EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7051 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Upper and Lower µ(σ1, σ2)-Continuous Multifunctions Nipaporn Chutiman1, 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, Pattani, 94000, Thailand Abstract. This paper presents new concepts of continuous multifunctions defined from a general- ized topological space into a bitopological space, called upper µ(σ1, σ2)-continuous multifunctions and lower µ(σ1, σ2)-continuous multifunctions. Furthermore, several characterizations and some properties concerning upper µ(σ1, σ2)-continuous multifunctions and lower µ(σ1, σ2)-continuous multifunctions are discussed. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper µ(σ1, σ2)-continuous multifunction, lower µ(σ1, σ2)-continuous multifunction 1. Introduction In 2002, Császár [1] introduced the concepts of generalized topological spaces and generalized neighborhood systems. The classes of topological spaces and neighborhood systems are contained in the classes of generalized topological spaces and generalized neighborhood systems, respectively. Moreover, Császár [1] introduced two kinds of gen- eralized continuous functions by utilizing the notions of generalized topological spaces and generalized neighborhood systems. In 2009, Kanibir and Reilly [2] extended the concept of generalized continuous functions to multifunctions and defined upper semi generalized continuous multifunctions and lower semi generalized continuous multifunc- tions. On the other hand, the present authors introduced and investigated four classes of multifunctions defined from a generalized topological space into a generalized topological space, namely upper β(µX , µY )-continuous multifunctions [3], lower β(µX , µY )-continuous multifunctions [3], upper α(µX , µY )-continuous multifunctions [4] and lower α(µX , µY )- continuous multifunctions [4]. Pue-on et al. [5] introduced and studied the concepts of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7051 Email addresses: nipaporn.c@msu.ac.th (N. Chutiman), 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) N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7051 2 of 8 upper (τ1, τ2)-continuous multifunctions and lower (τ1, τ2)-continuous multifunctions. Kla- narong et al. [6] investigated several characterizations of upper (τ1, τ2)-continuous multi- functions 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. [7] 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. Quite recently, Khampakdee et al. [8] pre- sented new classes of continuous multifunctions defined from an ideal topological space into a bitopological space, namely upper τ⋆(σ1, σ2)-continuous multifunctions and lower τ⋆(σ1, σ2)-continuous multifunctions. Furthermore, several characterizations and some properties of upper τ⋆(σ1, σ2)-continuous multifunctions and lower τ⋆(σ1, σ2)-continuous multifunctions were established in [8]. In this paper, we introduce the concepts of upper µ(σ1, σ2)-continuous multifunctions and lower µ(σ1, σ2)-continuous multifunctions. We also investigate several characterizations of upper µ(σ1, σ2)-continuous multifunctions and lower µ(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 [9] 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 [9] 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 [9] of A and is denoted by τ1τ2-Int(A). Lemma 1. [9] 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). A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [10] (resp. (τ1, τ2)s-open [11], (τ1, τ2)p-open [11], (τ1, τ2)β-open [11]) 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)p-closed). A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-δ-open [12] if A is the union N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7051 3 of 8 of (τ1, τ2)r-open sets of X. The complement of a τ1τ2-δ-open set is called τ1τ2-δ-closed [12]. The union of all τ1τ2-δ-open sets of X contained in A is called the τ1τ2-δ-interior [12] 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 [12] 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 [10] 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 [10] 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 [10] 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 [10] of A and is denoted by (τ1, τ2)θ-Int(A). Lemma 2. [10] For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: (1) If A is τ1τ2-open in X, then τ1τ2-Cl(A) = (τ1, τ2)θ-Cl(A). (2) (τ1, τ2)θ-Cl(A) is τ1τ2-closed in X. Let X be a nonempty set, and denote P(X) the power set of X. We call a class µ ⊆ P(X) a generalized topology (briefly, GT) if ∅ ∈ µ, and an arbitrary union of elements of µ belongs to µ [1]. A set X with a GT µ on it is said to be a generalized topological space (briefly, GTS) and is denoted by (X,µ). For a GTS (X,µ), the elements of µ are called µ-open sets and the complements of µ-open sets are called µ-closed sets. For A ⊆ X, we denote by cµ(A) the intersection of all µ-closed sets containing A and by iµ(A) the union of all µ-open sets contained in A. Then, we have iµ(iµ(A)) = iµ(A), cµ(cµ(A)) = cµ(A), and iµ(A) = X − cµ(X −A). According to [13], for A ⊆ X and x ∈ X, we have x ∈ cµ(A) if and only if x ∈ M ∈ µ implies M ∩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 µ(σ1, σ2)-continuous multifunctions In this section, we introduce the concepts of upper µ(σ1, σ2)-continuous multifunctions and lower µ(σ1, σ2)-continuous multifunctions. Furthermore, several characterizations of upper µ(σ1, σ2)-continuous multifunctions and lower µ(σ1, σ2)-continuous multifunctions are discussed. Definition 1. A multifunction F : (X,µ) → (Y, σ1, σ2) is called upper µ(σ1, σ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ⊆ V , there exists a µ-open set U of X containing x such that F (U) ⊆ V . A multifunction F : (X,µ) → (Y, σ1, σ2) is called upper µ(σ1, σ2)-continuous if F is upper µ(σ1, σ2)-continuous at each point x of X. N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7051 4 of 8 Theorem 1. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper µ(σ1, σ2)-continuous; (2) F+(V ) is µ-open in X for every σ1σ2-open set V of Y ; (3) F−(K) is µ-closed in X for every σ1σ2-closed set K of Y ; (4) cµ(F −(B)) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ; (5) F+(σ1σ2-Int(B)) ⊆ iµ(F +(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V and by (1), there exists a µ-open set U of X containing x such that F (U) ⊆ V . Thus, x ∈ U ⊆ F+(V ) and hence x ∈ iµ(F +(V )). Therefore, F+(V ) ⊆ iµ(F +(V )). This shows that F+(V ) is µ-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): Let B be any subset of Y . Then, σ1σ2-Cl(B) is σ1σ2-closed in Y and by (3), cµ(F−(B)) ⊆ cµ(F −(σ1σ2-Cl(B))) = F−(σ1σ2-Cl(B)). (4) ⇒ (5): Let B be any subset of Y . By (4), we have X − iµ(F +(B)) = cµ(X − F+(B)) = cµ(F −(Y −B)) ⊆ F−(σ1σ2-Cl(Y −B)) = F−(Y − σ1σ2-Int(B)) = X − F+(σ1σ2-Int(B)) and so F+(σ1σ2-Int(B)) ⊆ iµ(F +(B)). (5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that F (x) ⊆ V . Then, x ∈ F+(V ) = iµ(F +(V )). There exists a µ-open set U of X containing x such that U ⊆ F+(V ); hence F (U) ⊆ V . This shows that F is upper µ(σ1, σ2)-continuous. Definition 2. A multifunction F : (X,µ) → (Y, σ1, σ2) is called lower µ(σ1, σ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there ex- ists a µ-open set U of X containing x such that F (z) ∩ V ̸= ∅ for every z ∈ U . A multifunction F : (X,µ) → (Y, σ1, σ2) is called lower µ(σ1, σ2)-continuous if F is lower µ(σ1, σ2)-continuous at each point x of X. Theorem 2. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower µ(σ1, σ2)-continuous; (2) F−(V ) is µ-open in X for every σ1σ2-open set V of Y ; (3) F+(K) is µ-closed in X for every σ1σ2-closed set K of Y ; N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7051 5 of 8 (4) cµ(F +(B)) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (5) F (cµ(A)) ⊆ σ1σ2-Cl(F (A)) for every subset A of X; (6) F−(σ1σ2-Int(B)) ⊆ iµ(F −(B)) for every subset B of Y . Proof. We prove only the implications (4) ⇒ (5) and (5) ⇒ (6) being the proofs of the other similar to those of Theorem 1. (4) ⇒ (5): Let A be any subset of X. By (4), we have cµ(A) ⊆ cµ(F +(F (A))) ⊆ F+(σ1σ2-Cl(F (A))) and hence F (cµ(A)) ⊆ σ1σ2-Cl(F (A)). (5) ⇒ (6): Let B be any subset of Y . By (5), F (cµ(F +(Y −B))) ⊆ σ1σ2-Cl(F (F+(Y −B))) ⊆ σ1σ2-Cl(Y −B) = Y − σ1σ2-Int(B). Since F (cµ(F +(Y −B))) = F (cµ(X − F−(B))) = F (X − iµ(F −(B))), we have X − iµ(F −(B)) ⊆ F+(Y − σ1σ2-Int(B)) = X − F−(σ1σ2-Int(B)) and hence F−(σ1σ2-Int(B)) ⊆ iµ(F −(B)). Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)s-regular [7] if for each (τ1, τ2)s-closed set F and each x ̸∈ F , there exist disjoint (τ1, τ2)s-open sets U and V such that x ∈ U and F ⊆ V . Lemma 3. [7] A bitopological space (X, τ1, τ2) is (τ1, τ2)s-regular if and only if for each x ∈ X and each (τ1, τ2)s-open set U containing x, there exists a (τ1, τ2)s-open set V such that x ∈ V ⊆ (τ1, τ2)-sCl(V ) ⊆ U . Lemma 4. [7] Let (X, τ1, τ2) be a (τ1, τ2)s-regular space. Then, the following properties hold: (1) τ1τ2-Cl(A) = τ1τ2-δ-Cl(A) for every subset A of X. (2) Every τ1τ2-open set is τ1τ2-δ-open. Theorem 3. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper µ(σ1, σ2)-continuous; (2) F−(σ1σ2-δ-Cl(B)) is µ-closed in X for every subset B of Y ; (3) F−(K) is µ-closed in X for every σ1σ2-δ-closed set K of Y ; (4) F+(V ) is µ-open in X for every σ1σ2-δ-open set V of Y . N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7051 6 of 8 Proof. (1) ⇒ (2): Let B be any subset of Y . By Lemma 4, σ1σ2-δ-Cl(B) is σ1σ2-closed in Y . Since F is upper µ(σ1, σ2)-continuous, by Theorem 1 we have F−(σ1σ2-δ-Cl(B)) is µ-closed in X. (2) ⇒ (3): Let K be any σ1σ2-δ-closed set of Y . Then, σ1σ2-δ-Cl(K) = K and by (2), F−(K) is µ-closed in X. (3) ⇒ (4): This follows from the fact that F+(Y −B) = X −F−(B) for any subset B of Y . (4) ⇒ (1): Let V be any σ1σ2-open set of Y . Since (Y, σ1, σ2) is (σ1, σ2)s-regular, we have V is σ1σ2-δ-open in Y and by (4), F+(V ) is µ-open in X. Thus, F is upper µ(σ1, σ2)-continuous by Theorem 1. Theorem 4. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower µ(σ1, σ2)-continuous; (2) F+(σ1σ2-δ-Cl(B)) is µ-closed in X for every subset B of Y ; (3) F+(K) is µ-closed in X for every σ1σ2-δ-closed set K of Y ; (4) F−(V ) is µ-open in X for every σ1σ2-δ-open set V of Y . Proof. The proof is similar to that of Theorem 3. Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-regular [14] if for each τ1τ2-closed set F and each x ̸∈ F , there exist disjoint τ1τ2-open sets U and V such that x ∈ U and F ⊆ V . Lemma 5. [14] A bitopological space (X, τ1, τ2) is (τ1, τ2)-regular if and only if for each x ∈ X and each τ1τ2-open set U containing x, there exists a τ1τ2-open set V such that x ∈ V ⊆ τ1τ2-Cl(V ) ⊆ U . Lemma 6. [6] Let (X, τ1, τ2) be a (τ1, τ2)-regular space. Then, the following properties hold: (1) τ1τ2-Cl(A) = (τ1, τ2)θ-Cl(A) for every subset A of X. (2) Every τ1τ2-open set is (τ1, τ2)θ-open. Theorem 5. For a multifunction F : (X,µ) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)- regular, the