EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1634-1646 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and lower sβ(⋆)-continuous multifunctions Chawalit Boonpok1, Prapart Pue-on1,∗ 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 sβ(⋆)-continuous multifunctions. In particular, some characterizations of upper and lower sβ(⋆)-continuous multi- functions are investigated. Moreover, the relationships between sβ(⋆)-continuous multifunctions and almost sβ(⋆)-continuous multifunctions are discussed. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper sβ(⋆)-continuous multifunction, lower sβ(⋆)-continuous mul- tifunction 1. Introduction The concept of semi-continuity was first introduced by Levine [13]. In 1982, Mash- hour et al. [15] introduced and investigated the notion of precontinuous functions. Abd El-Monsef et al. [7] introduced the notion of β-continuous functions as a generalization of semi-continuous functions [13] and precontinuous functions [15]. Borśık and Doboš [4] introduced the notion of almost quasi-continuity which is weaker than that of quasi- continuity [14] and investigated a decomposition theorem of quasi-continuity. Popa and Noiri [17] investigated some characterizations of β-continuity and showed that almost quasi-continuity is equivalent to β-continuity. In 1993, Popa and Noiri [18] extended the concept of β-continuous functions to multifunctions and introduced the notions of upper and lower β-continuous multifunctions. Moreover, the relationships between β-continuous mulfunctions and quasi-continuous multifunctions were established in [17]. Noiri and Popa [16] introduced and studied the concepts of upper and lower almost β-continuous mul- functions. In 2003, Hatir et al. [8] introduced and investigated the notions of strong β-I -open sets and strongly β-I -continuous functions in ideal topological spaces. Hatir et al. [9] investigated further properties of strong β-I -open sets and strongly β-I - continuous functions. In 2019, Boonpok [2] introduced and studied the concepts of upper and lower ⋆-continuous multifunctions in ideal topological spaces. In [3], the present ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4732 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), prapart.p@msu.ac.th (P. Pue-on) https://www.ejpam.com 1634 © 2023 EJPAM All rights reserved. C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (3) (2023), 1634-1646 1635 author introduced and investigated the notions of upper and lower β(⋆)-continuous mul- tifunctions. The purpose of the present paper is to introduce the notions of upper and lower sβ(⋆)-continuous multifunctions. Furthermore, several characterizations of upper and lower sβ(⋆)-continuous multifunctions are investigated. Moreover, the relationships between sβ(⋆)-continuous multifunctions and almost sβ(⋆)-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 [12], 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 [11] if A⋆ ⊆ A. The interior of a subset A in (X, τ⋆(I )) is denoted by 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 , following [1] 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). Lemma 1. For a