EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3677-3686 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and Lower Rarely m-I-Continuous Multifunctions Takashi Noiri1, Valeriu Popa2 1 2949-1 Shiokita-Cho, Hinagu, Yatsushiro-Shi, Kumamoto-Ken, 869-5142 Japan 2 Department of Mathematics and Informatics, “Vasile Alecsandri” University of Bacǎu, Calea Marasesti 157, Bacǎu, 600115, Romania Abstract. In 1979, Popa [24] first introduced rarely continuous functions. In this paper, we introduce upper and lower rarely m-continuous multifunctions. Moreover, we extend this concept to a multifunction F : (X, τ, I) → (Y, σ), where (X, τ, I) is an ideal topologiccal space. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Rare set, m-open, m-structure, mIO(X), ideal topological space, rarely continuous function, upper/lower rarely m-I-continuous, multifunction 1. Introduction Semi-open sets, preopen sets, α-open sets, β-open set and b-open sets play an impor- tant part in the research of generalizations of continuous functions. By using these notions, various types of continuous multifunctions are introduced and studied. As an unified form of the above generalizations of open sets, in [27] and [28] the present authors introduced minimal structures and m-spaces. We recall the notions in the section 2. In 1979, Popa [24] first introduced the concept of rare continuity which was further studied by Long and Herrington [19] and Jafari [10], [11]. Several weak forms of rarely continuous functions, for example, rare quasi-continuity [26], rare α-continuity [13], rare pre-continuity [12] etc have been introduced and studied. Moreover these concepts are ex- tended to multifunctions: rare continuity [25], rare quasi-continuity [15], rare α-continuity [4], rare β-continuity [14]. The purpose of this paper is to introduce the concept of upper and lower rarely m- continuous multifunctions which unifies the above stated multifunctions, that is, rare quasi-continuity, rare pre-continuity, rare α-continuity, and rare β-continuity. As general- izations of open sets, the notion of I-open sets, semi-I-open sets, pre-I-open sets, α-I-open sets, β-I-open sets and b-I-open sets are introduced and studied in an ideal topological space (X.τ, I). In the last section, we extend the results of an upper and lower rarely DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5531 Email addresses: t.noiri@nifty.com (T. Noiri), vpopa@ub.ro (V. Popa) https://www.ejpam.com 3677 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3677-3686 3678 m-continuous multifunction F : (X,m) → (Y, σ) to a multifunction F : (X, τ, I) → (Y, σ). Throughout the present paper, (X, τ) and (Y, σ) (briefly X and Y ) always denote topological spaces and F : X → Y presents a multivalued function. For a multifunction F : X → Y , we shall denote the upper and lower inverse of a subset B of a space 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 ̸= ∅}. 