EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7044 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and Lower τ ⋆α(σ1, σ2)-Continuity 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, Patt Ronald 94000, Thailand Abstract. A new class of continuous multifunctions between an ideal topological space and a bitopological space, called upper (lower) τ⋆α(σ1, σ2)-continuous multifunctions, has been de- fined and studied. Moreover, several characterizations and some properties concerning upper τ⋆α(σ1, σ2)-continuous multifunctions and lower τ⋆α(σ1, σ2)-continuous multifunctions are estab- lished. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper τ⋆α(σ1, σ2)-continuous multifunction, lower τ⋆α(σ1, σ2)-continuous multifunction 1. Introduction In 1982, Noiri [1] introduced a class of functions defined between topological spaces, namely strongly semi-continuous functions. Mashhour et al. [2] called strongly semi- continuous functions α-continuous functions and investigated some characterizations of such functions. In 1986, Neubrunn [3] extended the concept of α-continuous functions to multifunctions and presented two classes of multifunctions defined from a topological space into a topological space, called upper α-continuous multifunctions and lower α- continuous multifunctions. In 1993, Popa and Noiri [4] obtained several characterizations and some basic properties of upper α-continuous multifunctions and lower α-continuous multifunctions. On the other hand, the present author introduced and investigated four classes of multifunctions defined from an ideal topological space into an ideal topological space, namely upper ⋆-continuous multifunctions [5], lower ⋆-continuous multifunctions [5], upper α(⋆)-continuous multifunctions [6], lower α(⋆)-continuous multifunctions [6], upper β(⋆)-continuous multifunctions [7], lower β(⋆)-continuous multifunctions [7], upper ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7044 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 (4) (2025), 7044 2 of 9 sβ(⋆)-continuous multifunctions [8], lower sβ(⋆)-continuous multifunctions [8], upper α-⋆- continuous multifunctions [9], lower α-⋆-continuous multifunctions [9], ı⋆-continuous mul- tifunctions [10] and pı-continuous multifunctions [11]. Pue-on et al. [12] introduced and studied two classes of multifunctions between bitopological spaces, namely upper (τ1, τ2)- continuous multifunctions and lower (τ1, τ2)-continuous multifunctions. Klanarong et al. [13] 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. [14] 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 [15], the present authors introduced and investigated the concepts of upper (τ1, τ2)α-continuous multifunctions and lower (τ1, τ2)α-continuous multifunctions. Quite recently, Khampakdee et al. [16] presented new classes of contin- uous multifunctions defined from an ideal topological space into a bitopological space, namely upper τ⋆(σ1, σ2)-continuous multifunctions and lower τ⋆(σ1, σ2)-continuous mul- tifunctions. In this paper, we introduce the concepts of multifunctions between an ideal topological space and a bitopological space, called upper τ⋆α(σ1, σ2)-continuous multi- functions and lower τ⋆α(σ1, σ2)-continuous multifunctions. We also investigate several characterizations of upper τ⋆α(σ1, σ2)-continuous multifunctions and lower τ⋆α(σ1, σ2)- continuous multifunctions. 