following properties are equivalent: (1) F is upper µ(σ1, σ2)-continuous; (2) F−((σ1, σ2)θ-Cl(B)) is µ-closed in X for every subset B of Y ; (3) F−(K) is µ-closed in X for every (σ1, σ2)θ-closed set K of Y ; (4) F+(V ) is µ-open in X for every (σ1, σ2)θ-open set V of Y . N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7051 7 of 8 Proof. (1) ⇒ (2): Let B be any subset of Y . By Lemma 6, (σ1, σ2)θ-Cl(B) is σ1σ2- closed in Y . Since F is upper µ(σ1, σ2)-continuous, by Theorem 1 F−((σ1, σ2)θ-Cl(B)) is µ-closed in X. (2) ⇒ (3): Let K be any (σ1, σ2)θ-closed set of Y . Then, (σ1, σ2)θ-Cl(K) = K and by (2), we have F−(K) is µ-closed in X. (3) ⇒ (4): This follows from the fact that F+(Y −B) = X −F−(B) for any subset B of Y . (4) ⇒ (1): Let V be any σ1σ2-open set of Y . Since (Y, σ1, σ2) is (σ1, σ2)-regular, we have V is (σ1, σ2)θ-open in Y and by (4), F+(V ) is µ-open in X. Thus, F is upper µ(σ1, σ2)-continuous by Theorem 1. Theorem 6. For a multifunction F : (X,µ) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)- regular, the following properties are equivalent: (1) F is lower µ(σ1, σ2)-continuous; (2) F+((σ1, σ2)θ-Cl(B)) is µ-closed in X for every subset B of Y ; (3) F+(K) is µ-closed in X for every (σ1, σ2)θ-closed set K of Y ; (4) F−(V ) is µ-open in X for every (σ1, σ2)θ-open set V of Y . Proof. The proof is similar to that of Theorem 5. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] Á. Császár. Generalized topology, generalized continuity. Acta Mathematica Hun- garica, 96(4):351–357, 2002. [2] A. Kanibir and I. L. Reilly. Generalized continuity for multifunctions. Acta Mathe- matica Hungarica, 122(3):283–292, 2009. [3] C. Boonpok. On upper and lower β(µX , µY )-continuous multifunctions. International Journal of Mathematics and Mathematical Sciences, 2012:931656, 2012. [4] N. Srisarakham and C. Boonpok. Characterizations of upper and lower α(µX , µY )- continuous multifunctions. Journal of Mathematics and Computer Science, 17:255– 265, 2017. [5] P. Pue-on, S. Sompong, and C. Boonpok. Upper and lower (τ1, τ2)-continuous multi- functions. International Journal of Mathematics and Computer Science, 19(4):1305– 1310, 2024. [6] C. Klanarong, S. Sompong, and C. Boonpok. (τ1, τ2)-continuity and (τ1, τ2)θ-closed sets. International Journal of Mathematics and Computer Science, 19(4):1299–1304, 2024. N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7051 8 of 8 [7] M. Thongmoon, S. Sompong, and C. Boonpok. (τ1, τ2)-continuous multifunctions and τ1τ2-δ-open sets. International Journal of Mathematics and Computer Science, 19(4):1369–1375, 2024. [8] J. Khampakdee, A. Sama-Ae, and C. Boonpok. Upper and lower continuous multi- functions defined between an ideal topological space and a bitopological space. Eu- ropean Journal of Pure and Applied Mathematics, 18(3):6565, 2025. [9] C. Boonpok, C. Viriyapong, and M. Thongmoon. On upper and lower (τ1, τ2)- precontinuous multifunctions. Journal of Mathematics and Computer Science, 18:282– 293, 2018. [10] C. Viriyapong and C. Boonpok. (τ1, τ2)α-continuity for multifunctions. Journal of Mathematics, 2020:6285763, 2020. [11] C. Boonpok. (τ1, τ2)δ-semicontinuous multifunctions. Heliyon, 6:e05367, 2020. [12] C. Boonpok and P. Pue-on. Characterizations of almost (τ1, τ2)-continuous multi- functions. International Journal of Analysis and Applications, 22:33, 2024. [13] Á. Császár. δ-and θ-modifications of generalized topologies. Acta Mathematica Hun- garica, 120:274–279, 2008. [14] M. Chiangpradit, S. Sompong, and C. Boonpok. On characterizations of (τ1, τ2)- regular spaces. International Journal of Mathematics and Computer Science, 19(4):1329–1334, 2024.