EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5633 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and Lower Near (τ1, τ2)-continuity Montri Thongmoon1, 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 classes of multifunctions called upper nearly (τ1, τ2)-continuous multifunctions and lower nearly (τ1, τ2)-continuous multifunctions. Furthermore, some character- izations of upper nearly (τ1, τ2)-continuous multifunctions and lower nearly (τ1, τ2)-continuous multifunctions are established. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper nearly (τ1, τ2)-continuous multifunction, lower nearly (τ1, τ2)- continuous multifunction 1. Introduction The field of the mathematical science which goes under the name of topology is concerned with all questions directly or indirectly related to continuity. Weaker and stronger forms of open sets play an important role in the generalization of different forms of continuity. Using different forms of open sets, several authors have introduced and investigated various types of continuity for functions and multifunctions. Carna- han [30] introduced the notion of N-closed sets in topological spaces. Noiri [44] stud- ied several properties of N-closed sets and some separation axioms. The concept of N-continuous functions was introduced by Malghan and Hanchinamani [43]. Noiri and Ergun [45] investigated some characterizations of N-continuous functions. Viriyapong and Boonpok [61] investigated some characterizations of (Λ, sp)-continuous functions by utilizing the notions of (Λ, sp)-open sets and (Λ, sp)-closed sets due to Boonpok and Khampakdee [12]. Dungthaisong et al. [36] introduced and studied the concept of g(m,n)-continuous functions. Duangphui et al. [35] introduced and investigated the no- tion of (µ, µ′)(m,n)-continuous functions. Furthermore, several characterizations of almost ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5633 Email addresses: montri.t@msu.ac.th (M. Thongmoon), 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) M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5633 2 of 13 (Λ, p)-continuous functions, strongly θ(Λ, p)-continuous functions, almost strongly θ(Λ, p)- continuous functions, θ(Λ, p)-continuous functions, weakly (Λ, b)-continuous functions, θ(⋆)-precontinuous functions, (Λ, p(⋆))-continuous functions, ⋆-continuous functions, θ-I - continuous functions, almost (g,m)-continuous functions, pairwise almost M -continuous functions, (τ1, τ2)-continuous functions, almost (τ1, τ2)-continuous functions, weakly (τ1, τ2)- continuous functions and faintly (τ1, τ2)-continuous functions were presented in [54], [57], [16], [48], [25], [11], [8], [10], [4], [1], [2], [26], [23], [18] and [55], respectively. Chiangpra- dit et al. [33] introduced and investigated the notion of weakly quasi (τ1, τ2)-continuous functions. Kong-ied et al. [42] introduced and studied the concept of almost quasi (τ1, τ2)- continuous functions. Thongmoon et al. [59] introduced and investigated the notion of rarely (τ1, τ2)-continuous functions. In 2003, Ekici [37] introduced and studied the concept of nearly continuous multifunc- tions as a generalization of semi-continuous multifunctions and N-continuous functions. Moreover, Ekici [38] introduced and investigated the notion of almost nearly continuous