EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2544-2556 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and lower weak sβ(⋆)-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. This paper is concerned with the concepts of upper and lower weakly sβ(⋆)-continuous multifunctions. Moreover, some characterizations of upper and lower weakly sβ(⋆)-continuous multifunctions are investigated. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper weakly sβ(⋆)-continuous multifunction, lower weakly sβ(⋆)- continuous multifunction 1. Introduction In topology, there has been recently significant interest in characterizing and inves- tigating the characterizations of some weak forms of continuity for functions and multi- functions. As weak forms of continuity in topological spaces, weak continuity [12], quasi- continuity [14], semi-continuity [13] and almost continuity in the sense of Husain [9] are well-known. It is shown in [15] that quasicontinuity is equivalent to semi-continuity. It will be shown that weak continuity, semi-continuity and almost continuity are respectively independent. Popa and Stan [23] introduced weak quasi-continuity which is implied by both weak continuity and quasicontinuity. Janković [10] introduced almost weak continu- ity as a generalization of both weak continuity and almost continuity. Noiri [16] obtained some characterizations of almost weak continuity and some relations between almost weak continuity and weak continuity. Popa [20] and Smithson [24] independently introduced the notion of weakly continuous multifunctions. The present authors introduced and studied other weak forms of continuous multifunctions: weakly quasicontinuous multifunctions [17], almost weakly continuous multifunctions [18], weakly α-continuous multifunctions [22], weakly β-continuous multifunctions [21]. These multifunctions have similar charac- terizations. The analogy in their definitions and results suggests the need of formulating a unified theory. Noiri and Popa [19] introduced and studied the notions of upper and lower weakly m-continuous multifunctions as a multifunction from a set satisfying certain min- imal condition into a topological space. In [2], the present author introduced and studied ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4734 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), jeeranunt.k@msu.ac.th (J. Khampakdee) https://www.ejpam.com 2544 © 2023 EJPAM All rights reserved. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (4) (2023), 2544-2556 2545 the concepts of upper and lower ⋆-continuous multifunctions in ideal topological spaces. Moreover, several characterizations of upper and lower ⋆-continuous multifunctions were investigated in [3]. The purpose of the present paper is to introduce the notions of upper and lower weakly sβ(⋆)-continuous multifunctions. Furthermore, some characterizations of upper and lower weakly 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 [11], 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 [10] if A⋆ ⊆ A. The interior of a subset A in (X, τ⋆(I )) is denoted by Int⋆(A). 