EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 1, 2024, 201-211 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and lower α-⋆-continuity Chawalit Boonpok1, Jeeranunt Khampakdee1,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand Abstract. Our main purpose is to introduce the concepts of upper and lower α-⋆-continuous mul- tifunctions. In particular, some characterizations of upper and lower α-⋆-continuous multifunctions are investigated. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: upper α-⋆-continuous multifunction, lower α-⋆-continuous multifunc- tion 1. Introduction The field of mathematical science called topology is concerned with all questions di- rectly or indirectly related to continuity. Continuity is an important concept for the study and investigation in topological spaces. This concept has been extended to the setting mul- tifunctions and has been generalized by weaker forms of open sets. In 1965, Nj̊astad [21] introduced a weak form of open sets called α-sets. Mashhour et al. [19] defined a function to be α-continuous if the inverse image of each open set is an α-set and obtained sev- eral characterizations of such functions. Noiri [22] investigated the relationships between α-continuous functions and several known functions, for example, almost continuous func- tions, η-continuous functions, δ-continuous functions or irresolute functions. In [23], the present author introduced the concept of almost α-continuity in topological spaces as a gen- eralization of α-continuity and almost continuity. Neubrunn [20] introduced the notion of upper (resp. lower) α-continuous multifunctions. These multifunctions are further investi- gated by the present authors [24]. Boonpok et al. [11] introduced and studied the notions of upper and lower (τ1, τ2)-precontinuous multifunctions. Viriyapong and Boonpok [26] introduced and investigated the concepts of upper and lower (τ1, τ2)α-continuous multi- functions. Moreover, several characterizations of upper and lower (τ1, τ2)δ-semicontinuous multifunctions were established in [6]. In [10], the authors investigated some character- izations of upper and lower almost weakly (τ1, τ2)-continuous multifunctions. Laprom ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4858 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), jeeranunt.k@msu.ac.th (J. Khampakdee) https://www.ejpam.com 201 © 2024 EJPAM All rights reserved. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 201-211 202 et al. [18] introduced and studied the notions of upper and lower β(τ1, τ2)-continuous multifunctions. The concept of ideal topological spaces was introduced and studied by Kuratowski [17] and Vaidyanathswamy [25]. Every topological space is an ideal topological space and all the results of ideal topological spaces are generalizations of the results established in topological spaces. In 1990, Janković and Hamlett [16] introduced the concept of I - open sets in ideal topological spaces. Abd El-Monsef et al. [14] further investigated I -open sets and I -continuous functions. Later, several authors studied ideal topological spaces giving several convenient definitions. Some authors obtained decompositions of continuity. For instance, Açikgöz