2. Preliminaries Throughout the present paper, spaces (X, τ1, τ2) and (Y, σ1, σ2) (or simply X and Y ) always mean bitopological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a bitopological space (X, τ1, τ2). The closure of A and the interior of A with respect to τi are denoted by τi-Cl(A) and τi-Int(A), respectively, for i = 1, 2. A subset A of a bitopological space (X, τ1, τ2) is called τ1τ2-closed [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). J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7044 3 of 9 A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [15] (resp. (τ1, τ2)s-open [18], (τ1, τ2)p-open [18], (τ1, τ2)β-open [18]) 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 τ1τ2-δ-open [19] 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 [19]. The union of all τ1τ2-δ-open sets of X contained in A is called the τ1τ2-δ-interior [19] 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 [19] 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 [15] 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 [15] 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 [15] 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 [15] of A and is denoted by (τ1, τ2)θ-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 [20], 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 [21] 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 [5] (resp. I ⋆-preopen [5], semi-I ⋆-open [22], semi-I ⋆-preopen [22]) if A = Int⋆(Cl⋆(A)) (resp. A ⊆ Int⋆(Cl⋆(A)), A ⊆ Cl⋆(Int⋆(A)), A ⊆ Cl⋆(Int⋆(Cl⋆(A)))). The complement of a R-I ⋆-open (resp. I ⋆-preopen, semi-I ⋆-open, semi-I ⋆-preopen) set is said to be R-I ⋆-closed (resp. I ⋆-preclosed, semi-I ⋆-closed, semi-I ⋆-preclosed). For a subset A of an ideal topological space (X, τ,I ), the intersection of all semi-I ⋆-closed sets containing A is called the semi-I ⋆-closure [22] of A and is denoted by sCl⋆(A) (sClI ⋆(A) [22]). The union of all semi-I ⋆-open sets contained in A is called the semi-I ⋆-interior [22] of A and is denoted by sInt⋆(A) (sIntI ⋆(A) [22]). Lemma 2. [22] For a subset A of an ideal topological space (X, τ,I ), the following properties hold: (1) sCl⋆(A) = A ∪ Int⋆(Cl⋆(A)). (2) sInt⋆(A) = A ∩ Cl⋆(Int⋆(A)). J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7044 4 of 9 A subset A of an ideal topological space (X, τ,I ) is called τ⋆-α-open [23] (α-I ⋆-open [24]) if A ⊆ Int⋆(Cl⋆(Int⋆(A))). The complement of a τ⋆-α-open set is called τ⋆-α-closed. Lemma 3. [24] For a subset A of an ideal topological space (X, τ,I ), the following properties are equivalent: (1) A is α-I ⋆-open in X. (2) G ⊆ A ⊆ Int⋆(Cl⋆(G)) for some ⋆-open set G. (3) G ⊆ A ⊆ sCl⋆(G) for some ⋆-open set G. (4) A ⊆ sCl⋆(Int⋆(A)). For a subset A of an ideal topological space (X, τ,I ), the intersection of all α-I ⋆- closed sets containing A is called the α-I ⋆-closure [24] of A and is denoted by αCl⋆(A) (αClI ⋆(A) [24]). The α-I ⋆-interior [24] of A is defined by the union of all α-I ⋆-open sets contained in A and is denoted by αInt⋆(A) (αIntI ⋆(A) [24]). Lemma 4. [24] For a subset A of an ideal topological space (X, τ,I ), the following properties hold: (1) A is α-I ⋆-closed in X if and only if sInt⋆(Cl⋆(A)) ⊆ A. (2) sInt⋆(Cl⋆(A)) = Cl⋆(Int⋆(Cl⋆(A))). (3) αCl⋆(A) = A ∪ Cl⋆(Int⋆(Cl⋆(A))). (4) αInt⋆(A) = A ∩ Int⋆(Cl⋆(Int⋆(A))). By a multifunction F : X → Y , we mean a point-to-set correspondence from X into Y , and we always assume that F (x) ̸= ∅ for all x ∈ X. For a multifunction F : X → Y , we shall denote the upper and lower inverse of a set B of Y by F+(B) and F−(B), respectively, that is, F+(B) = {x ∈ X | F (x) ⊆ B} and F−(B) = {x ∈ X | F (x) ∩ B ̸= ∅}. In particular, F−(y) = {x ∈ X | y ∈ F (x)} for each point y ∈ Y . For each A ⊆ X, F (A) = ∪x∈AF (x). 