EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5718 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and Lower Weakly s-(τ1, τ2)-continuous Multifunctions Prapart Pue-on1, 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 article presents new classes of multifunctions called upper weakly s-(τ1, τ2)- continuous multifunctions and lower weakly s-(τ1, τ2)-continuous multifunctions. Furthermore, several characterizations of upper weakly s-(τ1, τ2)-continuous multifunctions and lower weakly s-(τ1, τ2)-continuous multifunctions are discussed. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: τ1τ2-open set, upper weakly s-(τ1, τ2)-continuous multifunction, lower weakly s-(τ1, τ2)-continuous multifunction 1. Introduction In topology, there has been recently significant interest in characterizing and inves- tigating the characterizations of some weak forms of continuity for functions and mul- tifunctions. As weak forms of continuity in topological spaces, weak continuity [44], quasicontinuity [47], semi-continuity [45] and almost continuity in the sense of Husain [36] are well-known. Lee [43] studied the concept of semiconnected functions. Kohli [40] introduced the notion of s-continuous functions and investigated some character- izations of semilocally connected spaces in terms of s-continuous functions. The no- tion of s-continuity as a generalization of continuity and semiconnectedness. Moreover, Kohli [41] introduced the concepts of s-regular spaces and completely s-regular spaces and proved that s-regularity and complete s-regularity are preserved under certain s- continuous functions. Viriyapong and Boonpok [68] 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. [33] introduced ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5718 Email addresses: prapatr.p@msu.ac.th (P. Pue-on), 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) P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5718 2 of 16 and studied the concept of g(m,n)-continuous functions. Duangphui et al. [32] introduced and investigated the notion of (µ, µ′)(m,n)-continuous functions. Furthermore, several characterizations of almost (Λ, p)-continuous functions, strongly θ(Λ, p)-continuous func- tions, 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 slightly (τ1, τ2)s-continuous functions were presented in [60], [63], [16], [54], [25], [11], [8], [10], [4], [1], [2], [26], [23], [18] and [59], respectively. Srisarakham et al. [61] introduced and studied the concept of faintly (τ1, τ2)-continuous functions. Thongmoon et al. [66] introduced and investi- gated the notion of rarely (τ1, τ2)-continuous functions. Kong-ied at al. [42] introduced and studied the concept of almost quasi (τ1, τ2)-continuous functions. Chiangpradit et al. [30] introduced and investigated the notion of weakly quasi (τ1, τ2)-continuous functions. Prachanpol et al. [53] introduced and studied the concept of weakly δ(τ1, τ2)-continuous functions. In 1989, Lipski [46] extended the concept of s-continuous functions to the setting of multifunctions. Popa [49] introduced the concept of precontinuous multifunctions and showed that H-almost continuity and precontinuity are equivalent for multifunctions. Ewert and Lipski [35] introduced and studied the concept of s-quasi-continuous multi- functions. Popa and Noiri [52] introduced and investigated the notion of s-precontinuous multifunctions as a generalization of s-continuous multifunctions and precontinuous mul- tifunctions. In particular, Popa and Noiri [51] introduced and studied the notion of s- β-continuous multifunctions. Popa and Noiri [50] introduced and investigated the con- cept of s-m-continuous multifunctions as multifunctions defined on a set satisfying some minimal conditions. 