et al. [1] studied the concepts of α-I -continuity and α- I -openness in ideal topological spaces and investigated several characterizations of these functions. Hatir and Noiri [15] introduced the notions of semi-I -open sets, α-I -open sets and β-I -open sets via idealization and using these sets obtained new decompositions of continuity. In [4], the author introduced and studied the notions of upper and lower ⋆- continuous multifunctions. Boonpok [7] investigated some characterizations of upper and lower β(⋆)-continuous multifunctions. Furthermore, several characterizations of almost α-⋆-continuous multifunctions and weakly α-⋆-continuous multifunctions were established in [9] and [8], respectively. In this paper, we introduce the notions of upper and lower α-⋆-continuous multifunctions. Moreover, some characterizations of upper and lower α-⋆- continuous multifunctions are discussed. 2. Preliminaries Throughout the present paper, spaces (X, τ) and (Y, σ) (or simply X and Y ) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a topological space (X, τ). The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. 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 [17], 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 [16] 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 [12] (resp. semi-I -open [15]) 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 [12] (resp. semi-I - closed [15]). For a subset A of an ideal topological space (X, τ,I ), the intersection of all semi-I -closed (resp. semi⋆-I -closed) sets containing A is called the semi-I -closure [13] (resp. semi⋆-I -closure [13]) of A and is denoted by sClI (A) (resp. s⋆ClI (A)). The union of all semi-I -open (resp. semi⋆-I -open) sets contained in A is called the semi-I -interior C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 201-211 203 (resp. semi⋆-I -interior) of A and is denoted by sIntI (A) (resp. s⋆IntI (A)). Lemma 1. For a subset A of an ideal topological space (X, τ,I ), the following properties hold: (1) If A is an open set, then s⋆ClI (A) = Int(Cl⋆(A)). (2) If A is a ⋆-open set, then sClI (A) = Int⋆(Cl(A)). Proof. (1) Suppose that A is an open set. Then, A ⊆ Int(Cl⋆(A)) and by Lemma 13(1) of [13], we have s⋆ClI (A) = A ∪ Int(Cl⋆(A)) = Int(Cl⋆(A)). (2) Suppose that A is a ⋆-open set. Then, we have A ⊆ Int⋆(Cl(A)) and by Lemma 13(2) of [13], sClI (A) = A ∪ Int⋆(Cl(A)) = Int⋆(Cl(A)). Recall that a subset A of an ideal topological space (X, τ,I ) is said to be α-⋆-closed [2] if Cl⋆(Int(Cl⋆(A))) ⊆ A. The complement of an α-⋆-closed set is said to be α-⋆-open. Proposition 1. Let (X, τ,I ) be an ideal topological space and {Aγ | γ ∈ Γ} be a family of subsets of X. If Aγ is α-⋆-closed for each γ ∈ Γ, then ∩ γ∈Γ Aγ is α-⋆-closed. Proof. Suppose that Aγ is α-⋆-closed for each γ ∈ Γ. Then, we have X − Aγ is α-⋆- open for each γ ∈ Γ. Thus, ∪ γ∈Γ (X − Aγ) = X − ∩ γ∈Γ Aγ is α-⋆-open and hence ∩ γ∈Γ Aγ is α-⋆-closed. For a subset A of an ideal topological space (X, τ,I ), the intersection of all α-⋆-closed sets containing A is called the α-⋆-closure of A and is denoted by ⋆αCl(A). The α-⋆- interior of A is defined by the union of all α-⋆-open sets contained in A and is denoted by ⋆αInt(A). Proposition 2. For