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) [8]. (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 [7] (resp. pre⋆I -open [5], strong β-I -open [7]) 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 [7] (resp. pre⋆I -closed [5], strong β-I -closed [7]). 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))). Lemma 3. [4] Let (X, τ,I ) be an ideal topological space. If V is ⋆-open, then sClI (V ) = Int⋆(Cl(V )). C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (4) (2023), 2544-2556 2546 Lemma 4. [4] For a subset A of an ideal topological space (X, τ,I ), x ∈ sβClI (A) if and only if U ∩A ̸= ∅ for every strong β-I -open set U containing x. Lemma 5. [4] For a subset A of an ideal topological space (X, τ,I ), the following prop- erties are hold: (1) X − sβClI (A) = sβIntI (X −A). (2) X − sβIntI (A) = sβClI (X −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). Then F is said to be surjection if F (X) = Y , or equivalent, if for each y ∈ Y there exists x ∈ X such that y ∈ F (x) and F is called injection if x ̸= y implies F (x) ∩ F (y) = ∅. 3. Upper and lower weakly sβ(⋆)-continuous multifunctions In this section, we introduce the notions of upper and lower weakly sβ(⋆)-continuous multifunctions. Moreover, several characterizations of upper and lower weakly sβ(⋆)- continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ,I ) → (Y, σ,J ) is said to be: (1) upper weakly 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) ⊆ Cl⋆(V ); (2) lower weakly 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) ∩ Cl⋆(V ) ̸= ∅ for every z ∈ U ; (3) upper (resp. lower) weakly sβ(⋆)-continuous if F has this property 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 weakly sβ(⋆)-continuous at a point x ∈ X; (2) x ∈ Cl⋆(Int(Cl⋆(F+(Cl⋆(V ))))) for every ⋆-open set V of Y containing F (x); C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (4) (2023), 2544-2556 2547 (3) x ∈ sβIntI (F+(Cl⋆(V ))) for every ⋆-open set V of Y containing F (x). Proof. (1) ⇒ (2): Let V be any ⋆-open set of Y containing F (x). By (1), there exists a strong β-I -open set U of X containing x such that F (U) ⊆ Cl⋆(V ). Then, x ∈ U ⊆ F+(Cl⋆(V )). Since U is strong β-I -open, we have x ∈ U ⊆ Cl⋆(Int(Cl⋆(U))) ⊆ Cl⋆(Int(Cl⋆(F+(Cl⋆(V ))))). (2) ⇒ (3): Let V be any ⋆-open set of Y containing F (x). Thus, by (2), we have x ∈ Cl⋆(Int(Cl⋆(F+(Cl⋆(V ))))). Since x ∈ F+(Cl⋆(V )) and by Lemma 2, we obtain x ∈ F+(Cl⋆(V )) ∩ Cl⋆(Int(Cl⋆(F+(Cl⋆(V ))))) = sβIntI (F+(Cl⋆(V ))). (3) ⇒ (1): Let V be any ⋆-open set of Y containing F (x). By (3), we have x ∈ sβIntI (F+(Cl⋆(V ))) and so there exists a strong β-I -open set U of X containing x such that U ⊆ F+(Cl⋆(V )); hence F (U) ⊆ Cl⋆(V ). This shows that F is upper weakly sβ(⋆)-continuous at x. Theorem 2. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is lower weakly sβ(⋆)-continuous at a point x ∈ X; (2) x ∈ Cl⋆(Int(Cl⋆(F−(Cl⋆(V ))))) for every ⋆-open set V of Y such that F (x)∩V ̸= ∅; (3) x ∈ sβIntI (F−(Cl⋆(V ))) for every ⋆-open set V of Y such that F (x) ∩ V ̸= ∅. Proof. The proof is similar to that of Theorem 1. Definition 2. A function f : (X, τ,I ) → (Y, σ,J ) is said to be weakly 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) ⊆ Cl⋆(V ). A function f : (X, τ,I ) → (Y, σ,J ) is said to be weakly sβ(⋆)-continuous if f has this property at each point of X. Corollary 1. For a function f : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) f is weakly sβ(⋆)-continuous at a point x ∈ X; (2) x ∈ Cl⋆(Int(Cl⋆(f−1(Cl⋆(V ))))) for every ⋆-open set V of Y containing f(x); (3) x ∈ sβIntI (f−1(Cl⋆(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: C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (4) (2023), 2544-2556 2548 (1) F is upper weakly sβ(⋆)-continuous; (2) F+(V ) ⊆ Cl⋆(Int(Cl⋆(F+(Cl⋆(V ))))) for every ⋆-open set V of Y ; (3) Int⋆(Cl(Int⋆(F−(V )))) ⊆ F−(Cl⋆(V )) for every ⋆-open set V of Y ; (4) Int⋆(Cl(Int⋆(F−(Int⋆(K))))) ⊆ F−(K) for every ⋆-closed set K of Y ; (5) sβClI (F−(Int⋆(K))) ⊆ F−(K) for every ⋆-closed set K of Y ; (6) sβClI (F−(Int⋆(Cl⋆(B)))) ⊆ F−(Cl⋆(B)) for every subset B of Y ; (7) F+(Int⋆(B)) ⊆ sβIntI (F+(Cl⋆(Int⋆(B)))) for every subset B of Y ; (8) F+(V ) ⊆ sβIntI (F+(Cl⋆(V ))) for every ⋆-open set V of Y ; (9) sβClI (F−(V )) ⊆ F−(Cl⋆(V )) for every ⋆-open set V of Y . Proof. (1) ⇒ (2): Let V be any ⋆-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V and by Theorem 1, x ∈ sβIntI (F+(Cl⋆(V ))) and hence F+(V ) ⊆ Cl⋆(Int(Cl⋆(F+(Cl⋆(V ))))) by Lemma 2. (2) ⇒ (3): Let V be any ⋆-open set of Y . Thus, by (2), we have X − F−(Cl⋆(V )) = F+(Y − Cl⋆(V )) ⊆ Cl⋆(Int(Cl⋆(F+(Cl⋆(Y − Cl⋆(V )))))) = Cl⋆(Int(Cl⋆(F+(Y − Int⋆(Cl⋆(V )))))) ⊆ Cl⋆(Int(Cl⋆(F+(Y − V )))) = Cl⋆(Int(Cl⋆(X − F−(V )))) = X − Int⋆(Cl(Int⋆(F−(V )))) and hence Int⋆(Cl(Int⋆(F−(V )))) ⊆ F−(Cl⋆(V )). (3) ⇒ (4): Let K be any ⋆-closed set of Y . Then, Int⋆(K) is ⋆-open in Y and so Int⋆(Cl(Int⋆(F−(Int⋆(K))))) ⊆ F−(Cl⋆(Int⋆(K))) ⊆ F−(Cl⋆(K)) = F−(K). (4) ⇒ (5): Let K be any ⋆-closed set of Y . Then, we have Int⋆(Cl(Int⋆(F−(Int⋆(K))))) ⊆ F−(K) and F−(Int⋆(K)) ⊆ F−(K). Thus, by Lemma 2, sβI Cl(F−(Int⋆(K))) ⊆ F−(K). (5) ⇒ (6): Let B be any subset of Y . Then, Cl⋆(B) is ⋆-closed in Y and by (5), sβClI (F−(Int⋆(Cl⋆(B)))) ⊆ F−(Cl⋆(B)). (6) ⇒ (7): Let B be any subset of Y . By (6), F+(Int⋆(B)) = X − F−(Cl⋆(Y −B)) ⊆ X − sβClI (F−(Int⋆(Cl⋆(Y −B)))) C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (4) (2023), 2544-2556 2549 = sβIntI (F+(Cl⋆(Int⋆(B)))). (7) ⇒ (8): The proof is obvious. (8) ⇒ (9): Let V be any ⋆-open set of Y . Thus, by (8), we have sβClI (F−(V )) ⊆ sβClI (F−(Int⋆(Cl⋆(V )))) = sβClI (X − F+(Y − Int⋆(Cl⋆(V )))) = X − sβIntI (F+(Y − Int⋆(Cl⋆(V )))) = X − sβIntI (F+(Cl⋆(Y − Cl⋆(V )))) ⊆ X − F+(Y − Cl⋆(V )) = F−(Cl⋆(V )). (9) ⇒ (1): Let x ∈ X and V be any ⋆-open set of Y containing F (x). By (9), x ∈ F+(V ) ⊆ F+(Int⋆(Cl⋆(V ))) = X − F−(Cl⋆(Y − Cl⋆(V ))) ⊆ X − sβClI (F−(Y − Cl⋆(V ))) = sβIntI (F+(Cl⋆(V ))) and hence F is upper weakly sβ(⋆)-continuous by Theorem 1. Theorem 4. