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is lower τ⋆β(σ1, σ2)-continuous at x ∈ X if and only if x ∈ βInt⋆(F−(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 called 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) ⊆ V . A function f : (X, τ,I ) → (Y, σ1, σ2) is called 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 τ⋆β(σ1, σ2)-continuous at x ∈ X if and only if x ∈ βInt⋆(f−1(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 τ⋆β(σ1, σ2)-continuous at a point x ∈ X; (2) for each ⋆-open neighborhood U of x and each σ1σ2-open set V of Y with x ∈ F+(V ), F+(V ) ∩ U is not τ1τ2-nowhere dense; (3) for each ⋆-open neighborhood U of x and each σ1σ2-open set V of Y with x ∈ F+(V ), there exists a ⋆-open set G of X such that ∅ ̸= G ⊆ U and G ⊆ Cl⋆(F+(V )); (4) for each σ1σ2-open set V of Y with x ∈ F+(V ), there exists a τ⋆-semi-open set U of X containing x such that U ⊆ Cl⋆(F+(V )); (5) x ∈ Cl⋆(Int⋆(Cl⋆(F+(V )))) for every σ1σ2-open set V of Y with x ∈ F+(V ). Proof. (1) ⇒ (2) and (2) ⇒ (3): The proof are obvious. (3) ⇒ (4): Let V be any σ1σ2-open set of Y containing F (x). By U ⋆(x) we denote the family of all ⋆-open neighborhood of x. For each U ∈ U ⋆(x), there exists a ⋆-open set GU of X such that ∅ ̸= GU ⊆ U and GU ⊆ Cl⋆(F+(V )). Put W = ∪{GU | U ∈ U ⋆(x)}. Then, W is a ⋆-open set of X, x ∈ Cl⋆(W ) and W ⊆ Cl⋆(F+(V )). Moreover, we put U0 = W ∪ {x}. Then, W ⊆ U0 ⊆ Cl⋆(W ) and U0 is a τ⋆-semi-open set of X containing x and also U0 ⊆ Cl⋆(F+(V )). (4) ⇒ (5): Let V be any σ1σ2-open set of Y containing F (x). There exists a τ⋆-semi- open set U of X containing x such that U ⊆ Cl⋆(F+(V )). Thus, x ∈ U ⊆ Cl⋆(Int⋆(U)) ⊆ Cl⋆(Int⋆(Cl⋆(F+(V )))). (5) ⇒ (1): By utilizing Lemma 2, this can be proved similarly to that of Theorem 1. Theorem 4. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7048 6 of 9 (1) F is lower τ⋆β(σ1, σ2)-continuous at a point x ∈ X; (2) for each ⋆-open neighborhood U of x and each σ1σ2-open set V of Y with x ∈ F−(V ), F−(V ) ∩ U is not τ1τ2-nowhere dense; (3) for each ⋆-open neighborhood U of x and each σ1σ2-open set V of Y with x ∈ F−(V ), there exists a ⋆-open set G of X such that ∅ ̸= G ⊆ U and G ⊆ Cl⋆(F−(V )); (4) for each σ1σ2-open set V of Y with x ∈ F−(V ), there exists a τ⋆-semi-open set U of X containing x such that U ⊆ Cl⋆(F−(V )); (5) x ∈ Cl⋆(Int⋆(Cl⋆(F−(V )))) for every σ1σ2-open set V of Y with x ∈ F−(V ). 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: (1) f is τ⋆β(σ1, σ2)-continuous at a point x ∈ X; (2) for each ⋆-open neighborhood U of x and each σ1σ2-open set V of Y containing f(x), f−1(V ) ∩ U is not τ1τ2-nowhere dense; (3) for each ⋆-open neighborhood U of x and each σ1σ2-open set V of Y containing f(x), there exists a ⋆-open set G of X such that ∅ ̸= G ⊆ U and G ⊆ Cl⋆(f−1(V )); (4) for each σ1σ2-open set V of Y containing f(x), there exists a τ⋆-semi-open set U of X containing x such that U ⊆ Cl⋆(f−1(V )); (5) x ∈ Cl⋆(Int⋆(Cl⋆(f−1(V )))) for every σ1σ2-open set V of Y containing f(x). Theorem 5. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper τ⋆β(σ1, σ2)-continuous; (2) F+(V ) is τ⋆-β-open in X for every σ1σ2-open set V of Y ; (3) F−(K) is τ⋆-β-closed in X for every σ1σ2-closed set K of Y ; (4) βCl⋆(F−(B)) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (5) Int⋆(Cl⋆(Int⋆(F−(B)))) ⊆ F−(σ1σ2Cl(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). There exists a τ⋆-β-open set U of X containing x such that F (U) ⊆ V . Thus, x ∈ U ⊆ Cl⋆(Int⋆(Cl⋆(U))) ⊆ Cl⋆(Int⋆(Cl⋆(F+(V )))) and hence F+(V ) ⊆ Cl⋆(Int⋆(Cl⋆(F+(V )))). This shows that F+(V ) is τ⋆-β-open in X. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7048 7 of 9 (2) ⇒ (3): This follows from the fact that F+(Y −B) = X − F−(B) for every subset B of Y . (3) ⇒ (4): For any subset B of Y , σ1σ2-Cl(B) is σ1σ2-closed in Y and by (3), we have F−(σ1σ2-Cl(B)) is τ⋆-β-closed in X. Thus, βCl⋆(F−(B)) ⊆ F−(σ1σ2-Cl(B)). (4) ⇒ (5): Let B be any subset of Y . By (4) and Lemma 2, Int⋆(Cl⋆(Int⋆(F−(B)))) ⊆ βCl⋆(F−(B)) ⊆ F−(σ1σ2-Cl(B)). (5) ⇒ (2): Let V be any σ1σ2-open set of Y . Then, Y − V is σ1σ2-closed in Y and by (5), X − F+(V ) = F−(Y − V ) ⊇ Int⋆(Cl⋆(Int⋆(F−(Y − V )))) = Int⋆(Cl⋆(Int⋆(X − F+(V )))) = X − Cl⋆(Int⋆(Cl⋆(F+(V )))). Thus, F+(V ) ⊆ Cl⋆(Int⋆(Cl⋆(F+(V )))) and so F+(V ) is τ⋆-β-open in X. (2) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing F (x). By (2), we have F+(V ) is τ⋆-β-open in X. Put U = F+(V ). Then, U is a τ⋆-β-open set of X containing x such that F (U) ⊆ V . This shows that F is upper 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 τ⋆β(σ1, σ2)-continuous; (2) F−(V ) is τ⋆-β-open in X for every σ1σ2-open set V of Y ; (3) F+(K) is τ⋆-β-closed in X for every σ1σ2-closed set K of Y ; (4) βCl⋆(F+(B)) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (5) Int⋆(Cl⋆(Int⋆(F+(B)))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (6) F (Int⋆(Cl⋆(Int⋆(A)))) ⊆ σ1σ2-Cl(F (A)) for every subset A of X; (7) F (βCl⋆(A)) ⊆ σ1σ2-Cl(F (A)) for every subset A of X. Proof. It is shown similarly to the proof of Theorem 5 that the statements (1), (2), (3), (4) and (5) are equivalent. We shall prove only the following implications. (5) ⇒ (6): Let A be any subset of X. By (5), we have Int⋆(Cl⋆(Int⋆(F+(F (A))))) ⊆ F+(σ1σ2-Cl(F (A))) and hence F (Int⋆(Cl⋆(Int⋆(A)))) ⊆ σ1σ2-Cl(F (A)). (6) ⇒ (7): Let A be any subset of X. By (6) and Lemma 2, we have F (βCl⋆(A)) = F (A ∪ Int⋆(Cl⋆(Int⋆(A)))) P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7048 8 of 9 = F (A) ∪ F (Int⋆(Cl⋆(Int⋆(A)))) ⊆ σ1σ2-Cl(F (A)). (7) ⇒ (3): Let K be any σ1σ2-closed set of Y . Thus by (7), F (βCl⋆(F+(K))) ⊆ σ1σ2-Cl(F (F+(K))) ⊆ σ1σ2-Cl(K) = K. Thus, βCl⋆(F+(K)) ⊆ F+(K) and hence F+(K) is τ⋆-β-closed in X. Corollary 3. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is τ⋆β(σ1, σ2)-continuous; (2) f−1(V ) is τ⋆-β-open in X for every σ1σ2-open set V of Y ; (3) f−1(K) is τ⋆-β-closed in X for every σ1σ2-closed set K of Y ; (4) βCl⋆(f−1(B)) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ; (5) Int⋆(Cl⋆(Int⋆(f−1(B)))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ; (6) f(Int⋆(Cl⋆(Int⋆(A)))) ⊆ σ1σ2-Cl(f(A)) for every subset A of X; (7) f(βCl⋆(A)) ⊆ σ1σ2-Cl(f(A)) for every subset A of X. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] M. E. Abd El-Monsef, S. N. El-Deeb, and R. A. Mahmoud. β-open sets and β- continuous mappings. 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