a subset A of an ideal topological space (X, τ,I ), the following properties hold: (1) ⋆αCl(A) is α-⋆-closed. (2) A is α-⋆-closed if and only if A = ⋆αCl(A). Proof. (1) Follows from Proposition 1. (2) Follows from (1). Lemma 2. For a subset A of an ideal topological space (X, τ,I ), the following properties are equivalent: (1) A is α-⋆-open in X; (2) G ⊆ A ⊆ Int⋆(Cl(G)) for some ⋆-open set G; (3) G ⊆ A ⊆ sClI (G) for some ⋆-open set G; (4) A ⊆ sClI (Int⋆(A)). C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 201-211 204 Proof. (1) ⇒ (2): Suppose that A is an α-⋆-open set. Then, A ⊆ Int⋆(Cl(Int⋆(A))). Put G = Int⋆(A), then G is a ⋆-open set such that G ⊆ A ⊆ Int⋆(Cl(G)). (2) ⇒ (3): This follows from Lemma 1(2). (3) ⇒ (4): Suppose that G ⊆ A ⊆ sClI (G) for some ⋆-open set G. Then, we have G ⊆ Int⋆(A) and hence A ⊆ sClI (Int⋆(A)). (4) ⇒ (1): Suppose that A ⊆ sClI (Int⋆(A)). Since Int⋆(A) is ⋆-open in X and by Lemma 1(2), A ⊆ Int⋆(Cl(Int⋆(A))). Thus, A is α-⋆-open in X. Lemma 3. For a subset A of an ideal topological space (X, τ,I ), the following properties hold: (1) A is α-⋆-closed in X if and only if sIntI (Cl⋆(A)) ⊆ A. (2) sIntI (Cl⋆(A)) = Cl⋆(Int(Cl⋆(A))). (3) ⋆αCl(A) = A ∪ Cl⋆(Int(Cl⋆(A))). (4) ⋆αInt(A) = A ∩ Int⋆(Cl(Int⋆(A))). Proof. (1) Follows from Lemma 2. (2) Follows from Lemma 13(1) of [13]. (3) We observe that Cl⋆(Int(Cl⋆(A ∪ Cl⋆(Int(Cl⋆(A)))))) ⊆ Cl⋆(Int(Cl⋆(A ∪ (Cl⋆(A))))) ⊆ Cl⋆(Int(Cl⋆(A))) ⊆ A ∪ Cl⋆(Int(Cl⋆(A))). Thus, A ∪ Cl⋆(Int(Cl⋆(A))) is α-⋆-closed and hence ⋆αCl(A) ⊆ A ∪ Cl⋆(Int(Cl⋆(A))). On the other hand, since ⋆αCl(A) is α-⋆-closed, we have Cl⋆(Int(Cl⋆(A))) ⊆ Cl⋆(Int(Cl⋆(⋆αCl(A)))) ⊆ ⋆αCl(A) and hence A ∪ Cl⋆(Int(Cl⋆(A))) ⊆ ⋆αCl(A). Thus, ⋆αCl(A) = A ∪ Cl⋆(Int(Cl⋆(A))). (4) Since ⋆αInt(A) is α-⋆-open, we have ⋆αInt(A) ⊆ Int⋆(Cl(Int⋆(⋆αInt(A)))) ⊆ Int⋆(Cl(Int⋆(A))) and hence ⋆αInt(A) ⊆ A ∩ Int⋆(Cl(Int⋆(A))). On the other hand, we have A ∩ Int⋆(Cl(Int⋆(A))) ⊆ Int⋆(Cl(Int⋆(A))) = Int⋆(Cl(Int⋆(A) ∩ Int⋆(Cl(Int⋆(A))))) = Int⋆(Cl(Int⋆(A ∩ Int⋆(Cl(Int⋆(A)))))). Thus, A ∩ Int⋆(Cl(Int⋆(A))) is α-⋆-open and so A ∩ Int⋆(Cl(Int⋆(A))) ⊆ ⋆αInt(A). This shows that ⋆αInt(A) = A ∩ Int⋆(Cl(Int⋆(A))). C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 201-211 205 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 , following [3] 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 α-⋆-continuous multifunctions In this section, we introduce the notions of upper and lower α-⋆-continuous multifunc- tions. Moreover, several characterizations of upper and lower α-⋆-continuous multifunc- tions are discussed. Definition 1. A multifunction F : (X, τ,I ) → (Y, σ,J ) is said to be: (1) upper α-⋆-continuous at a point x of X if, for each ⋆-open set V such that F (x) ⊆ V , there exists an α-⋆-open set U of X containing x such that F (U) ⊆ V ; (2) lower α-⋆-continuous at a point x of X if, for each ⋆-open set V such that F (x) ∩ V ̸= ∅, there exists an α-⋆-open set U of X containing x such that F (z) ∩ V ̸= ∅ for each z ∈ U ; (3) upper (resp. lower) α-⋆-continuous if F is upper (resp. lower) α-⋆-continuous at each point of X. Theorem 1. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is upper α-⋆-continuous at x ∈ X; (2) x ∈ sClI (Int⋆(F+(V ))) for every α-⋆-open set V of Y containing F (x); (3) x ∈ ⋆αInt(F+(V )) for every α-⋆-open set V of Y containing F (x). Proof. (1) ⇒ (2): Let V be any ⋆-open set of Y containing F (x). Then, there exists an α-⋆-open set U of X containing x such that F (U) ⊆ V ; hence x ∈ U ⊆ F+(V ). Since U is α-⋆-open, by Lemma 2, we have x ∈ U ⊆ sClI (Int⋆(U)) ⊆ sClI (Int⋆(F+(V ))). (2) ⇒ (3): Let V be any ⋆-open set of Y containing F (x). Then by (2), we have x ∈ sClI (Int⋆(F+(V ))) and by Lemma 1(2), x ∈ Int⋆(Cl(Int⋆(F+(V )))). Thus, by Lemma 3(4), x ∈ ⋆αInt(F+(V )). (3) ⇒ (1): Let V be any ⋆-open set of Y containing F (x). By (3), x ∈ ⋆αInt(F+(V )) and so there exists an α-⋆-open set U of X containing x such that U ⊆ F+(V ); hence F (U) ⊆ V . This shows that F is upper α-⋆-continuous at x. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 201-211 206 Theorem 2. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is lower α-⋆-continuous at x ∈ X; (2) x ∈ sClI (Int⋆(F−(V ))) for every α-⋆-open set V of Y such that F (x) ∩ V ̸= ∅; (3) x ∈ ⋆αInt(F−(V )) for every α-⋆-open set V of Y such that F (x) ∩ V ̸= ∅. Proof. The proof is similar to that of Theorem 1. Definition 2. A subset N of an ideal topological space (X, τ,I ) is said to be a ⋆- neighbourhood (resp. α-⋆-neighbourhood) of x ∈ X if there exists a ⋆-open (resp. α-⋆-open) set V of X such that x ∈ V ⊆ N . Theorem 3. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is upper α-⋆-continuous; (2) F+(V ) is α-⋆-open in X for every ⋆-open set V of Y ; (3) F−(K) is α-⋆-closed in X for every ⋆-closed set K of Y ; (4) sIntI (Cl⋆(F−(B))) ⊆ F−(Cl⋆(B)) for every subset B of Y ; (5) ⋆αCl(F−(B)) ⊆ F−(Cl⋆(B)) for every subset B of Y ; (6) for each x ∈ X and each ⋆-neighbourhood V of F (x), F+(V ) is an α-⋆-neighbourhood of x; (7) for each x ∈ X and each ⋆-neighbourhood V of F (x), there exists an α-⋆-neighbourhood U of x such that F (U) ⊆ V . Proof. (1) ⇒ (2): Let V be any ⋆-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V . Since F is upper α-⋆-continuous at x, there exists an α-⋆-open set U of X containing x such that F (U) ⊆ V ; hence x ∈ U ⊆ F+(V ). By Lemma 2, x ∈ U ⊆ sClI (Int⋆(U)) ⊆ sClI (Int⋆(F+(V ))). Thus, F+(V ) ⊆ sClI (Int⋆(F+(V ))). It follows from Lemma 2 that F+(V ) is 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, Cl⋆(B) is ⋆-closed in Y and by (3), F−(Cl⋆(B)) is α-⋆-closed in X. Thus, by Lemma 3(1), sIntI (Cl⋆(F−(B))) ⊆ sIntI (Cl⋆(F−(Cl⋆(B)))) ⊆ F−(Cl⋆(B)). C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 201-211 207 (4) ⇒ (5): Let B be any subset of Y . By (4) and Lemma 3(3), ⋆αCl(F−(B)) = F−(B) ∪ sIntI (Cl⋆(F−(B))) ⊆ F−(Cl⋆(B)). (5) ⇒ (3): Let K be any ⋆-closed set of Y . By (5), we have ⋆αCl(F−(K)) ⊆ F−(Cl⋆(K)) = F−(K). This shows that F−(K) is α-⋆-closed in X. (2) ⇒ (6): Let x ∈ X and V be a ⋆-neighbourhood of F (x). Then, there exists a ⋆-open set G of Y such that F (x) ⊆ G ⊆ V . Thus, x ∈ F+(G) ⊆ F+(V ). By (2), F+(G) is α-⋆-open and hence F+(V ) is an α-⋆-neighbourhood of x. (6) ⇒ (7): Let x ∈ X and V be a ⋆-neighbourhood of F (x). By (6), we have F+(V ) is an α-⋆-neighbourhood of x. Put U = F+(V ), then U is an α-⋆-neighbourhood of x such that F (U) ⊆ V . (7) ⇒ (1): Let x ∈ X and V be any ⋆-open set of Y such that F (x) ⊆ V . Then, V is a ⋆-neighbourhood of F (x) and so there exists an α-⋆-neighbourhood U of x such that F (U) ⊆ V . Since U is an α-⋆-neighbourhood of x, there exists an α-⋆-open set G of X such that x ∈ G ⊆ U ; hence F (G) ⊆ V . This shows that F is upper α-⋆-continuous. Theorem 4. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is lower α-⋆-continuous; (2) F−(V ) is α-⋆-open in X for every ⋆-open set V of Y ; (3) F+(K) is α-⋆-closed in X for every ⋆-closed set K of Y ; (4) sIntI (Cl⋆(F+(B))) ⊆ F+(Cl⋆(B)) for every subset B of Y ; (5) ⋆αCl(F+(B)) ⊆ F+(Cl⋆(B)) for every subset B of Y ; (6) F (⋆αCl(A)) ⊆ Cl⋆(F (A)) for every subset A of X; (7) F (sIntI (Cl⋆(A))) ⊆ Cl⋆(F (A)) for every subset A of X; (8) F (Cl⋆(Int(Cl⋆(A)))) ⊆ 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+(Cl⋆(F (A))) and hence F (⋆αCl(A)) ⊆ Cl⋆(F (A)). (6) ⇒ (7): Let A be any subset of X. By (6) and Lemma 3, F (sIntI (Cl⋆(A))) = F (Cl⋆(Int(Cl⋆(A)))) C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 201-211 208 ⊆ F (A ∪ Cl⋆(Int(Cl⋆(A)))) = F (⋆αCl(A)) ⊆ Cl⋆(F (A)). (7) ⇒ (8): Let A be any subset of X. By (7) and Lemma 3(2), we have F (Cl⋆(Int(Cl⋆(A)))) = F (sIntI (Cl⋆(A))) ⊆ Cl⋆(F (A)). (8) ⇒ (1): Let x ∈ X and V be any ⋆-open set such that F (x) ∩ V ̸= ∅. Then, we have x ∈ F−(V ). We shall show that F−(V ) is α-⋆-open in X. By the hypothesis, F (Cl⋆(Int(Cl⋆(F+(Y − V ))))) ⊆ 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 α-⋆-open in X. Put U = F−(V ), then U is an α-⋆-open set of X containing x such that F (z)∩V ̸= ∅ for every z ∈ U . This shows that F is lower α-⋆-continuous. Definition 3. A function f : (X, τ,I ) → (Y, σ,J ) is called α-⋆-continuous if f−1(V ) is α-⋆-open in X for every ⋆-open set V of Y . Corollary 1. For a function f : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) f is α-⋆-continuous; (2) f−1(K) is α-⋆-closed in X for every ⋆-closed set K of Y ; (3) sIntI (Cl⋆(f−1(B))) ⊆ f−1(Cl⋆(B)) for any subset B of Y ; (4) ⋆αCl(f−1(B)) ⊆ f−1(Cl⋆(B)) for any subset B of Y ; (5) for each x ∈ X and each ⋆-neighbourhood V of f(x), f−1(V ) is an α-⋆-neighbourhood of x; (6) for each x ∈ X and each ⋆-neighbourhood V of f(x), there exists an α-⋆-neighbourhood U of x such that f(U) ⊆ V ; (7) f(⋆αCl(A)) ⊆ Cl⋆(f(A)) for every subset A of X; (8) f(sIntI (Cl⋆(A))) ⊆ Cl⋆(f(A)) for every subset A of X; (9) f(Cl⋆(Int(Cl⋆(A)))) ⊆ Cl⋆(f(A)) for every subset A of X. Definition 4. [5] A subset A of an ideal topological space (X, τ,I ) is said to be: (1) ⋆-paracompact if every cover of A by ⋆-open sets of X is refined by a cover of A which consists of ⋆-open sets of X and is ⋆-locally finite in X; C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 201-211 209 (2) ⋆-regular if for each x ∈ A and each ⋆-open set U of X containing x, there exists a ⋆-open set V of X such that x ∈ V ⊆ Cl(V ) ⊆ U . Lemma 4. [5] Let A be a subset of an ideal topological space (X, τ,I ). If A is a ⋆-regular ⋆-paracompact set of X and each ⋆-open set U containing A, then there exists a ⋆-open set V such that A ⊆ V ⊆ Cl(V ) ⊆ U . A multifunction F : (X, τ,I ) → (Y, σ,J ) is called punctually ⋆-paracompact (resp. punctually ⋆-regular) if for each x ∈ X, F (x) is ⋆-paracompact (resp. ⋆-regular). By Cl⋆α(F ) : (X, τ,I ) → (Y, σ,J ), we shall