subset A of an ideal topological space (X, τ,I ), the following properties hold: (1) If V ∈ τ , then V ∩ Cl⋆(A) ⊆ Cl⋆(V ∩A) [9]. (2) If F is closed in X, then Int⋆(A ∪ F ) ⊆ Int⋆(A) ∪ F . A subset A of an ideal topological space (X, τ,I ) is called semi-I -open [10] (resp. pre⋆I -open [5], strong β-I -open [8]) if A ⊆ Cl⋆(Int(A)) (resp. A ⊆ Int⋆(Cl(A)), A ⊆ Cl⋆(Int(Cl⋆(A)))). The complement of a semi-I -open (resp. pre⋆I -open, strong β-I - open) set is called semi-I -closed [10] (resp. pre⋆I -closed [5], strong β-I -closed [8]). The strong β-I -closure (resp. semi-I -closure) [6] of a subset A of an ideal topological space (X, τ,I ), denoted by sβClI (A) (resp. sClI (A)), is defined by the intersection of all C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (3) (2023), 1634-1646 1636 strong β-I -closed (resp. semi-I -closed) sets of X containing A. Let A be a subset of an ideal topological space (X, τ,I ). The union of all strong β-I -open sets of X contained in A is called the strong β-I -interior of A and is denoted by sβIntI (A). Lemma 2. For a subset A of an ideal topological space (X, τ,I ), the following properties hold: (1) sClI (A) = A ∪ Int⋆(Cl(A)) [6]. (2) sβClI (A) = A ∪ Int⋆(Cl(Int⋆(A))) [6]. (3) sβIntI (A) = A ∩ Cl⋆(Int(Cl⋆(A))). Proof. (3) We observe that A ∩ Cl⋆(Int(Cl⋆(A))) ⊆ Cl⋆(Int(Cl⋆(A))) = Cl⋆(Int(Cl⋆(A) ∩ Int(Cl⋆(A)))) ⊆ Cl⋆(Int(Cl⋆(A ∩ Int(Cl⋆(A))))) ⊆ Cl⋆(Int(Cl⋆(A ∩ Cl⋆(Int(Cl⋆(A)))))). Thus, A∩Cl⋆(Int(Cl⋆(A))) is strong β-I -open and so A∩Cl⋆(Int(Cl⋆(A))) ⊆ sβIntI (A). On the other hand, since sβIntI (A) is strong-β-I -open, we have sβIntI (A) ⊆ Cl⋆(Int(Cl⋆(sβIntI (A)))) ⊆ Cl⋆(Int(Cl⋆(A))) and hence sβIntI (A) ⊆ A ∩ Cl⋆(Int(Cl⋆(A))). Thus, sβIntI (A) = A ∩ Cl⋆(Int(Cl⋆(A))). Lemma 3. Let V be a subset of an ideal topological space (X, τ,I ). If V is ⋆-open, then sClI (V ) = Int⋆(Cl(V )). Proof. Suppose that V is ⋆-open. Then, we have V ⊆ Int⋆(Cl(V )), by Lemma 2, sClI (V ) = V ∪ Int⋆(Cl(V )) = Int⋆(Cl(V )). Lemma 4. Let A be a subset of an ideal topological space (X, τ,I ) and x ∈ X. Then, x ∈ sβClI (A) if and only if U ∩A ̸= ∅ for every strong β-I -open set U containing x. Proof. Let x ∈ sβClI (A). Suppose that U ∩A = ∅ for some strong β-I -open set U of X containing x. Then, A ⊆ X−U and X−U is strong β-I -closed. Since x ∈ sβClI (A), we have x ∈ sβClI (X − U) = X − U ; hence x ̸∈ U , which is a contradiction that x ∈ U . Thus, U ∩A ̸= ∅ for every strong β-I -open set U containing x. Conversely, assume that U ∩ A ̸= ∅ for every strong β-I -open set U of X containing x. We shall show that x ∈ sβClI (A). Suppose that x ̸∈ sβClI (A). Then, there exists a strong β-I -closed set F such that A ⊆ F and x ̸∈ F . Thus, X − F is a strong β-I -open set containing x such that (X − F ) ∩ A = ∅. This a contradiction to U ∩ A ̸= ∅; hence x ∈ sβClI (A). C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (3) (2023), 1634-1646 1637 Lemma 5. For a subset A of an ideal topological space (X, τ,I ), the following properties are hold: (1) X − sβClI (A) = sβIntI (X −A). (2) X − sβIntI (A) = sβClI (X −A). Proof. (1) Let x ∈ X − sβClI (A). Then, x ̸∈ sβClI (A) and there exists a strong β-I -open set V of X containing x such that V ∩ A = ∅. Thus, V ⊆ X − A and hence x ∈ sβIntI (X−A). This shows that X−sβClI (A) ⊆ sβIntI (X−A). On the other hand, let x ∈ sβIntI (X−A). Then, there exists a strong β-I -open set V of X containing x such that V ⊆ X−A and so V ∩A = ∅. By Lemma 4, x ̸∈ sβClI (A); hence x ∈ X−sβClI (A). Thus, sβIntI (X −A) ⊆ X − sβClI (A) and so X − sβClI (A) = sβIntI (X −A). (2) This follows from (1). 