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is lower weakly sβ(⋆)-continuous; (2) F−(V ) ⊆ Cl⋆(Int(Cl⋆(F−(Cl⋆(V ))))) for every ⋆-open set V of Y ; (3) Int⋆(Cl(Int⋆(F+(V )))) ⊆ F+(Cl⋆(V )) for every ⋆-open set V of Y ; (4) Int⋆(Cl(Int⋆(F+(Int⋆(K))))) ⊆ F+(K) for every ⋆-closed set K of Y ; (5) sβClI (F+(Int⋆(K))) ⊆ F+(K) for every ⋆-closed set K of Y ; (6) sβClI (F+(Int⋆(Cl⋆(B)))) ⊆ F+(Cl⋆(B)) for every subset B of Y ; (7) F−(Int⋆(B)) ⊆ sβIntI (F−(Cl⋆(Int⋆(B)))) for every subset B of Y ; (8) F−(V ) ⊆ sβIntI (F−(Cl⋆(V ))) for every ⋆-open set V of Y ; (9) sβClI (F+(V )) ⊆ F+(Cl⋆(V )) for every ⋆-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, σ,J ), the following properties are equivalent: C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (4) (2023), 2544-2556 2550 (1) f is weakly sβ(⋆)-continuous; (2) f−1(V ) ⊆ Cl⋆(Int(Cl⋆(f−1(Cl⋆(V ))))) for every ⋆-open set V of Y ; (3) Int⋆(Cl(Int⋆(f−1(V )))) ⊆ f−1(Cl⋆(V )) for every ⋆-open set V of Y ; (4) Int⋆(Cl(Int⋆(f−1(Int⋆(K))))) ⊆ f−1(K) for every ⋆-closed set K of Y ; (5) sβClI (f−1(Int⋆(K))) ⊆ f−1(K) for every ⋆-closed set K of Y ; (6) sβClI (f−1(Int⋆(Cl⋆(B)))) ⊆ f−1(Cl⋆(B)) for every subset B of Y ; (7) f−1(Int⋆(B)) ⊆ sβIntI (f−1(Cl⋆(Int⋆(B)))) for every subset B of Y ; (8) f−1(V ) ⊆ sβIntI (f−1(Cl⋆(V ))) for every ⋆-open set V of Y ; (9) sβClI (f−1(V )) ⊆ f−1(Cl⋆(V )) for every ⋆-open set V of Y . Recall that a subset A of an ideal topological space (X, τ,I ) is called R-I ⋆-open [2] (resp. I ⋆-preopen [2], I ⋆-semi-open [3]) if A = Int⋆(Cl⋆(A)) (resp. A ⊆ Int⋆(Cl⋆(A)), A ⊆ Cl⋆(Int⋆(A))). The complement of a R-I ⋆-open (resp. I ⋆-preopen, I ⋆-semi-open) set is called R-I ⋆-closed [2] (resp. I ⋆-preclosed [2], I ⋆-semi-closed [3]). Let A be a subset of an ideal topological space (X, τ,I ). A point x in an ideal topological space (X, τ,I ) is called a ⋆θ-cluster point of A [3] if Cl⋆(U)∩A ̸= ∅ for every ⋆-open set U of X containing x. The set of all ⋆θ-cluster points of A is called the ⋆θ-closure [3] of A and is denoted by ⋆θCl(A). A subset B of an ideal topological space (X, τ,I ) is called ⋆θ-closed [3] if ⋆θCl(B) = B. The complement of a ⋆θ-closed set is called ⋆θ-open [3]. Lemma 6. [3] For a subset A of an ideal topological space (X, τ,I ), the following prop- erties hold: (1) If A is ⋆-open in X, then Cl⋆(A) = ⋆θCl(A). (2) ⋆θCl(A) is ⋆-closed in X. Theorem 5. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is upper weakly sβ(⋆)-continuous; (2) sβClI (F−(Int⋆(⋆θCl(B)))) ⊆ F−(⋆θCl(B)) for every subset B of Y ; (3) sβClI (F−(Int⋆(Cl⋆(B)))) ⊆ F−(⋆θCl(B)) for every subset B of Y ; (4) sβClI (F−(Int⋆(Cl⋆(V )))) ⊆ F−(Cl⋆(V )) for every ⋆-open set V of Y ; (5) sβClI (F−(Int⋆(Cl⋆(V )))) ⊆ F−(Cl⋆(V )) for every J ⋆-preopen set V of Y ; (6) sβClI (F−(Int⋆(K))) ⊆ F−(K) for every R-J ⋆-closed set K of Y ; C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (4) (2023), 2544-2556 2551 (7) sβClI (F−(Int⋆(Cl⋆(V )))) ⊆ F−(Cl⋆(V )) for every strong β-J -open set V of Y ; (8) sβClI (F−(Int⋆(Cl⋆(V )))) ⊆ F−(Cl⋆(V )) for every J ⋆-semi-open set V of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . Thus, by Lemma 6, ⋆θCl(B) is ⋆-closed in Y and by Theorem 3, sβClI (F−(Int⋆(⋆θCl(B)))) ⊆ F−(⋆θCl(B)). (2) ⇒ (3): This is obvious since Cl⋆(B) ⊆ ⋆θCl(B) for every subset B of Y . (3) ⇒ (4): This is