multifunctions as a generalization of nearly continuous multifunctions and almost con- tinuous multifunctions [47]. Furthermore, several characterizations and some properties concerning (τ1, τ2)δ-semicontinuous multifunctions, almost weakly (τ1, τ2)-continuous mul- tifunctions, weakly quasi (Λ, sp)-continuous multifunctions, ⋆-continuous multifunctions, β(⋆)-continuous multifunctions, α-⋆-continuous multifunctions, almost α-⋆-continuous mul- tifunctions, almost quasi ⋆-continuous multifunctions, weakly α-⋆-continuous multifunc- tions, sβ(⋆)-continuous multifunctions, weakly sβ(⋆)-continuous multifunctions, θ(⋆)-quasi continuous multifunctions, almost ı⋆-continuous multifunctions, weakly (Λ, sp)-continuous multifunctions, α(Λ, sp)-continuous multifunctions, almost α(Λ, sp)-continuous multifunc- tions, weakly α(Λ, sp)-continuous multifunctions, almost β(Λ, sp)-continuous multifunc- tions, slightly (Λ, sp)-continuous multifunctions, (τ1, τ2)-continuous multifunctions, al- most (τ1, τ2)-continuous multifunctions, weakly (τ1, τ2)-continuous multifunctions, weakly quasi (τ1, τ2)-continuous multifunctions, almost quasi (τ1, τ2)-continuous multifunctions, c- (τ1, τ2)-continuous multifunctions, c-quasi (τ1, τ2)-continuous multifunctions and s-(τ1, τ2)p- continuous multifunctions were established in [5], [28], [62], [3], [7], [17], [24], [6], [21], [20], [15], [9], [19], [22], [39], [13], [27], [56], [14], [51], [41], [58], [52], [50], [40], [49] and [64], respectively. Noiri and Popa [46] introduced and studied the notion of almost nearly m- continuous multifunctions as multifunctions from a set satisfying some minimal conditions into a topological spaces. Carpintero et al. [31] introduced and studied the notion of nearly ω-continuous multifunctions as a weaker form of nearly continuous multifunctions. Rosas et al. [53] introduced and studied upper and lower almost nearly continuous multi- functions using notions of topological ideals. In this paper, we introduce the concepts of upper nearly (τ1, τ2)-continuous multifunctions and lower nearly (τ1, τ2)-continuous multi- functions. We also investigate several characterizations of upper nearly (τ1, τ2)-continuous multifunctions and lower nearly (τ1, τ2)-continuous multifunctions. M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5633 3 of 13 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 [29] if A = τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is called τ1τ2-open. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection of all τ1τ2-closed sets of X containing A is called the τ1τ2-closure [29] 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 [29] of A and is denoted by τ1τ2-Int(A). Lemma 1. [29] 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 τ1τ2-clopen [29] if A is both τ1τ2-open and τ1τ2-closed. A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [60] (resp. (τ1, τ2)s-open [5], (τ1, τ2)p-open [5], (τ1, τ2)β-open [5]) 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 called (τ1, τ2)r-closed (resp. (τ1, τ2)s-closed, (τ1, τ2)p- closed, (τ1, τ2)β-closed). A subset A of a bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-open [63] if A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)- open set is said to be α(τ1, τ2)-closed. A subset A of a bitopological space (X, τ1, τ2) is said to be N (τ1, τ2)-closed if every cover of A by (τ1, τ2)r-open sets of X has a finite subcover. Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is called a (τ1, τ2)θ-cluster point [60] 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 [60] 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 [60] 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 [60] of A and is denoted by (τ1, τ2)θ-Int(A). Lemma 2. [60] For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5633 4 of 13 (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. 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 nearly (τ1, τ2)-continuous multifunctions In this section, we introduce the notions of upper nearly (τ1, τ2)-continuous multifunc- tions and lower nearly (τ1, τ2)-continuous multifunctions. Moreover, several characteriza- tions of upper nearly (τ1, τ2)-continuous multifunctions and lower nearly (τ1, τ2)-continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper nearly (τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing F (x) and having N (σ1, σ2)-closed complement, there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ V . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper nearly (τ1, τ2)-continuous if F is upper nearly (τ1, τ2)-continuous at each point x of X. Theorem 1. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper nearly (τ1, τ2)-continuous at x ∈ X; (2) x ∈ τ1τ2-Int(F +(V )) for each σ1σ2-open set V of Y containing F (x) and having N (σ1, σ2)-closed complement; (3) x ∈ F−(σ1σ2-Cl(B)) for each subset B of Y having the N (σ1, σ2)-closed σ1σ2- closure such that x ∈ τ1τ2-Cl(F −(B)); (4) x ∈ τ1τ2-Int(F +(B)) for each subset B of Y such that Y −σ1σ2-Int(B) is N (σ1, σ2)- closed and x ∈ F+(σ1σ2-Int(B)). Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing F (x) and having N (σ1, σ2)-closed complement and x ∈ F+(V ). By (1), there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ V . Thus, x ∈ U ⊆ F+(V ). Since U is τ1τ2-open, we have x ∈ τ1τ2-Int(F +(V )). (2) ⇒ (3): Let B be any subset of Y having the N (σ1, σ2)-closed σ1σ2-closure. Then, σ1σ2-Cl(B) is σ1σ2-closed and Y −σ1σ2-Cl(B) is a σ1σ2-open set having N (σ1, σ2)-closed complement. Suppose that x ̸∈ F−(σ1σ2-Cl(B)). Then, we have x ∈ X − F−(σ1σ2-Cl(B)) = F+(Y − σ1σ2-Cl(B)) M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5633 5 of 13 and hence F (x) ⊆ Y − σ1σ2-Cl(B). Since Y − σ1σ2-Cl(B) is a σ1σ2-open set having N (σ1, σ2)-closed complement, by (2) we have x ∈ τ1τ2-Int(F +(Y − σ1σ2-Cl(B))) = τ1τ2-Int(X − F−(σ1σ2-Cl(B))) = X − τ1τ2-Cl(F −(σ1σ2-Cl(B))) ⊆ X − τ1τ2-Cl(F −(B)). Thus, x ̸∈ τ1τ2-Cl(F −(B)). (3) ⇒ (4): Let B be any subset of Y such that Y − σ1σ2-Int(B) is N (σ1, σ2)-closed. Suppose that x ̸∈ τ1τ2-Int(F +(B)). Then, we have x ∈ X − τ1τ2-Int(F +(B)) = τ1τ2-Cl(X − F+(B)) = τ1τ2-Cl(F −(Y −B)) and by (3), x ∈ F−(σ1σ2-Cl(Y − B)) = F−(Y − σ1σ2-Int(B)) = X − F+(σ1σ2-Int(B)). Thus, x ̸∈ F+(σ1σ2-Int(B)). (4) ⇒ (1): Let V be any σ1σ2-open set of Y containing F (x) and having N (σ1, σ2)- closed complement. Then, Y − σ1σ2-Int(V ) = Y − V which is N (σ1, σ2)-closed and x ∈ F+(σ1σ2-Int(V )). By (4), we have x ∈ τ1τ2-Int(F +(V )). Therefore, there exists a τ1τ2-open set U of X containing x such that x ∈ U ⊆ F+(V ). Thus, F (U) ⊆ V . This shows that F is upper nearly (τ1, τ2)-continuous at x. Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower nearly (τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x)∩V ̸= ∅ and having N (σ1, σ2)-closed complement, there exists a τ1τ2-open set U of X containing x such that F (z) ∩ V ̸= ∅ for each z ∈ U . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower nearly (τ1, τ2)-continuous if F is lower nearly (τ1, τ2)-continuous at each point x of X. Theorem 2. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower nearly (τ1, τ2)-continuous at x ∈ X; (2) x ∈ τ1τ2-Int(F −(V )) for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅ and having N (σ1, σ2)-closed complement; (3) x ∈ F+(σ1σ2-Cl(B)) for each subset B of Y having N (σ1, σ2)-closed σ1σ2-closure such that x ∈ τ1τ2-Cl(F +(B)); (4) x ∈ τ1τ2-Int(F −(B)) for each subset B of Y such that Y −σ1σ2-Int(B) is N (σ1, σ2)- closed and x ∈ F−(σ1σ2-Int(B)). Proof. The proof is similar to that of Theorem 1. Theorem 3. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5633 6 of 13 (1) F is upper nearly (τ1, τ2)-continuous; (2) F+(V ) is τ1τ2-open in X for each σ1σ2-open set V of Y having N (σ1, σ2)-closed complement; (3) F−(K) is τ1τ2-closed in X for every N (σ1, σ2)-closed and σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(F −(B)) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y having the N (σ1, σ2)- closed σ1σ2-closure; (5) F+(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F +(B)) for every subset B of Y such that Y−σ1σ2-Int(B) is N (σ1, σ2)-closed. Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing F (x) and having N (σ1, σ2)-closed complement and x ∈ F+(V ). Then, we have F (x) ⊆ V . By Theorem 1, x ∈ τ1τ2-Int(F +(V )). Thus, F+(V ) ⊆ τ1τ2-Int(F +(V )) and hence F+(V ) is τ1τ2-open in X. (2) ⇒ (3): The proof follows immediately from the fact that F+(Y −B) = Y −F−(B) for every subset B of Y . (3) ⇒ (4): Let B be any subset of Y having the N (σ1, σ2)-closed σ1σ2-closure. Then, σ1σ2-Cl(B) is σ1σ2-closed and by (3), F−(σ1σ2-Cl(B)) is τ1τ2-closed in X. Thus, F−(B) ⊆ F−(σ1σ2-Cl(B)) = τ1τ2-Cl(σ1σ2-Cl(B)) and hence τ1τ2-Cl(F −(B)) ⊆ F−(σ1σ2-Cl(B)). (4) ⇒ (5): Let B be any subset of Y such that Y − σ1σ2-Int(B) is N (σ1, σ2)-closed. Then by (4), we have X − τ1τ2-Int(F +(B)) = τ1τ2-Cl(X − F+(B)) = τ1τ2-Cl(F −(Y −B)) ⊆ τ1τ2-Cl(F −(Y − σ1σ2-Int(B))) ⊆ F−(Y − σ1σ2-Int(B)) = X − F+(σ1σ2-Int(B)). Thus, F+(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F +(B)). (5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing F (x) and having N (σ1, σ2)-closed complement. Thus by (5), x ∈ F+(V ) = F+(σ1σ2-Int(V )) ⊆ τ1τ2-Int(F +(V )). By Theorem 1, F is upper nearly (τ1, τ2)-continuous at x. This shows that F is upper nearly (τ1, τ2)-continuous. Theorem 4. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5633 7 of 13 (1) F is lower nearly (τ1, τ2)-continuous; (2) F−(V ) is τ1τ2-open in X for each σ1σ2-open set V of Y having N (σ1, σ2)-closed complement; (3) F+(K) is τ1τ2-open in X for every N (σ1, σ2)-closed and σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(F +(B)) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y having the N (σ1, σ2)- closed σ1σ2-closure; (5) F−(σ1σ2-Cl(B)) ⊆ τ1τ2-Int(F −(B)) for every subset B of Y such that Y−σ1σ2-Int(B) is N (σ1, σ2)-closed. Proof. The proof is similar to that of Theorem 3. Corollary 