3. Upper and lower sβ(⋆)-continuous multifunctions In this section, we introduce the notions of upper and lower sβ(⋆)-continuous multi- functions. Moreover, several characterizations of upper and lower sβ(⋆)-continuous mul- tifunctions are discussed. Definition 1. A multifunction F : (X, τ,I ) → (Y, σ,J ) is said to be: (1) upper sβ(⋆)-continuous at a point x ∈ X if, for each ⋆-open set V of Y containing F (x), there exists a strong β-I -open set U of X containing x such that F (U) ⊆ V ; (2) lower sβ(⋆)-continuous at a point x ∈ X if, for each ⋆-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a strong β-I -open set U of X containing x such that F (z) ∩ V ̸= ∅ for every z ∈ U ; (3) upper (resp. lower) sβ(⋆)-continuous if F has this property at each point of X. Theorem 1. A multifunction F : (X, τ,I ) → (Y, σ,J ) is upper sβ(⋆)-continuous at x ∈ X if and only if x ∈ sβIntI (F+(V )) for every ⋆-open set V of Y containing F (x). Proof. Let V be any ⋆-open set of Y containing F (x). Then, there exists a strong β-I -open set U of X containing x such that F (U) ⊆ V . Then, U ⊆ F+(V )). Since U is strong β-I -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 )))) = sβIntI (F+(V )). Conversely, let V be any ⋆-open set of Y containing F (x). By (2), x ∈ sβIntI (F+(V )) and so there exists a strong β-I -open set U of X containing x such that U ⊆ F+(V ); hence F (U) ⊆ V . This shows that F is upper sβ(⋆)-continuous at x. Theorem 2. A multifunction F : (X, τ,I ) → (Y, σ,J ) is lower sβ(⋆)-continuous at x ∈ X if and only if x ∈ sβIntI (F−(V )) for every ⋆-open set V of Y such that F (x)∩V ̸= ∅. C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (3) (2023), 1634-1646 1638 Proof. The proof is similar to that of Theorem 1. Definition 2. A function f : (X, τ,I ) → (Y, σ,J ) is called sβ(⋆)-continuous at a point x ∈ X if, for each ⋆-open set V of Y containing f(x), there exists a strong β-I -open set U of X containing x such that f(U) ⊆ V . A function f : (X, τ,I ) → (Y, σ,J ) is called sβ(⋆)-continuous if f has this property at each point of X. Corollary 1. A function f : (X, τ,I ) → (Y, σ,J ) is sβ(⋆)-continuous at x ∈ X if and only if x ∈ sβIntI (f−1(V )) for every ⋆-open set V of Y containing f(x). Theorem 3. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is upper sβ(⋆)-continuous; (2) F+(V ) is strong β-I -open in X for every ⋆-open set V of Y ; (3) F−(K) is strong β-I -closed in X for every ⋆-closed set K of Y ; (4) sβClI (F−(B)) ⊆ F−(Cl⋆(B)) for every subset B of Y ; (5) Int⋆(Cl(Int⋆(F−(B)))) ⊆ F−(Cl⋆(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let V be any ⋆-open set of Y and x ∈ F+(V ). There exists a strong β-I -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 strong β-I -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): For any subset B of Y , Cl⋆(B) is ⋆-closed in Y and by (3), we have F−(Cl⋆(B)) is strong β-I -closed in X. Thus, sβClI (F−(B)) ⊆ F−(Cl⋆(B)). (4) ⇒ (5): Let B be any subset of Y . By (4) and Lemma 2, Int⋆(Cl(Int⋆(F−(B)))) ⊆ sβClI (F−(B)) ⊆ F−(Cl⋆(B)). (5) ⇒ (2): Let V be any ⋆-open set of Y . Then, Y − V is ⋆-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 strong β-I -open in X. (2) ⇒ (1): Let x ∈ X and V be any ⋆-open set of Y containing F (x). By (2), we have F+(V ) is strong β-I -open in X. Put U = F+(V ). Then, U is a strong β-I -open set of X containing x such that F (U) ⊆ V . This shows that F is upper sβ(⋆)-continuous. C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (3) (2023), 1634-1646 1639 Theorem 4. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is lower sβ(⋆)-continuous; (2) F−(V ) is strong β-I -open in X for every ⋆-open set V of Y ; (3) F+(K) is strong β-I -closed in X for every ⋆-closed set K of Y ; (4) sβClI (F+(B)) ⊆ F+(Cl⋆(B)) for every subset B of Y ; (5) Int⋆(Cl(Int⋆(F+(B)))) ⊆ F+(Cl⋆(B)) for every subset B of Y ; (6) F (Int⋆(Cl(Int⋆(A)))) ⊆ Cl⋆(F (A)) for every subset A of X; (7) F (sβClI (A)) ⊆ Cl⋆(F (A)) for every subset A of X. Proof. It is shown similarly to the proof of Theorem 3 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+(Cl⋆(F (A))) and hence F (Int⋆(Cl(Int⋆(A)))) ⊆ Cl⋆(F (A)). (6) ⇒ (7): Let A be any subset of X. By (6) and Lemma 2, we have F (sβClI (A)) = F (A ∪ Int⋆(Cl(Int⋆(A)))) = F (A) ∪ F (Int⋆(Cl(Int⋆(A)))) ⊆ Cl⋆(F (A)). (7) ⇒ (3): Let K be any ⋆-closed set of Y . By (7), F (sβClI (F+(K))) ⊆ Cl⋆(F (F+(K))) ⊆ Cl⋆(K) = K. Thus, sβClI (F+(K)) ⊆ F+(K) and hence F+(K) is strong β-I -closed in X. Corollary 2. For a function f : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) f is sβ(⋆)-continuous; (2) f−1(V ) is strong β-I -open in X for every ⋆-open set V of Y ; (3) f−1(K) is strong β-I -closed in X for every ⋆-closed set K of Y ; (4) sβClI (f−1(B)) ⊆ f−1(Cl⋆(B)) for every subset B of Y ; (5) Int⋆(Cl(Int⋆(f−1(B)))) ⊆ f−1(Cl⋆(B)) for every subset B of Y ; (6) f(Int⋆(Cl(Int⋆(A)))) ⊆ Cl⋆(f(A)) for every subset A of X; (7) f(sβClI (A)) ⊆ Cl⋆(f(A)) for every subset A of X. C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (3) (2023), 1634-1646 1640 4. Upper and lower almost sβ(⋆)-continuous multifunctions We begin this section by introducing the notions of upper and lower almost sβ(⋆)- continuous multifunctions. Definition 3. A multifunction F : (X, τ,I ) → (Y, σ,J ) is said to be: (1) upper almost sβ(⋆)-continuous at a point x ∈ X if, for each ⋆-open set V of Y containing F (x), there exists a strong β-I -open set U of X containing x such that F (U) ⊆ Int⋆(Cl(V )); (2) lower almost sβ(⋆)-continuous at a point x ∈ X if, for each ⋆-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a strong β-I -open set U of X containing x such that F (z) ∩ Int⋆(Cl(V )) ̸= ∅ for every z ∈ U ; (3) upper (resp. lower) almost β(⋆)-continuous if F has this property at each point of X. Remark 1. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following implication holds: upper sβ(⋆)-continuity ⇒ upper almost sβ(⋆)-continuity. The converse of the implication is not true in general. We give an example for the implication as follows. Example 1. Let X = {1, 2, 3} with a topology τ = {∅, X} and an ideal I = {∅}. Let Y = {a, b, c} with a topology σ = {∅, {b}, Y } and an ideal J = {∅, {b}}. A multifunction F : (X, τ,I ) → (Y, σ,J ) is defined as follows: F (1) = {b} and F (2) = F (3) = {a, c}. Then, F is upper almost sβ(⋆)-continuous but F is not upper sβ(⋆)-continuous, since {a, c} is ⋆-open in Y but F+({a, c}) is not strong β-I -open in X. Theorem 5. A multifunction F : (X, τ,I ) → (Y, σ,J ) is upper almost sβ(⋆)-continuous at x ∈ X if and only if x ∈ sβIntI (F+(sClJ (V ))) for every ⋆-open set V of Y containing F (x). Proof. Let V be any ⋆-open set of Y containing F (x). Then, there exists a strong β-I -open set U of X containing x such that F (U) ⊆ Int⋆(Cl(V )) = sClJ (V ); hence U ⊆ F+(sClJ (V )). Since U is strong β-I -open, we have x ∈ U ⊆ Cl⋆(Int(Cl⋆(U))) ⊆ Cl⋆(Int(Cl⋆(F+(sClJ (V ))))). Since x ∈ F+(V ) ⊆ F+(sClJ (V )) and by Lemma 2, x ∈ F+(sClJ (V )) ∩ Cl⋆(Int(Cl⋆(sClJ (V )))) = sβIntI (F+(sClJ (V ))). Conversely, let V be any ⋆-open set of Y containing F (x). Then, we have x ∈ sβIntI (F+(sClJ (V ))) C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (3) (2023), 1634-1646 1641 and so there exists a strong β-I -open set U ofX containing x such that U ⊆ F+(sClJ (V )); hence F (U) ⊆ sClJ (V ) = Int⋆(Cl(V )). This shows that F is upper almost β(⋆)- continuous at x. Theorem 6. A multifunction F : (X, τ,I ) → (Y, σ,J ) is lower almost sβ(⋆)-continuous at x ∈ X if and only if x ∈ sβIntI (F−(sClJ (V ))) for every ⋆-open set V of Y such that F (x) ∩ V ̸= ∅. Proof. The proof is similar to that of Theorem 5. Definition 4. A function f : (X, τ,I ) → (Y, σ,J ) is called almost sβ(⋆)-continuous at a point x ∈ X if, for each ⋆-open set V of Y containing f(x), there exists a strong β-I -open set U of X containing x such that f(U) ⊆ Int⋆(Cl(V )). A function f : (X, τ,I ) → (Y, σ,J ) is called almost β(⋆)-continuous if f has this property at each point of X. Corollary 3. A function f : (X, τ,I ) → (Y, σ,J ) is almost sβ(⋆)-continuous at x ∈ X if and only if x ∈ sβIntI (f−1(sClJ (V ))) for every ⋆-open set V of Y containing f(x). Recall that a subset A of an ideal topological space (X, τ,I ) is said to be R⋆-I -open [2] if A = Int⋆(Cl(A)). The complement of a R⋆-I -open set is said to be R⋆-I -closed. Theorem 7. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is upper almost sβ(⋆)-continuous; (2) for each x ∈ X and each ⋆-open set V of Y containing F (x), there exists a strong β-I -open set U of X containing x such that F (U) ⊆ sClJ (V ); (3) for each x ∈ X and each R⋆-J -open set V of Y containing F (x), there exists a strong β-I -open set U of X containing x such that F (U) ⊆ V ; (4) F+(V ) is strong β-I -open in X for every R⋆-J -open set V of Y ; (5) F−(K) is strong β-I -closed in X for every R⋆-J -closed set K of Y ; (6) F+(V ) ⊆ sβIntI (F+(sClJ (V ))) for every ⋆-open set V of Y ; (7) sβClI (F−(sIntJ (K))) ⊆ F−(K) for every ⋆-closed set K of Y ; (8) sβClI (F−(Cl⋆(Int(K)))) ⊆ F−(K) for every ⋆-closed set K of Y ; (9) sβClI (F−(Cl⋆(Int(Cl⋆(B))))) ⊆ F−(Cl⋆(B)) for every subset B of Y ; (10) Int⋆(Cl( Int⋆(F−(Cl⋆(Int(K)))))) ⊆ F−(K) for every ⋆-closed set K of Y ; C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (3) (2023), 1634-1646 1642 (11) Int⋆(Cl(Int⋆(F−(sIntJ (K))))) ⊆ F−(K) for every ⋆-closed set K of Y ; (12) F+(V ) ⊆ Cl⋆(Int(Cl⋆(F+(sClJ (V ))))) for every ⋆-open set V of Y . Proof. (1) ⇒ (2) and (2) ⇒ (3): The proofs are obvious. (3) ⇒ (4): Let V be any ⋆-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V and so there exists a strong β-I -open set Ux of X containing x such that F (Ux) ⊆ V . Thus, x ∈ Ux ⊆ F+(V ) and hence F+(V ) = ∪x∈F+(V )Ux. This shows that F+(V ) is strong β-I -open in X. (4) ⇒ (5): This follows from the fact that F+(Y −B) = Y − F−(B) for every subset B of Y . (5) ⇒ (6): Let V be any ⋆-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V ⊆ sClJ (V ) and hence x ∈ F+(sClJ (V )) = X − F−(Y − sClJ (V )). Since Y − sClJ (V ) is R⋆-J - closed, we have F−(Y − sClJ (V )) is strong β-I -closed in X. Thus, F+(sClJ (V )) is a strong β-I -open set of X containing x and so x ∈ sβIntI (F+(sClJ (V ))). This shows that F+(V ) ⊆ sβIntI (F+(sClJ (V ))). (6) ⇒ (7): Let K be any ⋆-closed set of Y . Then, since Y −K is ⋆-open and by (6), X − F−(K) = F+(Y −K) ⊆ sβIntI (F+(sClJ (Y −K))) = sβIntI (F+(Y − sIntJ (K))) = sβIntI (X − F−(sIntJ (K))) = X − sβClI (F−(sIntJ (K))). Thus, sβClI (F−(sIntJ (K))) ⊆ F−(K). (7) ⇒ (8): The proof is obvious since sIntJ (K) = Cl⋆(Int(K)) for every ⋆-closed set K of Y . (8) ⇒ (9): The proof is obvious. (9) ⇒ (10): By (9) and Lemma 2, Int⋆(Cl(Int⋆(F−(Cl⋆(Int⋆(K)))))) ⊆ sβClI (F−(Cl⋆(Int(K)))) ⊆ sβClI (F−(Cl⋆(Int(Cl⋆(K))))) ⊆ F−(Cl⋆(K)) = F−(K). (10) ⇒ (11): The proof is obvious since sIntJ (K) = Cl⋆(Int(K)) for every ⋆-closed set K of Y . (11) ⇒ (12): Let V be any ⋆-open set of Y . Then, Y −V is ⋆-closed in Y and by (11), Int⋆(Cl(Int⋆(F−(sIntJ (Y − V ))))) ⊆ F−(Y − V ) = X − F+(V ). Moreover, we have Int⋆(Cl(Int⋆(F−(sIntJ (Y − V ))))) = Int⋆(Cl(Int⋆(F−(Y − sClJ (V ))))) = Int⋆(Cl(Int⋆(X − F+(sClJ (V ))))) C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (3) (2023), 1634-1646 1643 = X − Cl⋆(Int(Cl⋆(F+(sClJ (V ))))). Thus, F+(V ) ⊆ Cl⋆(Int(Cl⋆(F+(sClJ (V ))))). (12) ⇒ (1): Let x be any point of X and V be any ⋆-open set of Y containing F (x). Then, we have x ∈ F+(V ) ⊆ Cl⋆(Int(Cl⋆(F+(sClJ (V ))))) and hence x ∈ sβIntI (F+(sClJ (V ))). Thus, F is upper almost sβ(⋆)-continuous at x by Theorem 5. Theorem 8. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is lower almost sβ(⋆)-continuous; (2) for each x ∈ X and each ⋆-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a strong β-I -open set U of X containing x such that U ⊆ F−(sClJ (V )); (3) for each x ∈ X and each R⋆-J -open set V of Y such that F (x)∩V ̸= ∅, there exists a strong β-I -open set U of X containing x such that U ⊆ F−(V ); (4) F−(V ) is strong β-I -open in X for every R⋆-J -open set V of Y ; (5) F+(K) is strong β-I -closed in X for every R⋆-J -closed set K of Y ; (6) F−(V ) ⊆ sβIntI (F−(sClJ (V ))) for every ⋆-open set V of Y ; (7) sβClI (F+(sIntJ (K))) ⊆ F+(K) for every ⋆-closed set K of Y ; (8) sβClI (F+(Cl⋆(Int(K)))) ⊆ F+(K) for