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, s-(τ1, τ2)p- continuous multifunctions, slightly (τ1, τ2)-continuous multifunctions and slightly (τ1, τ2)p- continuous multifunctions were established in [5], [28], [69], [3], [7], [17], [24], [6], [21], [20], [15], [9], [19], [22], [37], [13], [27], [62], [14], [57], [39], [65], [58], [56], [38], [55], [73], [70] and [71], respectively. Noiri and Popa [48] introduced and studied the notion of weakly s-m- continuous multifunctions as a generalization of both weakly m-continuous multifunctions P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5718 3 of 16 and s-m-continuous multifunctions. The class of weakly s-m-continuous multifunctions contains weakly s-precontinuous multifunctions due to Ekici and Park [34]. In this paper, we introduce the concepts of upper weakly s-(τ1, τ2)-continuous multifunctions and lower weakly s-(τ1, τ2)-continuous multifunctions. We also investigate several characterizations of upper weakly s-(τ1, τ2)-continuous multifunctions and lower weakly s-(τ1, τ2)-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 [29] 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 [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 called α(τ1, τ2)-open [72] if A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)-open set is called α(τ1, τ2)- closed. A subset A of a bitopological space (X, τ1, τ2) is called (τ1, τ2)r-open [67] (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, α(τ1, τ2)-open) set is called (τ1, τ2)r-closed (resp. (τ1, τ2)s-closed, (τ1, τ2)p-closed). Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is called a (τ1, τ2)θ-cluster point [67] 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 [67] 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 [67] 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 [67] of A and is denoted by (τ1, τ2)θ-Int(A). P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5718 4 of 16 Lemma 2. [67] 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. 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 wekly s-(τ1, τ2)-continuous multifunctions In this section, we introduce the notions of upper weakly s-(τ1, τ2)-continuous multi- functions and lower weakly s-(τ1, τ2)-continuous multifunctions. Moreover, some charac- terizations of upper weakly s-(τ1, τ2)-continuous multifunctions and lower weakly s-(τ1, τ2)- continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper weakly s-(τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing F (x) and having σ1σ2-connected complement, there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper weakly s-(τ1, τ2)-continuous if F is upper weakly s-(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 weakly s-(τ1, τ2)-continuous; (2) F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y having σ1σ2- connected complement; (3) τ1τ2-Cl(F −(σ1σ2-Int(K))) ⊆ F−(K) for every σ1σ2-connected σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(F −(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y hav- ing the σ1σ2-connected σ1σ2-closure; (5) F+(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y such that Y − σ1σ2-Int(B) is σ1σ2-connected; (6) τ1τ2-Cl(F −(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y having the σ1σ2-connected σ1σ2-closure; P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5718 5 of 16 (7) τ1τ2-Cl(F −(V )) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y having the σ1σ2- connected σ1σ2-closure; (8) τ1τ2-Cl(F −(σ1σ2-Int(K))) ⊆ F−(K) for every σ1σ2-connected (σ1, σ2)r-closed set K of Y . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y having the σ1σ2-connected comple- ment and x ∈ F+(V ). Then, there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). Therefore, we have x ∈ U ⊆ F+(σ1σ2-Cl(V )). Since U is τ1τ2-open, we have x ∈ τ1τ2-Int(F +(σ1σ2-Cl(V ))) and hence F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(V ))). (2) ⇒ (3): LetK be any σ1σ2-connected σ1σ2closed set of Y . Then, Y −K is σ1σ2-open in Y having σ1σ2-connected complement. By (2), we have X − F−(K) = F+(Y −K) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(Y −K))) = τ1τ2-Int(F +(Y − σ1σ2-Int(K))) = τ1τ2-Int(X − F−(σ1σ2-Int(K))) = X − τ1τ2-Cl(F −(σ1σ2-Int(K))) and hence τ1τ2-Cl(F −(σ1σ2-Int(K))) ⊆ F−(K). (3) ⇒ (4): Let B be any subset of Y having the σ1σ2-connected σ1σ2-closure. Then, σ1σ2-Cl(B) is σ1σ2-closed σ1σ2-connected in Y and by (3), we have τ1τ2-Cl(F −(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 σ1σ2-connected. Then by (4), X − τ1τ2-Int(F +(σ1σ2-Cl(σ1σ2-Int(B)))) = τ1τ2-Cl(X − F+(σ1σ2-Cl(σ1σ2-Int(B)))) = τ1τ2-Cl(F −(Y − σ1σ2-Cl(σ1σ2-Int(B)))) = τ1τ2-Cl(F −(σ1σ2-Int(σ1σ2-Cl(Y −B)))) ⊆ F−(σ1σ2-Cl(Y −B)) = X − F+(σ1σ2-Int(B)). Thus, F+(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(σ1σ2-Int(B)))). (5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing F (x) and having σ1σ2-connected complement. By (5), we have x ∈ F+(V ) = F+(σ1σ2-Int(V )) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(V ))). Then, there exists a τ1τ2-open set U of X containing x such that U ⊆ F+(σ1σ2-Cl(V )). Thus, F (U) ⊆ σ1σ2-Cl(V ) and hence F is upper weakly s-(τ1, τ2)-continuous. (4) ⇒ (6) and (6) ⇒ (7): The proofs are obvious. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5718 6 of 16 (7) ⇒ (8): Let K be any σ1σ2-connected (σ1, σ2)r-closed set of Y . Then, we have K = σ1σ2-Cl(σ1σ2-Int(K)) is σ1σ2-connected and by (7), τ1τ2-Cl(F −(σ1σ2-Int(K))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(K))) = F−(K). (8) ⇒ (3): Let K be any σ1σ2-connected σ1σ2-closed set of Y . Since K is σ1σ2- connected, we have σ1σ2-Int(K) is σ1σ2-connected and hence σ1σ2-Cl(σ1σ2-Int(K)) is σ1σ2-connected. Let H = σ1σ2-Cl(σ1σ2-Int(K)). Then, H is a (σ1, σ2)r-closed σ1σ2- connected set of Y and σ1σ2-Int(H) = σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))) = σ1σ2-Int(K). By (8), we have τ1τ2-Cl(F −(σ1σ2-Int(K))) = τ1τ2-Cl(F −(σ1σ2-Int(H))) ⊆ F−(H) ⊆ F−(K). Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower weakly s- (τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x)∩V ̸= ∅ and having σ1σ2-connected complement, there exists a τ1τ2-open set U of X containing x such that σ1σ2-Cl(V ) ∩ F (z) ̸= ∅ for each z ∈ U . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower weakly s-(τ1, τ2)-continuous if F is lower weakly s-(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 weakly s-(τ1, τ2)-continuous; (2) F−(V ) ⊆ τ1τ2-Int(F −(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y having σ1σ2- connected complement; (3) τ1τ2-Cl(F +(σ1σ2-Int(K))) ⊆ F+(K) for every σ1σ2-connected σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(F +(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y hav- ing the σ1σ2-connected σ1σ2-closure; (5) F−(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F −(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y such that Y − σ1σ2-Int(B) is σ1σ2-connected; (6) τ1τ2-Cl(F +(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y having the σ1σ2-connected σ1σ2-closure; (7) τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y having the σ1σ2- connected σ1σ2-closure; (8) τ1τ2-Cl(F +(σ1σ2-Int(K))) ⊆ F+(K) for every σ1σ2-connected (σ1, σ2)r-closed set K of Y . P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5718 7 of 16 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 weakly s-(τ1, τ2)-continuous; (2) τ1τ2-Cl(F −(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y having the σ1σ2-connected σ1σ2-closure; (3) τ1τ2-Cl(F −(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y having the σ1σ2-connected σ1σ2-closure. Proof. (1) ⇒ (2): This follows from Theorem 1(4). (2) ⇒ (3): The proof is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open. (3) ⇒ (1): Since every σ1σ2-open set is (σ1, σ2)s-open, the proof follows from Theorem 1(7). Theorem 4. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly s-(τ1, τ2)-continuous; (2) τ1τ2-Cl(F +(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y having the σ1σ2-connected σ1σ2-closure; (3) τ1τ2-Cl(F +(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y having the σ1σ2-connected σ1σ2-closure. Proof. The proof is similar to that of Theorem 3. Theorem 5. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly s-(τ1, τ2)-continuous; (2) τ1τ2-Cl(F −(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B of Y having the σ1σ2-connected (σ1, σ2)θ-closure; (3) τ1τ2-Cl(F −(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B of Y having the σ1σ2-connected (σ1, σ2)θ-closure. Proof. (1) ⇒ (2): Let B be any subset of Y having the σ1σ2-connected (σ1, σ2)θ- closure. Then, (σ1, σ2)θ-Cl(B) is σ1σ2-connected σ1σ2-closed and by Theorem1, τ1τ2-Cl(F −(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)). P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5718 8 of 16 (2) ⇒ (3): The proof is obvious since σ1σ2-Cl(B) ⊆ (σ1, σ2)θ-Cl(B) for every subset B of Y . (3) ⇒ (1): Let K be any (σ1, σ2)r-closed σ1σ2-connected set of Y . Then, we have (σ1, σ2)θ-Cl(σ1σ2-Int(K)) = σ1σ2-Cl(σ1σ2-Int(K)) = K and by (3), τ1τ2-Cl(F −(σ1σ2-Int(K))) = τ1τ2-Cl(F −(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))))) ⊆ F−((σ1, σ2)θ-Cl(σ1σ2-Cl(K))) = F−(σ1σ2-Cl(σ1σ2-Cl(K))) = F−(K). Thus, τ1τ2-Cl(F −(σ1σ2-Int(K))) ⊆ F−(K) and by Theorem 1(8), F is upper weakly s- (τ1, τ2)-continuous. Theorem 6. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly s-(τ1, τ2)-continuous; (2) τ1τ2-Cl(F +(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y having the σ1σ2-connected (σ1, σ2)θ-closure; (3) τ1τ2-Cl(F +(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y having the σ1σ2-connected (σ1, σ2)θ-closure. Proof. The proof is similar to that of Theorem 5. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), by ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) [29] we denote a multifunction defined 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 3. [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 . Lemma 4. [29] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), ClF − ⊛ (V ) = F−(V ) for each σ1σ2-open set V of Y . Theorem 7. 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 weakly s-(τ1, τ2)- continuous if and only if ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is upper weakly s-(τ1, τ2)-continuous. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5718 9 of 16 Proof. Suppose that F is upper weakly s-(τ1, τ2)-continuous. Let V be any σ1σ2-open set of Y having σ1σ2-connected complement. By Theorem 1, Lemma 3 and Lemma 4, we have ClF+ ⊛ (V ) = F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(V ))) = τ1τ2-Int(ClF + ⊛ (σ1σ2-Cl(V ))). Thus by Theorem 1, ClF⊛ is upper weakly s-(τ1, τ2)-continuous. Conversely, suppose that ClF⊛ is upper weakly s-(τ1, τ2)-continuous. Let V be any σ1σ2-open set of Y having σ1σ2-connected complement. By Theorem 1, Lemma 3 and Lemma 4, we have F+(V ) = ClF+ ⊛ (V ) ⊆ τ1τ2-Int(ClF + ⊛ (σ1σ2-Cl(V ))) = τ1τ2-Int(F +(σ1σ2-Cl(V ))). By Theorem 1, F is upper weakly s-(τ1, τ2)-continuous. Theorem 8. 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 lower weakly s-(τ1, τ2)- continuous if and only if ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is lower weakly s-(τ1, τ2)-continuous. Proof. The proof is similar to that of Theorem 7. 4. Some results on weak s-(τ1, τ2)-continuity 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. [64] A bitopological space (X, τ1, τ2) is said to be s-τ1τ2-connected if X can- not be written as the union of two disjoint nonempty τ1τ2-open sets having τ1τ2-connected complements. Theorem 9. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an upper or lower weakly s-(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 s-σ1σ2-connected. Proof. Suppose that (Y, σ1, σ2) is not σ1σ2-connected. There exist nonempty σ1σ2- open sets U and V of Y having σ1σ2-connected complement 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 ) = F+(U ∪ V ) = X; (2) F+(U) ∩ F+(V ) = F+(U ∩ V ) = ∅; (3) F+(U) ̸= ∅ and F+(V ) ̸= ∅. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5718 10 of 16 Next, we shall show that F+(U) and F+(V ) are τ1τ2-open in X. (i) Let F be upper weakly s-(τ1, τ2)-continuous. By Theorem 1, F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(V ))) = τ1τ2-Int(F +(V )) since V is σ1σ2-clopen. Thus, F+(V ) = τ1τ2-Int(F +(V )) and hence F+(V ) is τ1τ2-open in X. Similarly, we obtain F+(U) is τ1τ2-open in X. This shows that (X, τ1, τ2) is not τ1τ2-connected. (ii) Let F be lower weakly s-(τ1, τ2)-continuous. Since V is a σ1σ2-clopen set with σ1σ2-connected complement, by Theorem 2 τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-Cl(V )) = F+(V ). Thus, F+(V ) = τ1τ2-Cl(F +(V )) and hence F+(V ) is τ1τ2-closed in X. Therefore, F+(U) is τ1τ2-open in X. Similarly, we obtain F+(V ) is τ1τ2-open in X. Consequently, this shows that (X, τ1, τ2) is not τ1τ2-connected. This completes the proof. 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 10. The set of all points x ∈ X at which a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is not upper weakly s-(τ1, τ2)-continuous is identical with the union of the τ1τ2-frontier of the upper inverse images of the σ1σ2-closure of σ1σ2-open sets containing F (x) and having σ1σ2-connected complement. Proof. Let x be a point of X at which F is not upper weakly s-(τ1, τ2)-continuous. Then, there exists a σ1σ2-open set V of Y containing F (x) and having σ1σ2-connected complement such that U ∩ (X − F+(σ1σ2-Cl(V ))) ̸= ∅ for every τ1τ2-open set U of X containing x. Then, we have x ∈ τ1τ2-Cl(X − F+(σ1σ2-Cl(V ))) and hence x ∈ τ1τ2-fr(F +(σ1σ2-Cl(V ))) since x ∈ F+(V ) ⊆ τ1τ2-Cl(F +(σ1σ2-Cl(V ))). Conversely, suppose that V is a σ1σ2-open set of Y containing F (x) and having σ1σ2- connected complement such that x ∈ τ1τ2-fr(F +(σ1σ2-Cl(V ))). If F is upper weakly s-(τ1, τ2)-continuous at x ∈ X, there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ); hence U ⊆ F+(σ1σ2-Cl(V )). Thus, x ∈ U ⊆ τ1τ2-Int(F +(σ1σ2-Cl(V ))). This contradicts that x ∈ τ1τ2-fr(F +(σ1σ2-Cl(V ))). Theorem 11. The set of all points x ∈ X at which a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is not lower weakly s-(τ1, τ2)-continuous is identical with the union of the τ1τ2-frontier of the lower inverse images of the σ1σ2-closure of σ1σ2-open sets meeting F (x) and having σ1σ2-connected complement. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5718 11 of 16 Proof. The proof is similar to that of Theorem 10. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be injective if x ̸= y implies that F (x) ∩ F (y) = ∅. Definition 6. [31] A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-T2 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 7. A bitopological space (X, τ1, τ2) is said to be strongly s-(τ1, τ2)-normal if for every disjoint τ1τ2-closed sets F and K of X, there exist τ1τ2-open sets U and V having τ1τ2-connected complements such that F ⊆ U , K ⊆ V and τ1τ2-Cl(U) ∩ τ1τ2-Cl(V ) = ∅. Theorem 12. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an injective upper weakly s-(τ1, τ2)- continuous multifunction into a strongly s-(τ1, τ2)-normal space (Y, σ1, σ2) and F (x) is σ1σ2-closed for each x ∈ X, then (X, τ1, τ2) is (τ1, τ2)-T2. Proof. For any distinct points x, y of X, we have F (x)∩F (y) = ∅ since F is injective. Since F (x) is σ1σ2-closed for each x ∈ X and (Y, σ1, σ2) is strongly s-(τ1, τ2)-normal, there exist σ1σ2-open sets V and W of Y having σ1σ2-connected complements such that F (x) ⊆ V , F (y) ⊆ W and σ1σ2-Cl(V ) ∩ σ1σ2-Cl(W ) = ∅. Since F is upper weakly s- (τ1, τ2)-continuous, there exist τ1τ2-open sets G,U of X containing x, y, respectively, such that F (G) ⊆ V and F (U) ⊆ W . Thus, G ∩ U = ∅ and hence (X, τ1, τ2) is (τ1, τ2)-T2. Definition 8. [64] A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper s- (τ1, τ2)-continuous at x ∈ X if for each σ1σ2-open set V of Y containing F (x) and having σ1σ2-connected 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 s-(τ1, τ2)- continuous if F has this property at each point x of X. Lemma 5. [64] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper s-(τ1, τ2)-continuous; (2) F+(V ) is τ1τ2-open in X for every σ1σ2-open set V of Y having σ1σ2-connected complement; (3) F−(K) is τ1τ2-closed in X for every σ1σ2-connected σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(F −(B)) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y having the σ1σ2-connected σ1σ2-closure; (5) F+(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F +(B)) for every subset B of Y such that Y−σ1σ2-Int(B) is σ1σ2-connected. Theorem 13. If F : (X, τ1, τ2) → (Y, σ1, σ2) is upper weakly s-(τ1, τ2)-continuous and satisfies F+(σ1σ2-Cl(V )) ⊆ F+(V ) for every σ1σ2-open set V of Y having σ1σ2-connected complement, then F is upper s-(τ1, τ2)-continuous. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5718 12 of 16 Proof. Let V be any σ1σ2-open set of Y having σ1σ2-connected complement. Since F is upper weakly s-(τ1, τ2)-continuous, by Theorem 1 we have F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(V ))) ⊆ τ1τ2-Int(F +(V )) and hence F+(V ) is τ1τ2-open in X. By Lemma 5, F is upper s-(τ1, τ2)-continuous. Definition 9. A bitopological space (X, τ1, τ2) is said to be s-(τ1, τ2)-normal if for each disjoint τ1τ2-closed sets F and K of X, there exist τ1τ2-open sets U and V having τ1τ2- connected complements such that F ⊆ U , K ⊆ V and U ∩ V = ∅. Theorem 14. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction such that F (x) is σ1σ2- closed in Y for each x ∈ X and (Y, σ1, σ2) is s-(σ1, σ2)-normal. Then, F is upper weakly s-(τ1, τ2)-continuous if and only if F is upper s-(τ1, τ2)-continuous. Proof. Suppose that F is upper weakly s-(τ1, τ2)-continuous. Let x ∈ X and G be any σ1σ2-open set of Y containing F (x) and having σ1σ2-connected complement. Since F (x) is σ1σ2-closed in Y , by the s-(σ1, σ2)-normality of (Y, σ1, σ2) there exist σ1σ2-open sets V and W having σ1σ2-connected complements such that F (x) ⊆ V , Y −G ⊆ W and V ∩W = ∅. Thus, F (x) ⊆ V ⊆ σ1σ2-Cl(V ) ⊆ σ1σ2-Cl(Y −W ) = Y −W . Since F is upper weakly s-(τ1, τ2)-continuous, there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ) ⊆ G. This shows that F is upper s-(τ1, τ2)-continuous. The converse is obvious. Definition 10. [64] A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower s- (τ1, τ2)-continuous at x ∈ X if for each σ1σ2-open set V of Y such that F (x)∩ V ̸= ∅ and having σ1σ2-connected 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 s-(τ1, τ2)-continuous if F has this property at each point x of X. Theorem 15. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction such that F (x) is σ1σ2- open in Y for each x ∈ X. Then, F is lower weakly s-(τ1, τ2)-continuous if and only if F is lower s-(τ1, τ2)-continuous. Proof. Suppose that F is lower weakly s-(τ1, τ2)-continuous. Let x ∈ X and V be any σ1σ2-open set of Y such that F (x)∩V ̸= ∅ and having σ1σ2-connected complement. Since F is lower weakly s-(τ1, τ2)-continuous, there exists a τ1τ2-open set U of X containing x such that σ1σ2-Cl(V ) ∩ F (z) ̸= ∅ for each z ∈ U . 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