EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5634 ISSN 1307-5543 – ejpam.com Published by New York Business Global s-(τ1, τ2)-continuity for Multifunctions Monchaya Chiangpradit1, 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 deals with the concepts of upper s-(τ1, τ2)-continuous multifunctions and lower s-(τ1, τ2)-continuous multifunctions. Moreover, several characterizations and some proper- ties concerning upper s-(τ1, τ2)-continuous multifunctions and lower s-(τ1, τ2)-continuous multi- functions are investigated. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper s-(τ1, τ2)-continuous multifunction, lower s-(τ1, τ2)-continuous multifunction 1. Introduction It is well-known that the branch of mathematics called topology is concerned with all questions directly or indirectly related to continuity. Continuity is an important concept for the study and investigation in the theory of classical point set topology. Generalization of this concept by using stronger and weaker forms of open sets such as semi-open sets [43], preopen sets [45], α-open sets [46], β-open sets [34] and θ-open sets [61] is one of the main research topics of general topology. Viriyapong and Boon- pok [63] investigated some characterizations of (Λ, sp)-continuous functions by utiliz- ing the notions of (Λ, sp)-open sets and (Λ, sp)-closed sets due to Boonpok and Kham- pakdee [12]. Dungthaisong et al. [33] introduced 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 and some prop- erties concerning 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, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5634 Email addresses: monchaya.c@msu.ac.th (M. Chiangpradit), 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. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5634 2 of 12 pairwise almost M -continuous functions, (τ1, τ2)-continuous functions, almost (τ1, τ2)- continuous functions, weakly (τ1, τ2)-continuous functions, faintly (τ1, τ2)-continuous func- tions, almost quasi (τ1, τ2)-continuous functions and weakly quasi (τ1, τ2)-continuous func- tions were presented in [56], [59], [16], [51], [25], [11], [8], [10], [4], [1], [2], [26], [23], [18], [57], [41] and [31], respectively. In 1965, Lee [42] studied the notion of semiconnected func- tions. Kohli [39] introduced the notion of s-continuous functions and investigated several characterizations of semilocally connected spaces in terms of s-continuous functions. The concept of s-continuity as a generalization of continuity and semiconnectedness. Moreover, Kohli [40] 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. In 1989, Lipski [44] extended the concept of s-continuous functions to the setting of multifunctions. Popa [47] introduced the concept of precontinuous multifunctions and showed that H-almost continuity and precontinuity are equivalent for multifunc- tions. Ewert and Lipski [35] introduced and investigated the concept of s-quasi-continuous multifunctions. Popa and Noiri [50] introduced a new class of multifunctions called s- precontinuous multifunctions is a generalization of s-continuous multifunctions and pre- continuous multifunctions. Viriyapong and Boonpok [64] introduced and studied the con- cept of weakly quasi (Λ, sp)-continuous multifunctions. Moreover, several characteriza- tions of (τ1, τ2)δ-semicontinuous multifunctions, almost weakly (τ1, τ2)-continuous mul- tifunctions, ⋆-continuous multifunctions, β(⋆)-continuous multifunctions, α-⋆-continuous multifunctions, almost α-⋆-continuous multifunctions, almost quasi ⋆-continuous multi- functions, weakly α-⋆-continuous multifunctions, sβ(⋆)-continuous multifunctions, weakly sβ(⋆)-continuous multifunctions, θ(⋆)-quasi continuous multifunctions, almost ı⋆-continuous multifunctions, weakly (Λ, sp)-continuous multifunctions, α(Λ, sp)-continuous multifunc- tions, almost α(Λ, sp)-continuous multifunctions, weakly α(Λ, sp)-continuous multifunc- tions, almost