every ⋆-closed set K of Y ; (9) sβClI (F+(Cl⋆(Int(Cl⋆(B))))) ⊆ F+(Cl⋆(B)) for every subset B of Y ; (10) Int⋆(Cl( Int⋆(F+(Cl⋆(Int(K)))))) ⊆ F+(K) for every ⋆-closed set K of Y ; (11) Int⋆(Cl( Int⋆(F+(sIntJ (K))))) ⊆ F+(K) for every ⋆-closed set K of Y ; (12) F−(V ) ⊆ Cl⋆( Int(Cl⋆(F−(sClJ (V ))))) for every ⋆-open set V of Y . Proof. The proof is similar to that of Theorem 7. Corollary 4. For a function f : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) f is almost sβ(⋆)-continuous; (2) for each x ∈ X and each ⋆-open set V of Y containing f(x), there exists a strong β-I -open set U of X containing x such that f(U) ⊆ sClJ (V ); C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (3) (2023), 1634-1646 1644 (3) for each x ∈ X and each R⋆-J -open set V of Y containing f(x), there exists a strong β-I -open set U of X containing x such that f(U) ⊆ V ; (4) f−1(V ) is strong β-I -open in X for every R⋆-J -open set V of Y ; (5) f−1(K) is strong β-I -closed in X for every R⋆-J -closed set K of Y ; (6) f−1(V ) ⊆ sβIntI (f−1(sClJ (V ))) for every ⋆-open set V of Y ; (7) sβClI (f−1(sIntJ (K))) ⊆ f−1(K) for every ⋆-closed set K of Y ; (8) sβClI (f−1(Cl⋆(Int(K)))) ⊆ f−1(K) for every ⋆-closed set K of Y ; (9) sβClI (f−1(Cl⋆(Int(Cl⋆(B))))) ⊆ f−1(Cl⋆(B)) for every subset B of Y ; (10) Int⋆(Cl( Int⋆(f−1(Cl⋆(Int(K)))))) ⊆ f−1(K) for every ⋆-closed set K of Y ; (11) Int⋆(Cl( Int⋆(f−1(sIntJ (K))))) ⊆ f−1(K) for every ⋆-closed set K of Y ; (12) f−1(V ) ⊆ Cl⋆( Int(Cl⋆(f−1(sClJ (V ))))) for every ⋆-open set V of Y . Theorem 9. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is upper almost sβ(⋆)-continuous; (2) sβClI (F−(V )) ⊆ F−(Cl⋆(V )) for every strong β-J -open set V of Y ; (3) sβClI (F−(V )) ⊆ F−(Cl⋆(V )) for every semi-J -open set V of Y ; (4) F+(V ) ⊆ sβIntI (F+(Int⋆(Cl(V )))) for every pre⋆J -open set V of Y . Proof. (1) ⇒ (2): Let V be any strong β-J -open set of Y . Since Cl⋆(V ) is R⋆-J - closed, by Theorem 7, F−(Cl⋆(V )) is strong β-I -closed in X and hence sβClI (F−(V )) ⊆ F−(Cl⋆(V )). (2) ⇒ (3): This is obvious since every semi-J -open set is strong β-J -open. (3) ⇒ (4): Let V be any pre⋆J -open set of Y . Then, we have V ⊆ Int⋆(Cl(V )) and Y − V ⊇ Cl⋆(Int(Y − V )). Since Cl⋆(Int(Y − V )) is semi-J -open in Y and by (3), X − F+(V ) = F−(Y − V ) ⊇ F−(Cl⋆(Int(Y − V ))) ⊇ sβClI (F−(Cl⋆(Int(Y − V )))) = sβClI (F−(Y − Int⋆(Cl(V )))) = sβClI (X − F+(Int⋆(Cl(V )))) = X − sβIntI (F+(Int⋆(Cl(V )))). REFERENCES 1645 Thus, F+(V ) ⊆ sβIntI (F+(Int⋆(Cl(V )))). (4) ⇒ (1): Let V be any R⋆-J -open set of Y . Then, V is pre⋆J -open in Y and by (4), F+(V ) ⊆ sβIntI (F+(Int⋆(Cl(V )))) = sβIntI (F+(V )) and hence F+(V ) is strong β-I -open in X. It follows from Theorem 7 that F is upper almost sβ(⋆)-continuous. Theorem 10. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is lower almost sβ(⋆)-continuous; (2) sβClI (F−(V )) ⊆ F−(Cl⋆(V )) for every strong β-J -open set V of Y ; (3) sβClI (F−(V )) ⊆ F−(Cl⋆(V )) for every semi-J -open set V of Y ; (4) F+(V ) ⊆ sβIntI (F+(Int⋆(Cl(V )))) for every pre⋆J -open set V of Y . Proof. The proof is similar to that of Theorem 9. Corollary 5. 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