2. Preliminaries Let (X, τ) be a topological space and A a subset of X. The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. A subset A is said to be regular open (resp. regular closed) if Int(Cl(A)) = A (resp. Cl(Int(A)) = A). Definition 1. Let (X, τ) be a topological space. A subset A of X is said to be (1) α-open [23] if A ⊂ Int(Cl(Int(A))), (2) semi-open [18] if A ⊂ Cl(Int(A)), (3) preopen [21] if A ⊂ Int(Cl(A)), (4) β-open [3] if A ⊂ Cl(Int(Cl(A))), (5) b-open [1] if A ⊂ Int(Cl(A)) ∪ Cl(Int(A)). The family of all semi-open (resp. preopen, α-open, β-open, b-open) sets in X is de- noted by SO(X) (resp. PO(X), α(X), β(X), BO(X)). Definition 2. A subfamily m of the power set P(X) of a nonempty set X is called a minimal structure (briefly m-structure) [27], [28] on X if ∅ ∈ m and X ∈ m. By (X,m), we denote a nonempty set X with a minimal structure m on X and call it an m-space. Each member of m is said to be m-open and the complement of an m-open set is said to be m-closed. By m(x), we denote the family {U : x ∈ U ∈ m}. Definition 3. Let (X,m) be an m-space. For a subset A of X, the m-closure of A and the m-interior of A are defined in [20] as follows: (1) mCl(A) = ∩{F : A ⊂ F,X \ F ∈ m}, (2) mInt(A) = ∪{U : U ⊂ A,U ∈ m}. Lemma 1. (Maki et al. [20]). Let (X,m) be an m-space. For subsets A and B of X, the following properties hold: (1) mCl(X \A) = X \mInt(A) and mInt(X \A) = X \mCl(A), (2) If (X \A) ∈ m, then mCl(A) = A and if A ∈ m, then mInt(A) = A, (3) mCl(∅) = ∅, mCl(X) = X, mInt(∅) = ∅ and mInt(X) = X, (4) If A ⊂ B, then mCl(A) ⊂ mCl(B) and mInt(A) ⊂ mInt(B), (5) mInt(A) ⊂ A ⊂ mCl(A), (6) mCl(mCl(A)) = mCl(A) and mInt(mInt(A)) = mInt(A). T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3677-3686 3679 Lemma 2. (Popa and Noiri [28]). Let (X,m) be an m-space and A a subset of X. Then x ∈ mCl(A) if and only if U ∩A ̸= ∅ for every U ∈ m containing x. Definition 4. An m-structure m on a nonempty set X is said to have property B [20] if the union of any family of subsets belonging to m belongs to m. Remark 1. Let (X, τ) be a topological space. Then the families τ , SO(X), PO(X), α(X), BO(X) and β(X) are m-structures and have property B. Lemma 3. (Popa and Noiri [29]). For an m-structure m on a nonempty set X, the following properties are equivalent: (1) m has property B; (2) If mInt(A) = A, then A ∈ m; (3) If mCl(A) = A, then A is m-closed. Definition 5. A subset A of a topological space (X, τ) is called a rare-set if Int(A) = ∅. Lemma 4. In a topological space (X, τ), Int(F ∪ R) ⊂ F for every rare set R and every closet set F . Proof. It is obvious that O∩Cl(A) ⊂ Cl(O∩A) of every subset A of X and any open set O of X. Hence Int(F ∪B) ⊂ (F ∪ Int(B)) for every subset B and every closed set F . Therefore, Int(F ∪R) ⊂ F for every rare set R and every closed set F . Definition 6. A function f : (X, τ) → (Y, σ) is said to be rarely continuous [24] at x ∈ X if for any open set V of Y such that f(x) ∈ V , there exist a rare set RV with RV ∩ V = ∅ and an open set U containing x such that f(U) ⊂ V ∪RV . 