obvious since Cl⋆(V ) = ⋆θCl(V ) for every ⋆-open set V of Y . (4) ⇒ (5): Let V be any J ⋆-preopen set of Y . Then, we have V ⊆ Int⋆(Cl⋆(V )) and so Cl⋆(V ) = Cl⋆(Int⋆(Cl⋆(V ))). Now, put G = Int⋆(Cl⋆(V )), then G is ⋆-open in Y and Cl⋆(G) = Cl⋆(V ). Thus, by (4), we have sβClI (F−(Int⋆(Cl⋆(V )))) ⊆ F−(Cl⋆(V )). (5) ⇒ (6): Let K be any R-J ⋆-closed set of Y . Then, Int⋆(K) is J ⋆-preopen in Y , by (5), sβClI (F−(Int⋆(K))) = sβClI (F−(Int⋆(Cl⋆(Int⋆(K))))) ⊆ F−(Cl⋆(Int⋆(K))) = F−(K). (6) ⇒ (7): Let V be any strong β-J -open set of Y . Then, V ⊆ Cl⋆(Int(Cl⋆(V ))). Since Cl⋆(V ) isR-J ⋆-closed in Y . Thus, by (6), sβClI (F−(Int⋆(Cl⋆(V )))) ⊆ F−(Cl⋆(V )). (7) ⇒ (8): This is obvious since every J ⋆-semi-open set is strong β-J -open. (8) ⇒ (1): Let V be any ⋆-open set of Y . Then, since V is J ⋆-semi-open set in Y , by (8), we have sβClI (F−(V )) ⊆ sβClI (F−(Int⋆(Cl⋆(V )))) ⊆ F−(Cl⋆(V )). By Theorem 3, F is upper weakly sβ(⋆)-continuous. Theorem 6. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is lower weakly sβ(⋆)-continuous; (2) sβClI (F+(Int⋆(⋆θCl(B)))) ⊆ F+(⋆θCl(B)) for every subset B of Y ; (3) sβClI (F+(Int⋆(Cl⋆(B)))) ⊆ F+(⋆θCl(B)) for every subset B of Y ; (4) sβClI (F+(Int⋆(Cl⋆(V )))) ⊆ F+(Cl⋆(V )) for every ⋆-open set V of Y ; (5) sβClI (F+(Int⋆(Cl⋆(V )))) ⊆ F+(Cl⋆(V )) for every J ⋆-preopen set V of Y ; (6) sβClI (F+(Int⋆(K))) ⊆ F+(K) for every R-J ⋆-closed set K of Y ; (7) sβClI (F+(Int⋆(Cl⋆(V )))) ⊆ F+(Cl⋆(V )) for every strongly β-J -open set V of Y ; (8) sβClI (F+(Int⋆(Cl⋆(V )))) ⊆ F+(Cl⋆(V )) for every J ⋆-semi-open set V of Y . Proof. The proof is similar to that of Theorem 5. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (4) (2023), 2544-2556 2552 Corollary 3. For a function f : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) f is weakly sβ(⋆)-continuous; (2) sβClI (f−1(Int⋆(⋆θCl(B)))) ⊆ f−1(⋆θCl(B)) for every subset B of Y ; (3) sβClI (f−1(Int⋆(Cl⋆(B)))) ⊆ f−1(⋆θCl(B)) for every subset B of Y ; (4) sβClI (f−1(Int⋆(Cl⋆(V )))) ⊆ f−1(Cl⋆(V )) for every ⋆-open set V of Y ; (5) sβClI (f−1(Int⋆(Cl⋆(V )))) ⊆ f−1(Cl⋆(V )) for every J ⋆-preopen set V of Y ; (6) sβClI (f−1(Int⋆(K))) ⊆ f−1(K) for every R-J ⋆-closed set K of Y ; (7) sβClI (f−1(Int⋆(Cl⋆(V )))) ⊆ f−1(Cl⋆(V )) for every strongly β-J -open set V of Y ; (8) sβClI (f−1(Int⋆(Cl⋆(V )))) ⊆ f−1(Cl⋆(V )) for every J ⋆-semi-open set V of Y . Theorem 7. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) F is upper weakly sβ(⋆)-continuous; (2) sβClI (F−(V )) ⊆ F−(Cl⋆(V )) for every J ⋆-preopen set V of Y ; (3) F+(V ) ⊆ sβIntI (F+(Cl⋆(V ))) for every J ⋆-preopen set V of Y . Proof. (1) ⇒ (2): Let V be any J ⋆-preopen set of Y . Since F is upper weakly sβ(⋆)- continuous, by Theorem 3, sβClI (F−(V )) ⊆ sβClI (F−(Int⋆(Cl⋆(V )))) ⊆ F−(Cl⋆(V )). (2) ⇒ (3): Let V be any J ⋆-preopen set of Y . Then, we have V ⊆ Int⋆(Cl⋆(V )) and Y − V ⊇ Cl⋆(Int⋆(Y − V )). Thus, by (3), X − F+(V ) = F−(Y − V ) ⊇ F−(Cl⋆(Int⋆(Y − V ))) ⊇ sβClI (F−(Int⋆(Y − V ))) = sβClI (F−(Y − Cl⋆(V ))) = sβClI (X − F+(Cl⋆(V ))) = X − sβIntI (F+(Cl⋆(V ))) and hence F+(V ) ⊆ sβIntI (F+(Cl⋆(V ))). (3) ⇒ (1): Let V be any ⋆-open