3. Upper and lower τ ⋆α(σ1, σ2)-continuous multifunctions In this section, we introduce the notions of upper τ⋆α(σ1, σ2)-continuous multifunctions and lower τ⋆α(σ1, σ2)-continuous multifunctions. Moreover, several characterizations of upper τ⋆α(σ1, σ2)-continuous multifunctions and lower τ⋆α(σ1, σ2)-continuous multifunc- tions discussed. Definition 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper τ⋆α(σ1, σ2)- continuous at a point x of X if for each σ1σ2-open set V such that F (x) ⊆ V , there exists a τ⋆-α-open set U of X containing x such that F (U) ⊆ V . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7044 5 of 9 is said to be upper τ⋆α(σ1, σ2)-continuous if F is upper τ⋆α(σ1, σ2)-continuous at each point of X. Theorem 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper τ⋆α(σ1, σ2)-continuous at x ∈ X; (2) x ∈ sCl⋆(Int⋆(F+(V ))) for every σ1σ2-open set V of Y containing F (x); (3) x ∈ αInt⋆(F+(V )) for every σ1σ2-open set V of Y containing F (x). Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing F (x). Then, there exists a τ⋆-α-open set U of X containing x such that F (U) ⊆ V ; hence x ∈ U ⊆ F+(V ). Since U is τ⋆-α-open, by Lemma 3 we have x ∈ U ⊆ sCl⋆(Int⋆(U)) ⊆ sCl⋆(Int⋆(F+(V ))). (2) ⇒ (3): Let V be any σ1σ2-open set of Y containing F (x). Then by (2), we have x ∈ sCl⋆(Int⋆(F+(V ))) and by Lemma 2, x ∈ Int⋆(Cl⋆(Int⋆(F+(V )))). Therefore, x ∈ αInt⋆(F+(V )) by Lemma 4. (3) ⇒ (1): Let V be any σ1σ2-open set of Y containing F (x). By (3), we have x ∈ αInt⋆(F+(V )) and so there exists a τ⋆-α-open set U of X containing x such that U ⊆ F+(V ); hence F (U) ⊆ V . This shows that F is upper τ⋆α(σ1, σ2)-continuous at x. Definition 2. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower τ⋆α(σ1, σ2)- continuous at a point x of X if for each σ1σ2-open set V such that F (x) ∩ V ̸= ∅, there exists a τ⋆-α-open set U containing x such that F (z) ∩ V ̸= ∅ for every z ∈ U . A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower τ⋆α(σ1, σ2)-continuous if F is lower τ⋆α(σ1, σ2)-continuous at each point of X. Theorem 2. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower τ⋆α(σ1, σ2)-continuous at x ∈ X; (2) x ∈ sCl⋆(Int⋆(F−(V ))) for every σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅; (3) x ∈ αInt⋆(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 subset N of an ideal topological space (X, τ,I ) is said to be a τ⋆-α- neighbourhood of x ∈ X if there exists a τ⋆-α-open set V of X such that x ∈ V ⊆ N . Theorem 3. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper τ⋆α(σ1, σ2)-continuous; (2) F+(V ) is τ⋆-α-open in X for every σ1σ2-open set V of Y ; J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7044 6 of 9 (3) F−(K) is τ⋆-α-closed in X for every σ1σ2-closed set K of Y ; (4) sInt⋆(Cl⋆(F−(B))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (5) αCl⋆(F−((B)) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (6) for each x ∈ X and each σ1σ2-neighbourhood V of F (x), F+(V ) is a τ⋆-α-neighbourhood of x; (7) for each x ∈ X and each σ1σ2-neighbourhood V of F (x), there exists a τ⋆-α- neighbourhood U of x such that F (U) ⊆ V . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V . Since F is upper τ⋆α(σ1, σ2)-continuous at x, there exists a τ⋆-α-open set U of X containing x such that F (U) ⊆ V ; hence x ∈ U ⊆ F+(V ). By Lemma 3, we have x ∈ U ⊆ sCl⋆(Int⋆(U)) ⊆ sCl⋆(Int⋆(F+(V ))). Thus, F+(V ) ⊆ sCl⋆(Int⋆(F+(V ))). It follows from Lemma 3 that F+(V ) is τ⋆-α-open in X. (2) ⇔ (3): This follows from the fact that F+(Y −B) = X −F−(B) for any 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), F−(σ1σ2-Cl(B)) is τ⋆-α-closed in X. By Lemma 4, we have sInt⋆(Cl⋆(F−(B))) ⊆ sInt⋆(Cl⋆(F−(Cl⋆(B)))) ⊆ F−(σ1σ2-Cl(B)). (4) ⇒ (5): Let B be any subset of Y . By (4) and Lemma 4, αCl⋆(F−(B)) = F−(B) ∪ sInt⋆(Cl⋆(F−(B))) ⊆ F−(σ1σ2-Cl(B)). (5) ⇒ (3): Let K be any σ1σ2-closed set