3. Rarely m-continuous multifunctions In this section, we define upper and lower rare m-continuity on a multifunction F : (X,m) → (Y, σ) and obtain their characterizations. Definition 7. Let (X,m) be an m-space and (Y, σ) a topological space. A multifunction F : (X,m) → (Y, σ) is said to be (1) upper rarely m-continuous at a point x ∈ X if for each open set V containing F (x), there exist a rare set RV with RV ∩ V = ∅ and an m-open set U ∈ m(x) such that F (U) ⊂ V ∪RV , (2) lower rarely m-continuous at a point x ∈ X if for each open set V meeting F (x), there exist a rare set RV with RV ∩ V = ∅ and an m-open set U ∈ m(x) such that F (u) ∩ (V ∪RV ) ̸= ∅ for each u ∈ U , Theorem 1. For a multifunction F : (X,m) → (Y, σ), the following properties are equiv- alent: (1) F is upper rarely m-continuous at x ∈ X; T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3677-3686 3680 (2) for each open set V of Y containing F(x), there exists a rare set RV with V ∩RV = ∅ such that x ∈ mInt(F+(V ∪RV )); (3) for each open set V of Y containing F(x), there exists a rare set RV with Cl(V ) ∩ RV = ∅ such that x ∈ mInt(F+(Cl(V ) ∪RV )); (4) for each regular open set V of Y containing F(x), there exists a rare set RV with V ∩RV = ∅ such that x ∈ mInt(F+(V ∪RV )); (5) for each open set V of Y containing F(x), there exists U ∈ m(x) such that Int[F (U) ∩ (Y \ V )] = ∅, (6) for each open set V of Y containing F(x), there exists U ∈ m(x) such that Int(F (U)) ⊂ Cl(V ). Proof. (1) ⇒ (2): Let V be any open set of Y containing F (x). By (1), there exist a rare set RV with RV ∩ V = ∅ and an m-open set U ∈ m(x) such that F (U) ⊂ V ∪ RV . Hence x ∈ U ⊂ F+(V ∪RV ). Since U is m-open, x ∈ mInt(F+(V ∪RV )). (2) ⇒ (3): Let V be any open set of Y such that F (x) ⊂ V . Then by (2), there exists a rare set RV with V ∩RV = ∅ such that x ∈ mInt(F+(V ∪RV )). Let SV = RV ∩(Y \Cl(V )), then SV ∩Cl(V ) = ∅ and SV is a rare set. Since Cl(V )∪SV = Cl(V )∪ [RV ∩(Y \Cl(V ))] = Cl(V ) ∪RV ⊃ V ∪RV . Therefore, x ∈ mInt(F+(V ∪RV )) ⊂ mInt(F+(Cl(V ) ∪ SV )). (3) ⇒ (4): Let V be any regular open set of Y containing F (x). By (3), there exists a rare set RV with Cl(V ) ∩ RV = ∅ such that x ∈ mInt(F+(Cl(V ) ∪ RV )). Let SV = RV ∪ (Cl(V ) \ V ). Then by Lemma 4, SV is a rare set and SV ∩ V = ∅. Therefore, x ∈ mInt(F+(V ∪ SV )). (4) ⇒ (5): V be any open set of Y containing F (x). Then F (x) ⊂ V ⊂ Int(Cl(V )) and Int(Cl(V )) is regular open. By (4), there exists a rare set RV with RV ∩ Int(Cl(V )) = ∅ and x ∈ mInt(F+(Int(Cl(V )) ∪ RV )). Hence there exists U ∈ m(x) such that x ∈ U ⊂ F+(Int(Cl(V ))∪RV ). Thus, F (U) ⊂ Int(Cl(V ))∪RV . Therefore, by using Lemma 4, we have Int[F (U) ∩ (Y \ V )] = Int(F (U)) ∩ Int(Y \ V ) ⊂ Int(Cl(V ) ∪RV ) ∩ (Y \ Cl(V )) ⊂ (Cl(V ) ∪ Int(RV )) ∩ (Y \ Cl(V )) = Cl(V ) ∩ (Y \ Cl(V )) = ∅. Therefore, we have Int[F (U) ∩ (Y \ V )] = ∅. (5) ⇒ (6): For each open set V of Y containing F (x), there exists U ∈ m(x) such that Int[F (U) ∩ (Y \ V )] = ∅. Hence Int[F (U)) ∩ (Y \ Cl(V )] = ∅ and Int(F (U)) ⊂ Cl(V ). (6) ⇒ (1): Let V be any open set of Y containing F (x). By (6), there exists U ∈ m(x) such that Int(F (U)) ⊂ Cl(V ). LetM = F (U)∩(Y \V ). Then