set of Y . Then, V is J ⋆-preopen in Y , by (4), F+(V ) ⊆ sβIntI (F+(Cl⋆(V ))). Thus, F is upper weakly sβ(⋆)-continuous by Theorem 3. Theorem 8. For a multifunction F : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (4) (2023), 2544-2556 2553 (1) F is lower weakly sβ(⋆)-continuous; (2) sβClI (F+(V )) ⊆ F+(Cl⋆(V )) for every J ⋆-preopen set V of Y ; (3) F−(V ) ⊆ sβIntI (F−(Cl⋆(V ))) for every J ⋆-preopen 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 weakly sβ(⋆)-continuous; (2) sβClI (f−1(V )) ⊆ f−1(Cl⋆(V )) for every J ⋆-preopen set V of Y ; (3) f−1(V ) ⊆ sβIntI (f−1(Cl⋆(V ))) for every J ⋆-preopen set V of Y . Definition 3. [4] 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 almost sβ(⋆)-continuity ⇒ upper weak 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 = {∅, {1}, {2}, {1, 2}, X} and an ideal I = {∅, {1}}. Let Y = {a, b, c} with a topology σ = {∅, {a}, {a, b}, Y } and an ideal J = {∅, {c}}. A multifunction F : (X, τ,I ) → (Y, σ,J ) is defined as follows: F (1) = {c} and F (2) = F (3) = {a, b}. Then, F is upper weakly sβ(⋆)-continuous but F is not upper almost sβ(⋆)-continuous, since {a, b} is ⋆-open in Y but F+({a, b}) is not strong β-I -open in X. Lemma 7. [2] For an ideal topological space (X, τ,I ), the following properties are equiv- alent: (1) (X, τ,I ) is ⋆-I -normal. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (4) (2023), 2544-2556 2554 (2) For each ⋆-closed set F and each ⋆-open set V containing F , there exists a ⋆-open set U such that F ⊆ U ⊆ Cl⋆(U) ⊆ V . Theorem 9. For a multifunction F : (X, τ,I ) → (Y, σ,J ) such that F (x) is ⋆-closed in Y for each x ∈ X and (Y, σ,J ) is a ⋆-J -normal space, the following properties are equivalent: (1) F is upper sβ(⋆)-continuous; (2) F is upper almost sβ(⋆)-continuous; (3) F is upper weakly sβ(⋆)-continuous. Proof. We show only the implication (3) ⇒ (1) since the others are obvious. Suppose that F is upper weakly sβ(⋆)-continuous. Let x ∈ X and V be any ⋆-open set of Y such that F (x) ⊆ V . Since F (x) is ⋆-closed in Y and Y is ⋆-J -normal, there exists a ⋆-open set G of Y such that F (x) ⊆ G ⊆ Cl⋆(G) ⊆ V . Since F is upper weakly sβ(⋆)-continuous, there exists a strong β-I -open set U of X containing x such that F (U) ⊆ Cl⋆(G); hence F (U) ⊆ V . This shows that F is upper sβ(⋆)-continuous. Theorem 10. For a multifunction F : (X, τ,I ) → (Y, σ,J ) such that F (x) is ⋆-open in Y for each x ∈ X, the following properties are equivalent: (1) F is lower sβ(⋆)-continuous; (2) F is lower almost sβ(⋆)-continuous; (3) F is lower weakly sβ(⋆)-continuous. Proof. (1) ⇒ (2) and (2) ⇒ (3): The proofs of these implications are obvious. (3) ⇒ (1): Suppose that F is lower weakly sβ(⋆)-continuous. Let x ∈ X and V be any ⋆-open set such that F (x)∩ V ̸= ∅. Then, there exists a strong β-I -open set U of X containing x such that F (z) ∩ Cl⋆(V ) ̸= ∅ for each z ∈ U . Since F (z) is ⋆-open, we have F (z) ∩ V ̸= ∅ for each z ∈ U and so F is lower sβ(⋆)-continuous. Definition 4. [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 )). 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