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper nearly (τ1, τ2)-continuous if F−(K) is τ1τ2-closed in X for every N (σ1, σ2)-closed set K of Y . Proof. Let V be any σ1σ2-open set of Y having N (σ1, σ2)-closed complement. Then, Y −V is N (σ1, σ2)-closed. By the hypothesis, F−(Y −V ) = X −F+(V ) is τ1τ2-closed in X and hence F+(V ) is τ1τ2-open in X. It follows from Theorem 3 that F is upper nearly (τ1, τ2)-continuous. Corollary 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower nearly (τ1, τ2)-continuous if F+(K) is τ1τ2-closed in X for every N (σ1, σ2)-closed set K of Y . Proof. The proof is similar to that of Corollary 1. Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-regular [32] if for each τ1τ2-closed set F and each point x ∈ X − F , there exist disjoint τ1τ2-open sets U and V such that x ∈ U and F ⊆ V . Theorem 5. Let (Y, σ1, σ2) be a (σ1, σ2)-regular space. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper nearly (τ1, τ2)-continuous; (2) F−((σ1, σ2)θ-Cl(B)) is τ1τ2-closed in X for every subset B of Y such that (σ1, σ2)θ-Cl(B) is N (σ1, σ2)-closed; (3) F−(K) is τ1τ2-closed in X for every N (σ1, σ2)-closed and (σ1, σ2)θ-closed set K of Y ; M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5633 8 of 13 (4) F+(V ) is τ1τ2-open in X for each (σ1, σ2)θ-open set V of Y having N (σ1, σ2)-closed complement. Proof. (1) ⇒ (2): Let B be any subset of Y such that (σ1, σ2)θ-Cl(B) is N (σ1, σ2)- closed. Then, (σ1, σ2)θ-Cl(B) is N (σ1, σ2)-closed and σ1σ2-closed. Thus by Theorem 3, F−((σ1, σ2)θ-Cl(B)) is τ1τ2-closed in X . (2) ⇒ (3): Let K be any N (σ1, σ2)-closed and (σ1, σ2)θ-closed set of Y . Then, we have K = (σ1, σ2)θ-Cl(K) is N (σ1, σ2)-closed and by (2), F−(K) is τ1τ2-closed in X. (3) ⇒ (4): Let V be any (σ1, σ2)θ-open set of Y having N (σ1, σ2)-closed complement. Then, Y − V is N (σ1, σ2)-closed and (σ1, σ2)θ-closed. By (3), F−(Y − V ) = X − F+(V ) is τ1τ2-closed in X and hence F+(V ) is τ1τ2-open in X. (4) ⇒ (1): Let V be any σ1σ2-open set of Y having N (σ1, σ2)-closed complement. Since (Y, σ1, σ2) is (σ1, σ2)-regular, V is (σ1, σ2)θ-open in Y and having N (σ1, σ2)-closed complement. By (4), we have F+(V ) is τ1τ2-open in X and by Theorem 3, F is upper nearly (τ1, τ2)-continuous. Theorem 6. Let (Y, σ1, σ2) be a (σ1, σ2)-regular space. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower nearly (τ1, τ2)-continuous; (2) F+((σ1, σ2)θ-Cl(B)) is τ1τ2-closed in X for every subset B of Y such that (σ1, σ2)θ-Cl(B) is N (σ1, σ2)-closed; (3) F+(K) is τ1τ2-closed in X for every N (σ1, σ2)-closed (σ1, σ2)θ-closed set K of Y ; (4) F−(V ) is τ1τ2-open in X for each (σ1, σ2)θ-open set V of Y having N (σ1, σ2)-closed complement. Proof. The proof is similar to that of Theorem 5. 4. Some results on near (τ1, τ2)-continuity Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-T2 [34] if for any pair of distinct points x, y in X, there exist disjoint τ1τ2-open sets U and V of X containing x and y, respectively. Definition 3. A bitopological space (X, τ1, τ2) is called N (τ1, τ2)-normal if for each dis- joint τ1τ2-closed sets K and H of X, there exist τ1τ2-open sets U and V having N (σ1, σ2)- closed complements such that K ⊆ U , H ⊆ V and U ∩ V = ∅. M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5633 9 of 13 Theorem 7. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an upper nearly (τ1, τ2)-continuous multi- function satisfying the following conditions: (1) F (x) is σ1σ2-closed in Y for each x ∈ X, (2) F (x) ∩ F (y) = ∅ for each distinct points x, y ∈ X, and (3) (Y, σ1, σ2) is an N (σ1, σ2)-normal space, then (X, τ1, τ2) is (τ1, τ2)-T2. Proof. Let x and y be distinct points of X. Then, we have F (x) ∩ F (y) = ∅. Since F (x) and F (y) are σ1σ2-closed and (Y, σ1, σ2) is N (σ1, σ2)-normal, there exist disjoint σ1σ2-open sets U and V having N (σ1, σ2)-closed complements such that F (x) ⊆ U and F (y) ⊆ V . By Theorem 3, F+(U) and F+(V ) are τ1τ2-open in X containing x and y, respectively, such that F+(U) ∩ F+(V ) = ∅. This shows that (X, τ1, τ2) is (τ1, τ2)-T2. Theorem 8. Let (X, τ1, τ2) be a bitopological space. If for each pair of distinct points x and x′ in X, there exists a multifunction F from (X, τ1, τ2) into an N (σ1, σ2)-normal space (Y, σ1, σ2) satisfying the following conditions: (1) F (x) and F (x′) are σ1σ2-closed in Y , (2) F is upper nearly (τ1, τ2)-continuous at x and x′, and (3) F (x) ∩ F (x′) = ∅, then (X, τ1, τ2) is (τ1, τ2)-T2. Proof. Let x and x′ be distinct points of X. Then, we have F (x) ∩ F (x′) = ∅. Since F (x) and F (x′) are σ1σ2-closed and (Y, σ1, σ2) is N (σ1, σ2)-normal, there exist disjoint σ1σ2-open sets V and V ′ having N (σ1, σ2)-closed complements such that F (x) ⊆ V and F (x′) ⊆ V ′. Since F is upper nearly (τ1, τ2)-continuous at x and x′, there exist τ1τ2-open sets U and U ′ of X containing x and x′, respectively, such that F (U) ⊆ V and F (U ′) ⊆ V ′. This implies that U ∩ U ′ = ∅. Thus, (X, τ1, τ2) is (τ1, τ2)-T2. Definition 4. A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-dense on X if τ1τ2-Cl(A) = X. Theorem 9. Let (X, τ1, τ2) be a bitopological space and (Y, σ1, σ2) be an N (σ1, σ2)-normal space. If the following four conditions are satisfied: (1) F : (X, τ1, τ2) → (Y, σ1, σ2) is upper nearly (τ1, τ2)-continuous, (2) G : (X, τ1, τ2) → (Y, σ1, σ2) is upper nearly (τ1, τ2)-continuous, (3) F (x) and G(x) are σ1σ2-closed in Y for each x ∈ X, and (4) A = {x ∈ X | F (x) ∩G(x) ̸= ∅}, M. Thongmoon, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5633 10 of 13 then A is τ1τ2-closed. If F (x) ∩ G(x) ̸= ∅ for each point x in a τ1τ2-dense set D of X, then F (x) ∩G(x) ̸= ∅ for each point x ∈ X. Proof. Suppose that x ̸∈ A. Then, F (x) ∩ G(x) = ∅. Since F (x) and G(x) are σ1σ2-closed and (Y, σ1, σ2) is N (σ1, σ2)-normal, there exist σ1σ2-open sets V and W in Y having N (σ1, σ2)-closed complements such that F (x) ⊆ V , G(x) ⊆ W and V ∩W = ∅. Since F is upper nearly (τ1, τ2)-continuous at x, there exists a τ1τ2-open set U ′ of X containing x such that F (U ′) ⊆ V . Since G is upper nearly (τ1, τ2)-continuous at x, there exists a τ1τ2-open set U ′′ of X containing x such that F (U ′′) ⊆ W . Now set U = U ′ ∩U ′′, then U is τ1τ2-open in X and U ∩A = ∅. Thus, x ̸∈ τ1τ2-Cl(A) and hence A = τ1τ2-Cl(A). This shows that A is τ1τ2-closed. On the other hand, if F (x) ∩G(x) ̸= ∅ on a τ1τ2-dense set D of X, then we have X = τ1τ2-Cl(D) ⊆ τ1τ2-Cl(A) = A. Thus, F (x) ∩G(x) ̸= ∅ for each x ∈ X. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] C. Boonpok. Almost (g,m)-continuous functions. International Journal of Mathe- matical Analysis, 4(40):1957–1964, 2010. 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