Int(M) ⊂ Int(F (U))∩Int(Y \ V ) = Int(F (U))∩(Y \Cl(V )) = ∅. HenceM is a rare set andM∩V = ∅. LetN = Cl(V )\V . Then N is a closed rare set such that N ∩V = ∅. Therefore, RV = M ∪N is a rare set and RV ∩V = ∅ by using Lemma 4: Int(RV ) = Int(Int(RV )) = Int(Int(M ∪N)) ⊂ Int(N) = ∅. Now, we have the following: F (U) = [F (U) \ Int(F (U)] ∪ Int(F (U))] ⊂ [F (U) \ Int(F (U)] ∪ Cl(V ) = [(F (U)) ∩ (V ∪ (Y \ V )) \ Int(F (U)] ∪ [(Cl(V ) \ V ) ∪ V ] = (F (U) ∩ V ) ∪ (F (U) ∩ (Y \ V )) \ Int(F (U)] ∪ [N ∪ V ] ⊂ [V ∪ (F (U) ∩ (Y \ V ))] ∪ [N ∪ V ] = V ∪ (M ∪N) T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3677-3686 3681 = V ∪RV . Consequently, there exists a rare set RV = M ∪ N such that RV ∩ V = ∅ and F (U) ⊂ V ∪RV . Thus, F is upper rarely m-continuous at x ∈ X. Theorem 2. For a multifunction F : (X,m) → (Y, σ), the following properties are equiv- alent: (1) F is lower rarely m-continuous at x ∈ X; (2) for each open set V of Y such that F (x) ∩ V ̸= ∅, there exists a rare set RV with V ∩RV = ∅ such that x ∈ mInt(F−(V ∪RV )); (3) for each open set V of Y such that F (x) ∩ V ̸= ∅, there exists a rare set RV with Cl(V ) ∩RV = ∅ such that x ∈ mInt(F−(Cl(V ) ∪RV )); (4) for each regular open set V of Y such that F (x) ∩ V ̸= ∅, there exists a rare set RV with V ∩RV = ∅ such that x ∈ mInt(F−(V ∪RV )). Proof. The proofs of (1) ⇒ (2), (2) ⇒ (3) and (3) ⇒ (4) are similar with Theorem 1. (4) ⇒ (1): Let V be an open set in Y such that F (x) ∩ V ̸= ∅. Then F (x) ∩ Int(Cl(V )) ̸= ∅. By (4), there exists a rare set RV with Int(Cl(V )) ∩ RV = ∅ such that x ∈ mInt(F−(Int(Cl(V )) ∪ RV )). Since Cl(V ) \ V is a closed rare set, by Lemma 4, (Cl(V ) \ V ) ∪RV is a rare set. Therefore, SV = [Int(Cl(V )) \ V ] ∪RV is a rare set. And Int(Cl(V ))∪RV = V ∪ [Int(Cl(V ))\V ]∪RV = V ∪SV . Therefore, x ∈ mInt(F−(V ∪SV )). Hence there exists U ∈ m(x) such that U ⊂ F−(V ∪ SV ) and F (u) ∩ (V ∪ SV ) ̸= ∅ for every u ∈ U . Remark 2. If m = τ (resp. α(X), β(X), SO(X)), then we have characterizations in [25] (resp. [4], [14], [15]). By Theorem 1, we have the following characterizations of rare m-continuity for a function f : (X,m) → (Y, σ) Corollary 1. For a function f : (X,m) → (Y, σ), the following properties are equivalent: (1) f is rarely m-continuous at x ∈ X; (2) for each open set V of Y containing f(x), there exists a rare set RV with V ∩RV = ∅ such that x ∈ mInt(f−1(V ∪RV )); (3) for each open set V of Y containing f(x), there exists a rare set RV with Cl(V ) ∩ RV = ∅ such that x ∈ mInt(f−1(Cl(V ) ∪RV )); (4) for each regular open set V of Y containing f(x), there exists a rare set RV with V ∩RV = ∅ such that x ∈ mInt(F−1(V ∪RV )); (5) for each open set V of Y containing f(x), there exists U ∈ m(x) such that Int[f(U)∩ (Y \ V )] = ∅, (6) for each open set V of Y containing f(x), there exists U ∈ m(x) such that Int(f(U)) ⊂ Cl(V ). Remark 3. If m = τ (resp. α, PO(X), SO(X)), then by Corollary 1, we obtain the characterizations of rare continuity [19] (resp. rare α-continuity [13], rare pre-continuity [12], rare quasicontinuity [26]). T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3677-3686 3682 4. Ideal Topological Spaces Definition 8. A nonempty collection I of subsets of a set X is called an ideal on X [17], [30] if it satisfies the following two conditions: (1) A ∈ I and B ⊂ A implies B ∈ I, (2) A ∈ I and B ∈ I implies 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). Let (X, τ, I) be an ideal topological space. For any subset A of X, A⋆(I, τ) = {x ∈ X : U ∩ A /∈ I for every U ∈ τ(x)}, where τ(x) = {U ∈ τ : x ∈ U}, is called the local function of A with respect to τ and I [16]. Hereafter A⋆(I, τ) is simply denoted by A⋆. It is well known that Cl⋆(A) = A ∪ A⋆ defines a Kuratowski closure operator on X and the topology generated by Cl⋆ is denoted by τ⋆. Lemma 5. Let (X, τ, I) be an ideal topological space and A, B be subsets of X. Then the following properties hold: (1) A ⊂ B implies Cl⋆(A) ⊂ Cl⋆(B), (2) Cl⋆(X) = X and Cl⋆(∅) = ∅, (3) Cl⋆(A) ∪ Cl⋆(B) ⊂ Cl⋆(A ∪B). Definition 9. Let (X, τ, I) be an ideal topological space. A subset A of X is said to be (1) α-I-open [8] if A ⊂ Int(Cl⋆(Int(A))), (2) semi-I-open [8] if A ⊂ Cl⋆(Int(A)), (3) pre-I-open [2] if A ⊂ Int(Cl⋆(A)), (4) b-I-open [5] if A ⊂ Int(Cl⋆(A)) ∪ Cl⋆(Int(A)), (5) β-I-open [9] if A ⊂ Cl(Int(Cl⋆(A))), (6) weakly semi-I-open [6] if A ⊂ Cl⋆(Int(Cl(A))), (7) weakly b-I-open [22] if A ⊂ Cl(Int(Cl⋆(A))) ∪ Cl⋆(Int(Cl(A))), (8) strongly β-I-open [7] if A ⊂ Cl⋆(Int(Cl⋆(A))). Among the sets in Definition 9, we have the following relations: DIAGRAM open ⇒ α-I-open ⇒ semi-I-open ⇒ weakly semi-I-open ⇓ ⇓ ⇓ pre-I-open ⇒ b-I-open ⇒ weakly b-I-open ⇓ ⇑ strongly β-I-open ⇒ β-I-open The family of all α-I-open (resp. semi-I-open, pre-I-open, b-I-open, β-I-open, weakly semi-I-open, weakly b-I-open, strongly β-I-open) sets in an ideal topological space (X, τ, I) is denoted by αIO(X) (resp. SIO(X), PIO(X), BIO(X), βIO(X), WSIO(X), WBIO(X), SβIO(X)). T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3677-3686 3683 Remark 4. If I = {∅}, then A⋆ = Cl(A) and Cl⋆(A) = A⋆ ∪A = Cl(A). Therefore, (1) τ⋆ = τ , αIO(X) = α(X), SIO(X) = SO(X), PIO(X) = PO(X), BIO(X) = BO(X) and βIO(X) = β(X). (2) WSIO(X), WBIO(X), SβIO(X) and βIO(X) are coincide with β(X). Definition 10. By mIO(X), we denote each one of the families τ⋆, αIO(X), SIO(X), PIO(X), BIO(X), βIO(X), WSIO(X), WBIO(X), SβIO(X). Lemma 6. Let (X, τ, I) be an ideal topological space. Then mIO(X) is an m-structure and has property B. Proof. The proof follows from Lemma 5(1)(2). As an example, we shall show that αIO(X) has property B. Let Aα be an α-I-open set for each α ∈ Λ. Then Aα ⊂ Int(Cl⋆(Int(Aα))) ⊂ Int(Cl⋆(Int(∪α∈ΛAα))) for each α ∈ Λ and hence ∪α∈ΛAα ⊂ Int(Cl⋆(Int(∪α∈ΛAα))). Therefore, ∪α∈ΛAα is α-I-open. Remark 5. It is shown in Theorem 3.4 of [8] (resp. Theorem 2.10 of [2], Theorem 2.1 of [6], Theorem 2.7 of [22], Proposition 3 of [7]) that SIO(X) (resp. PIO(X), WSIO(X), WBIO(X), SβIO(X)) has property B. Definition 11. Let (X, τ, I) be an ideal topological space. For a subset A of X, the mIO(X)-closure mClI(A) and the mIO(X)-interior