denote a multifunction defined as follows: [Cl⋆α(F )](x) = ⋆αClJ (F (x)) for each x ∈ X. Lemma 5. If F : (X, τ,I ) → (Y, σ,J ) is punctually ⋆-regular and punctually ⋆- paracompact, then [Cl⋆α(F )]+(V ) = F+(V ) for every ⋆-open set V of Y . Proof. Let V be any ⋆-open set of Y and x ∈ [Cl⋆α(F )]+(V ). Then, ⋆αClJ (F (x)) ⊆ V . Thus, F (x) ⊆ V and hence x ∈ F+(V ). Therefore, [Cl⋆α(F )]+(V ) ⊆ F+(V ). On the other hand, let V be any ⋆-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V . Since F (x) is punctually ⋆-regular and punctually ⋆-paracompact, by Lemma 4, there exists a ⋆-open set G such that F (x) ⊆ G ⊆ Cl(G) ⊆ V ; hence ⋆αClJ (F (x)) ⊆ Cl(G) ⊆ V . This shows that x ∈ [Cl⋆α(F )]+(V ). Therefore, F+(V ) ⊆ [Cl⋆α(F )]+(V ). Consequently, we obtain [Cl⋆α(F )]+(V ) = F+(V ). Theorem 5. Let F : (X, τ,I ) → (Y, σ,J ) be punctually ⋆-regular and punctually ⋆- paracompact. Then F is upper α-⋆-continuous if and only if Cl⋆α(F ) : (X, τ,I ) → (Y, σ,J ) is upper α-⋆-continuous. Proof. Suppose that F is upper α-⋆-continuous. Let x ∈ X and V be any ⋆-open set of Y such that ⋆αClJ (F (x)) ⊆ V . By Lemma 5, we have x ∈ [Cl⋆α(F )]+(V ) = F+(V ). Since F is upper α-⋆-continuous, there exists an α-⋆-open set U of X containing x such that F (U) ⊆ V . Sine F (z) is punctually ⋆-regular and punctually ⋆-paracompact for each z ∈ U , by Lemma 4, there exists a ⋆-open set G such that F (z) ⊆ G ⊆ Cl(G) ⊆ V . Thus, ⋆αClJ (F (z)) ⊆ Cl(G) ⊆ V and hence ⋆αClJ (F (U)) ⊆ V . This shows that Cl⋆α(F ) is upper α-⋆-continuous. Conversely, suppose that Cl⋆α(F ) is upper α-⋆-continuous. Let x ∈ X and V be any ⋆-open set of Y such that F (x) ⊆ V . By Lemma 5, we have x ∈ F+(V ) = [Cl⋆α(F )]+(V ) and hence ⋆αClJ (F (x)) ⊆ V . Since Cl⋆α(F ) is upper α-⋆-continuous, there exists an α-⋆- open set U of X containing x such that ⋆αClJ (F (U)) ⊆ V ; hence F (U) ⊆ V . This shows that F is upper α-⋆-continuous. Lemma 6. For a multifunction F : (X, τ,I ) → (Y, σ,J ), it follows that for each ⋆-open set V of Y [Cl⋆α(F )]−(V ) = F−(V ). REFERENCES 210 Proof. Suppose that V is any ⋆-open set of Y . Let x ∈ [Cl⋆α(F )]−(V ). Then, we have ⋆αClJ (F (x)) ∩ V ̸= ∅ and hence F (x) ∩ V ̸= ∅. Thus, x ∈ F−(V ). This shows that [Cl⋆α(F )]−(V ) ⊆ F−(V ). On the other hand, let x ∈ F−(V ). Then, ∅ ≠ F (x) ∩ V ⊆ ⋆αClJ (F (x)) ∩ V. Therefore, x ∈ [Cl⋆α(F )]−(V ). Thus, F−(V ) ⊆ [Cl⋆α(F )]−(V ) and hence [Cl⋆α(F )]−(V ) = F−(V ). Theorem 6. A multifunction F : (X, τ,I ) → (Y, σ,J ) is lower α-⋆-continuous if and only if Cl⋆α(F ) : (X, τ,I ) → (Y, σ,J ) is lower α-⋆-continuous. Proof. By utilizing Lemma 6, this can be proved similarly to that of Theorem 5. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] A. Açikgöz, T. Noiri, and Ş. Yüksel. On α-I -continuous and α-I -open functions. Acta Mathematica Hungarica, 105:27–37, 2004. [2] A. Açikgöz, T. Noiri, and Ş. Yüksel. On α-operfect sets and α-⋆-closed sets. Acta Mathematica Hungarica, 105(1-2):146–153, 2010. [3] C. Berge. Espaces topologiques fonctions multivoques. Dunod, Paris, 1959. [4] C. Boonpok. On continuous multifunctions in ideal topological spaces. Lobachevskii Journal of Mathematics, 40(1):24–35, 2019. [5] C. Boonpok. 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