β(Λ, sp)-continuous multifunctions, slightly (Λ, sp)-continuous multifunc- tions, (τ1, τ2)-continuous multifunctions, almost (τ1, τ2)-continuous multifunctions, weakly (τ1, τ2)-continuous multifunctions, weakly quasi (τ1, τ2)-continuous multifunctions, c-(τ1, τ2)- continuous multifunctions, c-quasi (τ1, τ2)-continuous multifunctions and almost quasi (τ1, τ2)-continuous multifunctions were established in [5], [28], [3], [7], [17], [24], [6], [21], [20], [15], [9], [19], [22], [36], [13], [27], [58], [14], [54], [38], [60], [55], [37], [52] and [53], respectively. Popa and Noiri [49] introduced and studied the notion of s-β-continuous multifunctions. In particular, Popa and Noiri [48] introduced and investigated the con- cept of s-m-continuous multifunctions as multifunctions defined on a set satisfying some minimal conditions. Quite recently, Viriyapong et al. [66] introduced and studied the con- cept of s-(τ1, τ2)p-continuous multifunctions. In this paper, we introduce the concepts of upper s-(τ1, τ2)-continuous multifunctions and lower s-(τ1, τ2)-continuous multifunctions. We also investigate several characterizations of upper s-(τ1, τ2)-continuous multifunctions and lower s-(τ1, τ2)-continuous multifunctions. M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5634 3 of 12 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 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 called (τ1, τ2)r- open [62] (resp. (τ1, τ2)s-open [5], (τ1, τ2)p-open [5], (τ1, τ2)β-open [5], α(τ1, τ2)-open) [65]) 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))), A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(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, (τ1, τ2)β-closed, α(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 s-(τ1, τ2)-continuous multifunctions In this section, we introduce the notions of upper s-(τ1, τ2)-continuous multifunctions and lower s-(τ1, τ2)-continuous multifunctions. Moreover, some characterizations of up- per s-(τ1, τ2)-continuous multifunctions and lower s-(τ1, τ2)-continuous multifunctions are discussed. M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5634 4 of 12 Definition 1. 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 is upper 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 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. Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y having σ1σ2-connected complement and x ∈ F+(V ). Then, there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ V . Therefore, we have x ∈ U ⊆ F+(V ) and hence x ∈ τ1τ2-Int(F +(V )). Thus, F+(V ) ⊆ τ1τ2-Int(F +(V )) and so F+(V ) is τ1τ2-open in X. (2) ⇒ (3): The proof follows immediately 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 having the σ1σ2-connected σ1σ2-closure. Thus by (3), τ1τ2-Cl(F −(B)) ⊆ τ1τ2-Cl(F −(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. By (4), we have X − τ1τ2-Int(F +(B)) = τ1τ2-Cl(X − F+(B)) = τ1τ2-Cl(F −(Y −B)) ⊆ F−(σ1σ2-Cl(Y −B)) = F−(Y − σ1σ2-Int(B)) = X − F+(σ1σ2-Int(B)) and hence 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 σ1σ2-connected complement. By (5), F+(V ) = F+(σ1σ2-Int(V )) ⊆ τ1τ2-Int(F +(V )). Then, there exists a τ1τ2-open set U of X containing x such that U ⊆ F+(V ). Thus, F (U) ⊆ V . This shows that F is upper s-(τ1, τ2)-continuous. M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5634 5 of 12 Definition 2. 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 is lower 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 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. Proof. The proof is similar to that of Theorem 1. Corollary 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper s-(τ1, τ2)-continuous if F−(B) is τ1τ2-closed in X for every σ1σ2-connected set B of Y . Proof. Let V be any σ1σ2-open set of Y having σ1σ2-connected complement. Then, Y −V is σ1σ2-connected and σ1σ2-closed. By the hypothesis, F−(Y −V ) is τ1τ2-closed in X. Thus, F+(V ) is τ1τ2-open in X and by Theorem 1, F is upper s-(τ1, τ2)-continuous. Corollary 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower s-(τ1, τ2)-continuous if F+(B) is τ1τ2-closed in X for every σ1σ2-connected set B of Y . Proof. The proof is similar to that of Corollary 1. Definition 3. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be s-(τ1, τ2)-continuous if for x ∈ X and 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 . Corollary 3. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is s-(τ1, τ2)-continuous; (2) f−1(V ) is τ1τ2-open in X for every σ1σ2-open set V of Y having σ1σ2-connected complement; M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5634 6 of 12 (3) f−1(K) is τ1τ2-closed in X for every σ1σ2-connected σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(f −1(B)) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y having the σ1σ2-connected σ1σ2-closure; (5) f−1(σ1σ2-Int(B)) ⊆ τ1τ2-Int(f −1(B)) for every subset B of Y such that Y−σ1σ2-Int(B) is σ1σ2-connected. 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 4. [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 2. [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 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 . Theorem 3. 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 s-(τ1, τ2)-continuous if and only if ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is upper s-(τ1, τ2)-continuous. Proof. We put G = ClF⊛. Suppose that F is upper s-(τ1, τ2)-continuous. Let x ∈ X and V be any σ1σ2-open set of Y containing G(x) and having σ1σ2-connected complement. By Lemma 3, we have x ∈ G+(V ) = F+(V ) and hence there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ V . Since F (z) is σ1σ2-paracompact and σ1σ2- regular for each z ∈ U , by Lemma 2 there exists a τ1τ2-open set W of Y such that F (z) ⊆ W ⊆ σ1σ2-Cl(W ) ⊆ V ; hence G(z) ⊆ σ1σ2-Cl(W ) ⊆ V for each z ∈ U . Thus, G(U) ⊆ V . This shows that G is upper s-(τ1, τ2)-continuous. Conversely, suppose that G is upper s-(τ1, τ2)-continuous. Let x ∈ X and V be any σ1σ2-open set of Y containing F (x) and having σ1σ2-connected complement. By Lemma 3, we have x ∈ F+(V ) = G+(V ) and hence G(x) ⊆ V . There exists a τ1τ2-open set U of X containing x such that G(U) ⊆ V . Thus, U ⊆ G+(V ) = F+(V ) and so F (U) ⊆ V . This shows that F is upper s-(τ1, τ2)-continuous. M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5634 7 of 12 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 4. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower s-(τ1, τ2)-continuous if and only if ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is lower s-(τ1, τ2)-continuous. Proof. By using Lemma 4 this is shown similarly as in Theorem 3. 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 5. The set of all points x of X at which a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is not upper s-(τ1, τ2)-continuous is identical with the union of the τ1τ2-frontier of the upper inverse images 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 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+(V )) ̸= ∅ for every τ1τ2-open set U of X containing x. Therefore, we have x ∈ τ1τ2-Cl(X − F+(V )) and hence x ∈ τ1τ2-fr(F +(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 +(V )). If F is upper s-(τ1, τ2)-continuous at x ∈ X, then there exists a τ1τ2-open set U of X containing x such that U ⊆ F+(V ); hence x ∈ τ1τ2-Int(F +(V )). This is a contradiction and so F is not upper s-(τ1, τ2)-continuous at x. Theorem 6. The set of all points x of X at which a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is not lower s-(τ1, τ2)-continuous is identical with the union of the τ1τ2-frontier of the lower inverse images of σ1σ2-open sets meeting F (x) and having σ1σ2-connected complement. Proof. The proof is similar to that of Theorem 5. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the graph G(F ) = {(x, F (x)) | x ∈ X} is said to be strongly (τ1, τ2)-closed if for each (x, y) ∈ (X × Y ) − G(F ), there exists a τ1τ2-open set U of X containing x and a σ1σ2-open set V of Y containing y such that [U × σ1σ2-Cl(V )] ∩G(F ) = ∅. Lemma 5. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) has a strongly (τ1, τ2)-closed graph if and only if for each (x, y) ∈ (X × Y )−G(F ), there exists a τ1τ2-open set U of X containing x and a σ1σ2-open set V of Y containing y such that F (U) ∩ σ1σ2-Cl(V ) = ∅. M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5634 8 of 12 Proof. This proof is obvious. Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-regular [30] 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 . Definition 5. A bitopological space (X, τ1, τ2) is said to be locally τ1τ2-connected if for each x ∈ X and each τ1τ2-open set G of X containing x, there exists a τ1τ2-open τ1τ2- connected set V such that x ∈ V ⊆ G. Theorem 7. Let (Y, σ1, σ2) be a (σ1, σ2)-regular and locally σ1σ2-connected space. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an upper s-(τ1, τ2)-continuous multifunction such that F (x) is σ1σ2-closed for each x ∈ X, then G(F ) is strongly (τ1, τ2)-closed. Proof. Let (x, y) ∈ (X × Y ) − G(F ). Then, y ∈ Y − F (x). Since (Y, σ1, σ2) is (σ1, σ2)-regular, there exist disjoint σ1σ2-open sets V and V ′ of Y such that F (x) ⊆ V and y ∈ V ′. Moreover, since (Y, σ1, σ2) is locally σ1σ2-connected, there exists a σ1σ2-open σ1σ2- connected set W of Y such that y ∈ W ⊆ σ1σ2-Cl(W ) ⊆ V ′. Since F is upper s-(τ1, τ2)- continuous and Y − σ1σ2-Cl(W ) is a σ1σ2-open set having σ1σ2-connected complement, there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ Y −σ1σ2-Cl(W ). Thus, F (U) ∩ σ1σ2-Cl(W ) = ∅ and by Lemma 5, G(F ) is strongly (τ1, τ2)-closed. Definition 6. [54] A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower (τ1, τ2)- continuous if for each x ∈ X and each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a τ1τ2-open set U of X containing x such that F (z) ∩ V ̸= ∅ for each z ∈ U . Lemma 6. [54] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower (τ1, τ2)-continuous; (2) F−(V ) is τ1τ2-open in X for every σ1σ2-open set V of Y ; (3) F+(K) is τ1τ2-closed in X for every σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(F +(B)) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ; (5) F (τ1τ2-Cl(A)) ⊆ σ1σ2-Cl(F (A)) for every subset A of X; (6) F−(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F −(B)) for every subset B of Y . Theorem 8. If F : (X, τ1, τ2) → (Y, σ1, σ2) is lower s-(τ1, τ2)-continuous and F (A) is σ1σ2-connected for every subset A of X, then F is lower (τ1, τ2)-continuous. Proof. Let A be any subset of X. Since σ1σ2-Cl(F (A)) is σ1σ2-closed and σ1σ2- connected, by Theorem 2 we have F+(σ1σ2-Cl(F (A))) = τ1τ2-Cl(F +(σ1σ2-Cl(F (A)))) and A ⊆ F+(F (A)) ⊆ F+(σ1σ2-Cl(F (A))). Thus, F (σ1σ2-Cl(A)) ⊆ σ1σ2-Cl(F (A)). It follows from Lemma 6 that F is lower (τ1, τ2)-continuous. Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-connected [29] if X cannot be written as the union of two disjoint nonempty τ1τ2-open sets. M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5634 9 of 12 Definition 7. A bitopological space (X, τ1, τ2) is said to be s-τ1τ2-connected if X cannot be written as the union of two disjoint nonempty τ1τ2-open sets having τ1τ2-connected complement. Theorem 9. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an upper or lower 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 s-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 ) ̸= ∅. Next, we shall show that F+(U) and F+(V ) are τ1τ2-open in X. (i) Let F be upper s- (τ1, τ2)-continuous. By Theorem 1, F+(U) and F+(V ) are τ1τ2-open in X. (ii) Let F be lower s-(τ1, τ2)-continuous. Since V is a σ1σ2-clopen set with σ1σ2-connected complement, by Theorem 2, 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. 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