EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5650 ISSN 1307-5543 – ejpam.com Published by New York Business Global Almost Near (τ1, τ2)-continuity for 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 classes of multifunctions called upper almost nearly (τ1, τ2)- continuous multifunctions and lower almost nearly (τ1, τ2)-continuous multifunctions. Moreover, several characterizations and some properties concerning upper almost nearly (τ1, τ2)-continuous multifunctions and lower almost nearly (τ1, τ2)-continuous multifunctions are established. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper almost nearly (τ1, τ2)-continuous multifunction, lower almost nearly (τ1, τ2)-continuous multifunction 1. Introduction In 1968, Singal and Singal [56] introduced the concept of almost continuous functions as a generalization of continuity. Popa [46] defined almost quasi-continuous functions as a generalization of almost continuity and quasi-continuity [42]. Munshi and Bassan [43] studied the notion of almost semi-continuous functions. Maheshwari et al. [40] introduced the concept of almost feebly continuous functions as a generalization of almost continu- ity. In 1984, Malghan and Hanchinamani [41] introduced the concept of N-continuous functions. Noiri and Ergun [44] investigated some characterizations of N-continuous func- tions. Ekici [34] introduced and studied the concept of nearly continuous multifunc- tions as a generalization of semi-continuous multifunctions and N-continuous functions. Viriyapong and Boonpok [65] investigated some characterizations of (Λ, sp)-continuous functions by utilizing the notions of (Λ, sp)-open sets and (Λ, sp)-closed sets due to Boon- pok and Khampakdee [12]. Dungthaisong et al. [33] introduced and studied the concept of g(m,n)-continuous functions. Duangphui et al. [32] 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.v18i1.5650 Email addresses: nipaporn.c@msu.ac.th (N. Chutiman), areeyuth.s@snru.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 (1) (2025), 5650 2 of 18 (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 and weakly (τ1, τ2)-continuous functions were presented in [57], [60], [16], [49], [25], [11], [8], [10], [4], [1], [2], [26], [23] and [18], respectively. Srisarakham et al. [58] introduced and stud- ied the concept of faintly (τ1, τ2)-continuous functions. Kong-ied et al. [39] introduced and investigated the notion of almost quasi (τ1, τ2)-continuous functions. Chiangpradit et al. [31] introduced and studied the concept of weakly quasi (τ1, τ2)-continuous functions. Thongmoon et al. [63] introduced and investigated the notion of rarely (τ1, τ2)-continuous functions. In 2004, Ekici [35] introduced and investigated the notion of almost nearly contin- uous multifunctions as a generalization of nearly continuous multifunctions and almost continuous multifunctions [47]. In 2009, Noiri and Popa [45] introduced and studied the notion of almost nearlym-continuous multifunctions as multifunctions from a set satisfying some minimal conditions into a topological spaces. Carpintero et al. [30] introduced and studied the notion of nearly ω-continuous multifunctions as a weaker form of nearly con- tinuous multifunctions. Moreover, several characterizations and some properties concern- ing (τ1, τ2)δ-semicontinuous multifunctions, almost weakly (τ1, τ2)-continuous multifunc- tions, weakly quasi (Λ, sp)-continuous multifunctions, ⋆-continuous multifunctions, β(⋆)- continuous multifunctions, α-⋆-continuous multifunctions, almost α-⋆-continuous multi- functions, 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], [66], [3], [7], [17], [24], [6], [21], [20], [15], [9], [19], [22], [36], [13], [27], [59], [14], [52], [38], [62], [53], [51], [37], [50] and [70], respectively. Rosas et al. [54] introduced and studied upper almost nearly continuous mul- tifunctions and lower almost nearly continuous multifunctions using notions of topological ideals. Rychlewicz [55] introduced and studied the notion of nearly quasi-continuous mul- tifunctions as a generalization of almost nearly continuous multifunctions and almost quasi continuous multifunctions [48]. In this paper, we introduce the concepts of upper almost nearly (τ1, τ2)-continuous multifunctions and lower almost nearly (τ1, τ2)-continuous mul- tifunctions. We also investigate several characterizations of upper almost nearly (τ1, τ2)- continuous multifunctions and lower almost nearly (τ1, τ2)-continuous multifunctions. N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5650 3 of 18 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). A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [64] (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 [69] 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. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection of all (τ1, τ2)p-closed (resp. (τ1, τ2)s-closed, α(τ1, τ2)- closed) sets of X containing A is called the (τ1, τ2)p-closure [68] (resp. (τ1, τ2)s-closure [5], α(τ1, τ2)-closure [67]) of A and is denoted by (τ1, τ2)-pCl(A) (resp. (τ1, τ2)-sCl(A), α(τ1, τ2)-Cl(A)). The union of all (τ1, τ2)p-open (resp. (τ1, τ2)s-open, α(τ1, τ2)-open) sets of X contained in A is called the (τ1, τ2)p-interior [68] (resp. (τ1, τ2)s-interior [5], α(τ1, τ2)-interior [67]) of A and is denoted by (τ1, τ2)-pInt(A) (resp. (τ1, τ2)-sInt(A), α(τ1, τ2)-Int(A)). A subset A of a bitopological space (X, τ1, τ2) is said to be N (τ1, τ2)- closed [61] if every cover of A by (τ1, τ2)r-open sets of X has a finite subcover. Lemma 1. For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: (1) (τ1, τ2)-sCl(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∪A [5]; (2) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A [52]. Lemma 2. Let (X, τ1, τ2) be a bitopological space. If V is a τ1τ2-open set of X hav- ing N (τ1, τ2)-closed complement, then τ1τ2-Int(τ1τ2-Cl(V )) is a (τ1, τ2)r-open set having N (τ1, τ2)-closed complement. Proof. It is obvious that τ1τ2-Int(τ1τ2-Cl(V )) is a (τ1, τ2)r-open set. Let us denote K = X − τ1τ2-Int(τ1τ2-Cl(V )). Of course, K ⊆ X − V . Let {Uγ | γ ∈ Γ} be a τ1τ2-open cover of the set K. Then, {Uγ | γ ∈ Γ} ∪ (X −K) is a τ1τ2-open cover of the set X − V . Thus, there exist indexes γ1, γ2, ..., γk such that X − V ⊆ k ∪ i=1 τ1τ2-Int(τ1τ2-Cl(Uγi)) ∪ τ1τ2-Int(τ1τ2-Cl(X −K)). N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5650 4 of 18 Since τ1τ2-Int(τ1τ2-Cl(X−K)) = τ1τ2-Int(τ1τ2-Cl(V )), τ1τ2-Int(τ1τ2-Cl(X−K))∩K = ∅. It was shown that K ⊆ k ∪ i=1 τ1τ2-Int(τ1τ2-Cl(Uγi)). The proof of N (τ1, τ2)-closedness of the set K is finished. Lemma 3. Let (X, τ1, τ2) be a bitopological space. If V is a (τ1, τ2)p-open set of X hav- ing N (τ1, τ2)-closed complement, then τ1τ2-Int(τ1τ2-Cl(V )) is a (τ1, τ2)r-open set having N (τ1, τ2)-closed complement. Proof. It is evident that τ1τ2-Int(τ1τ2-Cl(V )) is a (τ1, τ2)r-open set. Since V is (τ1, τ2)p- open, V ⊆ τ1τ2-Int(τ1τ2-Cl(V )) and hence X − τ1τ2-Int(τ1τ2-Cl(V )) ⊆ X − V . By the hypothesis, X − V is N (τ1, τ2)-closed and X − τ1τ2-Int(τ1τ2-Cl(V )) is (τ1, τ2)r-closed. Thus, it follows from Lemma 2 that X − τ1τ2-Int(τ1τ2-Cl(V )) is N (τ1, τ2)-closed. 