of Y . Thus by (5), we have αCl⋆(F−(K)) ⊆ F−(σ1σ2-Cl(K)) = F−(K) and hence F−(K) is τ⋆-α-closed in X. (2) ⇒ (6): Let x ∈ X and V be a σ1σ2-neighbourhood of F (x). Then, there exists a σ1σ2-open set G of Y such that F (x) ⊆ G ⊆ V . Thus, x ∈ F+(G) ⊆ F+(V ). By (2), F+(G) is τ⋆-α-open in X and so F+(V ) is a τ⋆-α-neighbourhood of x. (6) ⇒ (7): Let x ∈ X and V be a ⋆-neighbourhood of F (x). By (6), we have F+(V ) is a τ⋆-α-neighbourhood of x. Put U = F+(V ), then U is a τ⋆-α-neighbourhood of x such that F (U) ⊆ V . (7) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that F (x) ⊆ V . Then, V is a σ1σ2-neighbourhood of F (x) and so there exists a τ⋆-α-neighbourhood U of x such that F (U) ⊆ V . Since U is a τ⋆-α-neighbourhood of x, there exists a τ⋆-α-open set G of X such that x ∈ G ⊆ U ; hence F (G) ⊆ V . This shows that F is upper τ⋆α(σ1, σ2)-continuous. Theorem 4. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7044 7 of 9 (1) F is lower τ⋆α(σ1, σ2)-continuous; (2) F−(V ) is τ⋆-α-open in X for every σ1σ2-open set V of Y ; (3) F+(K) is τ⋆-α-closed in X for every σ1σ2-closed set K of Y ; (4) sInt⋆(Cl⋆(F+(B))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (5) αCl⋆(F+(B)) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (6) F (αCl⋆(A)) ⊆ σ1σ2-Cl(F (A)) for every subset A of X; (7) F (sInt⋆(Cl⋆(A))) ⊆ σ1σ2-Cl(F (A)) for every subset A of X; (8) F (Cl⋆(Int⋆(Cl⋆(A)))) ⊆ σ1σ2-Cl(F (A)) for every subset A of X. Proof. The proofs except for the following are similar to the proof of Theorem 3. (5) ⇒ (6): Let A be any subset of X. Since A ⊆ F+(F (A)), we have αCl⋆(A) ⊆ αCl⋆(F+(F (A))) ⊆ F+(σ1σ2-Cl(F (A))) and so F (αCl⋆(A)) ⊆ σ1σ2-Cl(F (A)). (6) ⇒ (7): Let A be any subset of X. By (6) and Lemma 4, F (sInt⋆(Cl⋆(A))) = F (Cl⋆(Int⋆(Cl⋆(A)))) ⊆ F (A ∪ Cl⋆(Int⋆(Cl⋆(A)))) = F (αCl⋆(A)) ⊆ σ1σ2-Cl(F (A)). (7) ⇒ (8): Let A be any subset of X. By (7) and Lemma 4, we have F (Cl⋆(Int⋆(Cl⋆(A)))) = F (sInt⋆(Cl⋆(A))) ⊆ σ1σ2-Cl(F (A)). (8) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set such that F (x) ∩ V ̸= ∅. Then, we have x ∈ F−(V ). We shall show that F−(V ) is τ⋆-α-open in X. By the hypoth- esis, F (Cl⋆(Int⋆(Cl⋆(F+(Y − V ))))) ⊆ σ1σ2-Cl(F (F+(Y − V ))) ⊆ Y − V and hence Cl⋆(Int⋆(Cl⋆(F+(Y − V )))) ⊆ F+(Y − V ) = X − F−(V ). Thus, F−(V ) ⊆ Int⋆(Cl⋆(Int⋆(F−(V )))) and so F−(V ) is τ⋆-α-open in X. Put U = F−(V ). Then, U is a τ⋆-α-open set of X containing x such that F (z) ∩ V ̸= ∅ for every z ∈ U . This shows that F is lower τ⋆α(σ1, σ2)-continuous. Definition 4. A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be τ⋆α(σ1, σ2)-continuous if for every σ1σ2-open set V of Y , f−1(V ) is τ⋆-α-open in X. J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7044 8 of 9 Corollary 1. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is τ⋆α(σ1, σ2)-continuous; (2) f−1(K) is τ⋆-α-closed in X for every σ1σ2-closed set K of Y ; (3) sInt⋆(Cl⋆(f−1(B))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ; (4) αCl⋆(f−1(B)) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ; (5) for each x ∈ X and each σ1σ2-neighbourhood V of f(x), f−1(V ) is a τ⋆-α-neighbourhood of x; (6) for each x ∈ X and each σ1σ2-neighbourhood V of f(x), there exists a τ⋆-α- neighbourhood U of x such that f(U) ⊆ V ; (7) f(αCl⋆(A)) ⊆ σ1σ2-Cl(f(A)) for every subset A of X; (8) f(sInt⋆(Cl⋆(A))) ⊆ σ1σ2-Cl(f(A)) for every subset A of X; (9) f(Cl⋆(Int⋆(Cl⋆(A)))) ⊆ σ1σ2-Cl(f(A)) for every subset A of X. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] T. Noiri. A function which preserves connected spaces. Časopis Pěstování Matem- atiky, 107:393–396, 1982. [2] A. S. Mashhour, I. A. Hasanein, and S. N. 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