mIntI(A) are defined as follows: (1) mClI(A) = ∩{F : A ⊂ F,X \ F ∈ mIO(X)}, (2) mIntI(A) = ∪{U : U ⊂ A,U ∈ mIO(X)}. Let (X, τ, I) be an ideal topological space and mIO(X) the m-structure on X. If mIO(X) = αIO(X) (resp. SIO(X), PIO(X), BIO(X), βIO(X), WSIO(X), WBIO(X), SβIO(X)), then we have (1) mClI(A) = αClI(A) (resp. sClI(A), pClI(A), bClI(A), βClI(A), wsClI(A), wbClI(A), sβClI(A)), (2) mIntI(A) = αIntI(A) (resp. sIntI(A), pIntI(A), bIntI(A), βIntI(A), wsIntI(A), wbIntI(A), sβIntI(A)). 5. Rarely m-I-continuous multifunctions In this section, by using the results in Section 3, we obtain properties of upper/lower rarely m-I-continuous multifunction F : (X, τ, I) → (Y, σ). Definition 12. Let (X, τ, I) be an ideal topological space and (Y, σ) a topological space. A multifunction F : (X, τ, I) → (Y, σ) is said to be (1) upper rarely m-I-continuous at a point x ∈ X if for each open set V containing F (x), there exist a rare set RV with RV ∩ V = ∅ and an mI-open set U ∈ mIO(X) containing x such that F (U) ⊂ V ∪RV , (2) lower rarely m-I-continuous at a point x ∈ X if for each open set V meeting F (x), there exist a rare set RV with RV ∩ V = ∅ and an m-open set U ∈ mIO(X) containing x T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3677-3686 3684 such that F (u) ∩ (V ∪RV ) ̸= ∅ for each u ∈ U , Lemma 7. A multifunction F : (X, τ, I) → (Y, σ) is upper/lower rarely m-I-continuous at x ∈ X if and only if a multifunction F : (X,mIO(X)) → (Y, σ) is upper/lower rarely m-continuous at x ∈ X. Proof. This is obvious from Definitions 7 and 12 Theorem 3. For a multifunction F : (X, τ, I) → (Y, σ), the following properties are equivalent: (1) F is upper rarely m-I-continuous at x ∈ X; (2) for each open set V of Y containing F(x), there exists a rare set RV with V ∩RV = ∅ such that x ∈ mIntI(F +(V ∪RV )); (3) for each open set V of Y containing F(x), there exists a rare set RV with Cl(V ) ∩ RV = ∅ such that x ∈ mIntI(F +(Cl(V ) ∪RV )); (4) for each regular open set V of Y containing F(x), there exists a rare set RV with V ∩RV = ∅ such that x ∈ mIntI(F +(V ∪RV )); (5) for each open set V of Y containing F(x), there exists U ∈ mIO(X) containing x such that Int[F (U) ∩ (Y \ V )] = ∅, (6) for each open set V of Y containing F(x), there exists U ∈ mIO(X) containing x such that Int(F (U)) ⊂ Cl(V ). Theorem 4. For a multifunction F : (X, τ, I) → (Y, σ), the following properties are equivalent: (1) F is lower rarely m-I-continuous at x ∈ X; (2) for each open set V of Y such that F (x) ∩ V ̸= ∅, there exists a rare set RV with V ∩RV = ∅ such that x ∈ mIntI(F −(V ∪RV )); (3) for each open set V of Y such that F (x) ∩ V ̸= ∅, there exists a rare set RV with Cl(V ) ∩RV = ∅ such that x ∈ mIntI(F −(Cl(V ) ∪RV )); (4) for each regular open set V of Y such that F (x) ∩ V ̸= ∅, there exists a rare set RV with V ∩RV = ∅ such that x ∈ mIntI(F −(V ∪RV )). By Theorem 3, we have the following characterizations of rare m-I-continuity of func- tion f : (X, τ, I) → (Y, σ) Corollary 2. 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