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 almost nearly (τ1, τ2)-continuous multifunctions In this section, we introduce the notions of upper almost nearly (τ1, τ2)-continuous multifunctions and lower almost nearly (τ1, τ2)-continuous multifunctions. Furthermore, several characterizations of upper almost nearly (τ1, τ2)-continuous multifunctions and lower almost nearly (τ1, τ2)-continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called upper almost 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) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called upper almost nearly (τ1, τ2)-continuous if F is upper almost 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 almost nearly (τ1, τ2)-continuous at x ∈ X; (2) x ∈ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))) for each σ1σ2-open set V of Y containing F (x) and having N (σ1, σ2)-closed complement; (3) x ∈ τ1τ2-Int(F +((σ1, σ2)-sCl(V ))) for each σ1σ2-open set V of Y containing F (x) and having N (σ1, σ2)-closed complement; N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5650 5 of 18 (4) x ∈ τ1τ2-Int(F +(V )) for each (σ1, σ2)r-open set V of Y containing F (x) and having N (σ1, σ2)-closed complement; (5) for each (σ1, σ2)r-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 . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing F (x) and having N (σ1, σ2)-closed complement. By (1), there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). Thus, we have x ∈ U ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V )) and hence x ∈ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))). (2) ⇒ (3): This follows from Lemma 1. (3) ⇒ (4): Let V be any (σ1, σ2)r-open set of Y containing F (x) and having N (σ1, σ2)- closed complement. It follows from Lemma 1 that V = σ1σ2-Int(σ1σ2-Cl(V )) = (σ1, σ2)-sCl(V ). (4) ⇒ (5): Let V be any (σ1, σ2)r-open set of Y containing F (x) and having N (σ1, σ2)- closed complement. By (4), x ∈ τ1τ2-Cl(F +(V )) and therefore there exists a τ1τ2-open set U of X such that x ∈ U ⊆ F+(V ); hence F (U) ⊆ V . (5) ⇒ (1): Let V be any σ1σ2-open set of Y containing F (x) and having N (σ1, σ2)- closed complement. By Lemma 2, σ1σ2-Int(σ1σ2-Cl(V )) is a (σ1, σ2)r-open set of Y con- taining F (x) and having N (σ1, σ2)-closed complement. Thus by (5), there exists a τ1τ2- open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). This shows that F is upper almost nearly (τ1, τ2)-continuous. Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower almost 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 σ1σ2-Int(σ1σ2-Cl(V )) ∩ F (z) ̸= ∅ for each z ∈ U . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower almost nearly (τ1, τ2)-continuous if F is lower almost 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 almost nearly (τ1, τ2)-continuous at x ∈ X; (2) x ∈ τ1τ2-Int(F −(σ1σ2-Int(σ1σ2-Cl(V )))) for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅ and having N (σ1, σ2)-closed complement; (3) x ∈ τ1τ2-Int(F −((σ1, σ2)-sCl(V ))) for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅ and having N (σ1, σ2)-closed complement; N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5650 6 of 18 (4) x ∈ τ1τ2-Int(F −(V )) for each (σ1, σ2)r-open set V of Y such that F (x)∩ V ̸= ∅ and having N (σ1, σ2)-closed complement; (5) for each (σ1, σ2)r-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 every z ∈ U . 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: (1) F is upper almost nearly (τ1, τ2)-continuous; (2) F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))) for each σ1σ2-open set V of Y having N (σ1, σ2)-closed complement; (3) τ1τ2-Cl(F −(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F−(K) for every N (σ1, σ2)-closed and σ1σ2- closed set K of Y ; (4) τ1τ2-Cl(F −(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F−(σ1σ2-Cl(B)) for every every sub- set B of Y having the N (σ1, σ2)-closed σ1σ2-closure; (5) F+(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) for every every subset B of Y such that Y − σ1σ2-Int(B) is N (σ1, σ2)-closed; (6) F+(V ) is τ1τ2-open in X for each (σ1, σ2)r-open set V of Y having N (σ1, σ2)-closed complement; (7) F−(K) is τ1τ2-closed in X for every N (σ1, σ2)-closed and (σ1, σ2)r-closed set K of Y . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing F (x) having N (σ1, σ2)- closed complement and x ∈ F+(V ). Then, F (x) ⊆ V . By Theorem 1, we have x ∈ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))) and hence F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))). (2) ⇒ (3): Let K be any N (σ1, σ2)-closed and σ1σ2-closed set K of Y . Then, Y −K is a σ1σ2-open set of Y having N (σ1, σ2)-closed complement. By (2), we have X − F−(K) = F+(Y −K) ⊆ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(Y −K)))) = τ1τ2-Int(X − F−(σ1σ2-Cl(σ1σ2-Int(K)))) N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5650 7 of 18 = X − τ1τ2-Cl(F −(σ1σ2-Cl(σ1σ2-Int(K)))). Thus, τ1τ2-Cl(F −(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F−(K). (3) ⇒ (4): Let B be any subset of Y having the N (σ1, σ2)-closed σ1σ2-closure. Then, σ1σ2-Cl(B) is a σ1σ2-closed and N (σ1, σ2)-closed set of Y . Thus by (3), τ1τ2-Cl(F −(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(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. Since Y − σ1σ2-Int(B) is σ1σ2-closed and N (σ1, σ2)-closed. Then by (4), we have F+(σ1σ2-Int(B)) = X − F−(Y − σ1σ2-Int(B)) = X − F−(σ1σ2-Cl(Y −B)) ⊆ X − τ1τ2-Cl(F −(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(Y −B))))) = X − τ1τ2-Cl(F −(Y − σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) = τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))). (5) ⇒ (6): Let V be any (σ1, σ2)r-open set V of Y having N (σ1, σ2)-closed comple- ment. Then, Y − σ1σ2-Int(V ) is N (σ1, σ2)-closed. Thus by (5), we have F+(V ) ⊆ τ1τ2-Int(F +(V )) and hence F+(V ) is τ1τ2-open in X. (6) ⇒ (7): Let K be any N (σ1, σ2)-closed and (σ1, σ2)r-closed set of Y . Then, Y −K is a (σ1, σ2)r-open set of Y having N (σ1, σ2)-closed complement. By (6), we have F+(Y −K) = X − F−(K) is τ1τ2-open in X and hence F−(K) is τ1τ2-closed in X. (7) ⇒ (1): Let x ∈ X and V be any (σ1, σ2)r-open set of Y containing F (x) and having N (σ1, σ2)-closed complement. Then, Y − V is (σ1, σ2)r-closed and N (σ1, σ2)-closed. By (7), F−(Y −V ) = X −F+(V ) is τ1τ2-closed in X. Thus, F+(V ) is τ1τ2-open in X. Then, there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ V . It follows from Theorem 1 that F is upper almost nearly (τ1, τ2)-continuous at x. This shows that F is upper almost nearly (τ1, τ2)-continuous. Theorem 4. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower nearly almost (τ1, τ2)-continuous; (2) F−(V ) ⊆ τ1τ2-Int(F −(σ1σ2-Int(σ1σ2-Cl(V )))) for each σ1σ2-open set V of Y having N (σ1, σ2)-closed complement; (3) τ1τ2-Cl(F +(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ F+(K) for every N (σ1, σ2)-closed and σ1σ2- closed set K of Y ; (4) τ1τ2-Cl(F +(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ F+(σ1σ2-Cl(B)) for every every sub- set B of Y having the N (σ1, σ2)-closed σ1σ2-closure; N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5650 8 of 18 (5) F−(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F −(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) for every every subset B of Y such that Y − σ1σ2-Int(B) is N (σ1, σ2)-closed; (6) F−(V ) is τ1τ2-open in X for each (σ1, σ2)r-open set V of Y having N (σ1, σ2)-closed complement; (7) F+(K) is τ1τ2-closed in X for every N (σ1, σ2)-closed and (σ1, σ2)r-closed set K of Y . Proof. The proof is similar to that of Theorem 3. Corollary 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper almost 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 (σ1, σ2)r-open set of Y having N (σ1, σ2)-closed complement. Then, Y − V is N (σ1, σ2)-closed and (σ1, σ2)r-closed. By the hypothesis, X − F+(V ) = F−(Y − V ) is τ1τ2-closed in X and hence F+(V ) is τ1τ2-open in X. It follows from Theorem 3 that F is upper almost nearly (τ1, τ2)-continuous. Corollary 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower almost 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. Theorem 5. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost nearly (τ1, τ2)-continuous; (2) τ1τ2-Cl(F −(V )) ⊆ F−(σ1σ2-Cl(V )) for every every (σ1, σ2)β-open set V of Y having the N (σ1, σ2)-closed σ1σ2-closure; (3) τ1τ2-Cl(F −(V )) ⊆ F−(σ1σ2-Cl(V )) for every every (σ1, σ2)s-open set V of Y having the N (σ1, σ2)-closed σ1σ2-closure; (4) F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))) for every every (σ1, σ2)p-open set V of Y having N (σ1, σ2)-closed complement. Proof. (1) ⇒ (2): Let V be any (σ1, σ2)β-open set of Y having the N (σ1, σ2)-closed σ1σ2-closure. Then, σ1σ2-Cl(V ) is (σ1, σ2)r-closed in Y . Since F is upper almost nearly (τ1, τ2)-continuous, by Theorem 3 we have F−(σ1σ2-Cl(V )) is τ1τ2-closed in X. Thus, τ1τ2-Cl(F −(V )) ⊆ τ1τ2-Cl(F −(σ1σ2-Cl(V ))) = F−(σ1σ2-Cl(V )). (2) ⇒ (3): The proof is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open. N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5650 9 of 18 (3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y having N (σ1, σ2)-closed complement. Then by Lemma 3, σ1σ2-Int(σ1σ2-Cl(V )) is a (σ1, σ2)r-open set having N (σ1, σ2)-closed complement. Then, Y − σ1σ2-Int(σ1σ2-Cl(V )) is a (σ1, σ2)r-closed and N (σ1, σ2)-closed set. Therefore, Y − σ1σ2-Int(σ1σ2-Cl(V )) is a (σ1, σ2)s-open set having the N (σ1, σ2)- closed σ1σ2-closure. By (3), we have X − τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))) = τ1τ2-Cl(F −(Y − σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(Y − σ1σ2-Int(σ1σ2-Cl(V )))) = X − F+(σ1σ2-Int(σ1σ2-Cl(V ))) ⊆ X − F+(V ) and hence F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))). (4) ⇒ (1): Let V be any (σ1, σ2)r-open set of Y having N (σ1, σ2)-closed complement. Then, V is a (σ1, σ2)p-open set having N (σ1, σ2)-closed complement. By (4), we have F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))) = τ1τ2-Int(F +(V )) and hence F+(V ) is τ1τ2-open in X. Thus by Theorem 3, F is upper almost nearly (τ1, τ2)-continuous. Theorem 6. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost nearly (τ1, τ2)-continuous; (2) τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-Cl(V )) for every every (σ1, σ2)β-open set V of Y having the N (σ1, σ2)-closed σ1σ2-closure; (3) τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-Cl(V )) for every every (σ1, σ2)s-open set V of Y having the N (σ1, σ2)-closed σ1σ2-closure; (4) F−(V ) ⊆ τ1τ2-Int(F −(σ1σ2-Int(σ1σ2-Cl(V )))) for every every (σ1, σ2)p-open set V of Y having N (σ1, σ2)-closed complement. Proof. The proof is similar to that of Theorem 5. Lemma 4. For a bitopological space (X, τ1, τ2), the following properties hold: (1) α(τ1, τ2)-Cl(U) = τ1τ2-Cl(U) for every (τ1, τ2)β-open set U of X; (2) (τ1, τ2)-pCl(U) = τ1τ2-Cl(U) for every (τ1, τ2)s-open set U of X. Corollary 3. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost nearly (τ1, τ2)-continuous; (2) τ1τ2-Cl(F −(V )) ⊆ F−(α(σ1, σ2)-Cl(V )) for every every (σ1, σ2)β-open set V of Y having the N (σ1, σ2)-closed σ1σ2-closure; N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5650 10 of 18 (3) τ1τ2-Cl(F −(V )) ⊆ F−((σ1, σ2)-pCl(V )) for every every (σ1, σ2)s-open set V of Y having the N (σ1, σ2)-closed σ1σ2-closure. Corollary 4. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost nearly (τ1, τ2)-continuous; (2) τ1τ2-Cl(F +(V )) ⊆ F+(α(σ1, σ2)-Cl(V )) for every every (σ1, σ2)β-open set V of Y having the N (σ1, σ2)-closed σ1σ2-closure; (3) τ1τ2-Cl(F +(V )) ⊆ F+((σ1, σ2)-pCl(V )) for every every (σ1, σ2)s-open set V of Y having the N (σ1, σ2)-closed σ1σ2-closure. Theorem 7. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost nearly (τ1, τ2)-continuous; (2) for each x ∈ X and for every σ1σ2-closed and N (σ1, σ2)-closed set K of Y such that x ∈ F+(Y −K), there exists a τ1τ2-closed set H of X such that x ∈ X −H and F−(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ H; (3) F+(σ1σ2-Int(σ1σ2-Cl(V ))) is τ1τ2-open in X for every σ1σ2-open set V of Y having N (σ1, σ2)-closed complement; (4) F−(σ1σ2-Cl(σ1σ2-Int(K))) is τ1τ2-closed in X for every σ1σ2-closed and N (σ1, σ2)- closed set K of Y . Proof. (1) ⇒ (2): Let x ∈ X and K be any σ1σ2-closed and N (σ1, σ2)-closed set of Y such that x ∈ F+(Y −K). Then, Y −K is a σ1σ2-open set having N (σ1, σ2)-closed complement. Since F is upper almost nearly (τ1, τ2)-continuous, there exists a τ1τ2-open set U of X containing x such that U ⊆ F+(σ1σ2-Int(σ1σ2-Cl(Y −K))) = X − F−(σ1σ2-Cl(σ1σ2-Int(K))). It is clear that H = X − U is τ1τ2-closed in X and F−(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ H. (2) ⇒ (1): The proof is similar to the proof (1) ⇒ (2). (1) ⇒ (3): Let V be any σ1σ2-open set of Y having N (σ1, σ2)-closed complement and x ∈ F+(σ1σ2-Int(σ1σ2-Cl(V ))). Then, we have σ1σ2-Int(σ1σ2-Cl(V )) is a σ1σ2-open set of Y having N (σ1, σ2)-closed complement. Thus by (1), there exists a τ1τ2-open set U of X containing x such that U ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))). Since U is τ1τ2-open, we have x ∈ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))) and hence F+(σ1σ2-Int(σ1σ2-Cl(V ))) ⊆ τ1τ2-Int(F +(σ1σ2-Int(σ1σ2-Cl(V )))). Thus, F+(σ1σ2-Int(σ1σ2-Cl(V ))) is τ1τ2-open in X. (3) ⇒ (1): The proof is clear. N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5650 11 of 18 (3) ⇒ (4): Let K be any σ1σ2-closed N (σ1, σ2)-closed set of Y . Then, Y −K is a σ1σ2- open set having N (σ1, σ2)-closed complement. By (3), F+(σ1σ2-Int(σ1σ2-Cl(Y −K))) is τ1τ2-open in X. Since σ1σ2-Int(σ1σ2-Cl(Y −K)) = Y − σ1σ2-Cl(σ1σ2-Int(K)), it follows that F+(σ1σ2-Int(σ1σ2-Cl(Y −K))) = F+(Y − σ1σ2-Cl(σ1σ2-Int(K))) = X − F−(σ1σ2-Cl(σ1σ2-Int(K))). Thus, F−(σ1σ2-Cl(σ1σ2-Int(K))) is τ1τ2-closed in X. (4) ⇒ (3): It can be obtained similarly as (3) ⇒ (4). Theorem 8. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost nearly (τ1, τ2)-continuous; (2) for each x ∈ X and for every σ1σ2-closed and N (σ1, σ2)-closed set K of Y such that x ∈ F−(Y −K), there exists a τ1τ2-closed set H of X such that x ∈ X −H and F+(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ H; (3) F−(σ1σ2-Int(σ1σ2-Cl(V ))) is τ1τ2-open in X for every σ1σ2-open set V of Y having N (σ1, σ2)-closed complement; (4) F+(σ1σ2-Cl(σ1σ2-Int(K))) is τ1τ2-closed in X for every σ1σ2-closed and N (σ1, σ2)- closed set K of Y . Proof. The proof is similar to that of Theorem 7. Recall that a net (xγ) in a topological space (X, τ) is said to be eventually in the set U ⊆ X if there exists an index γ0 ∈ ∇ such that xγ ∈ U for all γ ≥ γ0. A net (xγ) is called (τ1, τ2)-converge to a point x if for every τ1τ2-open set V containing x, there exists an index γ0 ∈ ∇ such that xγ ∈ V for all γ ≥ γ0. Theorem 9. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper almost nearly (τ1, τ2)- continuous if and only if for each x ∈ X and for each net (xγ) which (τ1, τ2)-converges to x in X and for each σ1σ2-open set V of Y having N (σ1, σ2)-closed complement such that x ∈ F+(V ), the net (xγ) is eventually in F+(σ1σ2-Int(σ1σ2-Cl(V ))). Proof. Let (xγ) be a net which (τ1, τ2)-converges to x in X and V be any σ1σ2-open set of Y having N (σ1, σ2)-closed complement such that x ∈ F+(V ). Since F is upper almost nearly (τ1, τ2)-continuous, there exists a τ1τ2-open set U of X containing x such that U ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))). Since (xγ) (τ1, τ2)-converges to x, it follows that there exists an index γ0 ∈ ∇ such that xγ ∈ U for all γ ≥ γ0. Therefore, xγ ∈ U ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))) for all γ ≥ γ0. Thus, the net (xγ) is eventually in F+(σ1σ2-Int(σ1σ2-Cl(V ))). N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5650 12 of 18 Conversely, suppose that F is not upper almost nearly (τ1, τ2)-continuous. Then, there exists a point x of X and a σ1σ2-open set V of Y having N (σ1, σ2)-closed complement with x ∈ F+(V ) such that U ̸⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))) for each τ1τ2-open set U of X containing x. Let xU ∈ U and xU ̸∈ F+(σ1σ2-Int(σ1σ2-Cl(V ))) for each τ1τ2-open set U of X containing x. Then, for each τ1τ2-neighbourhood net (xU ), (xU ) (τ1, τ2)-converges to x, but (xU ) is not eventually in F+(σ1σ2-Int(σ1σ2-Cl(V ))). This is a contradiction. Thus, F is upper almost nearly (τ1, τ2)-continuous. Theorem 10. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower almost nearly (τ1, τ2)- continuous if and only if for each x ∈ X and for each net (xγ) which (τ1, τ2)-converges to x in X and for each σ1σ2-open set V of Y having N (σ1, σ2)-closed complement such that x ∈ F−(V ), the net (xγ) is eventually in F−(σ1σ2-Int(σ1σ2-Cl(V ))). Proof. The proof is similar to that of Theorem 9. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), a multifunction ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is defined in [29] as follows: ClF⊛(x) = σ1σ2-Cl(F (x)) for each x ∈ X. Definition 3. [29] A subset A of a bitopological space (X, τ1, τ2) is said to be: (1) τ1τ2-paracompact if every cover of A by τ1τ2-open sets of X is refined by a cover of A which consists of τ1τ2-open sets of X and is τ1τ2-locally finite in X; (2) τ1τ2-regular if for each x ∈ A and each τ1τ2-open set U of X containing x, there exists a τ1τ2-open set V of X such that x ∈ V ⊆ τ1τ2-Cl(V ) ⊆ U . Lemma 5. [29] If A is a τ1τ2-regular τ1τ2-paracompact set of a bitopological space (X, τ1, τ2) and U is a τ1τ2-open neighbourhood of A, then there exists a τ1τ2-open set V of X such that A ⊆ V ⊆ τ1τ2-Cl(V ) ⊆ U . Lemma 6. [29] If F : (X, τ1, τ2) → (Y, σ1, σ2) is a multifunction such that F (x) is τ1τ2- regular and τ1τ2-paracompact for each x ∈ X, then ClF+ ⊛ (V ) = F+(V ) for each σ1σ2-open set V of Y . Theorem 11. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction such that F (x) is σ1σ2-paracompact and σ1σ2-regular for each x ∈ X. Then, F is upper almost nearly (τ1, τ2)-continuous if and only if G : (X, τ1, τ2) → (Y, σ1, σ2) is upper almost nearly (τ1, τ2)- continuous, where G denote ClF⊛. Proof. Suppose that F is upper almost nearly (τ1, τ2)-continuous. Let V be any (σ1, σ2)r-open set of Y having N (σ1, σ2)-connected complement. It follows from Lemma 6 and Theorem 3 that G+(V ) = F+(V ) is τ1τ2-open in X. By Theorem 3, G is upper almost nearly (τ1, τ2)-continuous. N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5650 13 of 18 Conversely, suppose that G is upper almost nearly (τ1, τ2)-continuous. Let V be any (σ1, σ2)r-open set of Y having N (σ1, σ2)-connected complement. By Lemma 6 and Theo- rem 3, F+(V ) = G+(V ) is τ1τ2-open in X. Thus by Theorem 3, F is upper almost nearly (τ1, τ2)-continuous. Lemma 7. [29] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), ClF − ⊛ (V ) = F−(V ) for each σ1σ2-open set V of Y . Theorem 12. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower almost nearly (τ1, τ2)- continuous if and only if G : (X, τ1, τ2) → (Y, σ1, σ2) is lower almost nearly (τ1, τ2)- continuous, where G denote ClF⊛. Proof. By using Lemma 7 this can be shown similarly as in Theorem 11. 4. Several characterizations The τ1τ2-frontier [26] of a subset A of a bitopological space (X, τ1, τ2), denoted by τ1τ2-fr(A), is defined by τ1τ2-fr(A) = τ1τ2-Cl(A) ∩ τ1τ2-Cl(X −A) = τ1τ2-Cl(A)− τ1τ2-Int(A). Theorem 13. The set of all points x of X at which a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is not upper almost nearly (τ1, τ2)-continuous is identical with the union of the τ1τ2-frontier of the upper inverse images of (σ1, σ2)r-open sets containing F (x) and having N (σ1, σ2)- closed complement. Proof. Let x be a point of X at which F is not upper almost nearly (τ1, τ2)-continuous. Then, by Theorem 1 there exists a (σ1, σ2)r-open set V of Y containing F (x) and having N (σ1, σ2)-closed complement such that U ∩ (X − F+(V )) ̸= ∅ for every τ1τ2-open set U of X containing x. Thus, x ∈ τ1τ2-Cl(X − F+(V )). On the other hand, we have x ∈ F+(V ) ⊆ τ1τ2-Cl(F +(V )) and hence x ∈ τ1τ2-fr(F +(V )). Conversely, suppose that V is a (σ1, σ2)r-open set of Y containing F (x) and having N (σ1, σ2)-closed complement such that x ∈ τ1τ2-fr(F +(V )). If F is upper almost nearly (τ1, τ2)-continuous at x ∈ X. Then by Theorem 1, we have x ∈ τ1τ2-Int(F +(V )). This is a contradiction and hence F is not upper almost nearly (τ1, τ2)-continuous at x. Theorem 14. The set of all points x of X at which a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is not lower almost nearly (τ1, τ2)-continuous is identical with the union of the τ1τ2-frontier of the lower inverse images of (σ1, σ2)r-open sets meeting F (x) and having N (σ1, σ2)- closed complement. N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5650 14 of 18 Proof. The proof is similar to that of Theorem 13. Recall that 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. Definition 4. [29] A bitopological space (X, τ1, τ2) is said to be τ1τ2-connected if X cannot be written as the union of two disjoint nonempty τ1τ2-open sets. Definition 5. A bitopological space (X, τ1, τ2) is said to be N (τ1, τ2)-connected if X cannot be written as the union of two disjoint nonempty τ1τ2-open sets having N (τ1, τ2)- closed complements. Theorem 15. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an upper or lower almost nearly (τ1, τ2)- continuous surjective multifunction such that F (x) is σ1σ2-connected for each x ∈ X and (X, τ1, τ2) is τ1τ2-connected, then (Y, σ1, σ2) is N (σ1, σ2)-connected. Proof. Suppose that (Y, σ1, σ2) is not N (σ1, σ2)-connected. There exist nonempty σ1σ2-open sets U and V of Y having N (σ1, σ2)-closed complements such that U ∩ V = ∅ and U ∪ V = Y . Since F (x) is σ1σ2-connected for each x ∈ X, either F (x) ⊆ U or F (x) ⊆ V . If x ∈ F+(U∪V ), then F (x) ⊆ U∪V and hence x ∈ F+(U)∪F+(V ). Moreover, since F is surjective, there exist x and y in X such that F (x) ⊆ U and F (y) ⊆ V ; hence x ∈ F+(U) and y ∈ F+(V ). Therefore, we obtain the following: (1) F+(U) ∪ F+(V ) = X; (2) F+(U) ∩ F+(V ) = ∅; (3) F+(U) ̸= ∅ and F+(V ) ̸= ∅. Next, we show that F+(U) and F+(V ) are τ1τ2-open in X. (i) Let F be upper almost nearly (τ1, τ2)-continuous. Since U and V are σ1σ2-clopen in Y , σ1σ2-Int(σ1σ2-Cl(U)) = U and σ1σ2-Int(σ1σ2-Cl(V )) = V . Thus, U and V are (σ1, σ2)r-open sets having N (σ1, σ2)- closed complements. Since F is upper almost nearly (τ1, τ2)-continuous, by Theorem 3 F+(U) and F+(V ) are τ1τ2-open sets. (ii) Let F be lower almost nearly (τ1, τ2)-continuous. 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