EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6008 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and Lower Contra-(τ1, τ2)-continuity Nongluk Viriyapong1, 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 between bitopological spaces, namely upper contra-(τ1, τ2)-continuous multifunctions and lower contra-(τ1, τ2)-continuous multifunctions. Moreover, several characterizations and some properties concerning upper contra-(τ1, τ2)-continuous multifunctions and lower contra-(τ1, τ2)-continuous multifunctions are investigated. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper contra-(τ1, τ2)-continuous multifunction, lower contra-(τ1, τ2)- continuous multifunction 1. Introduction Weaker and stronger forms of open sets in topological spaces such as semi-open sets, preopen sets, α-open sets, β-open sets, δ-open sets and θ-open sets play an important role in the researches of generalizations of continuity. By using these sets many au- thors introduced and investigated various types of continuity. Viriyapong and Boon- pok [1] 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 [2]. Dungthaisong et al. [3] introduced and studied the concept of g(m,n)-continuous functions. Duangphui et al. [4] 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 functions, almost strongly θ(Λ, p)-continuous func- tions, θ(Λ, p)-continuous functions, weakly (Λ, b)-continuous functions, θ(⋆)-precontinuous functions, (Λ, p(⋆))-continuous functions, ⋆-continuous functions, θ-I -continuous func- tions, almost (g,m)-continuous functions, pairwise almostM -continuous functions, (τ1, τ2)- continuous functions, almost (τ1, τ2)-continuous functions, weakly (τ1, τ2)-continuous func- tions and slightly (τ1, τ2)s-continuous functions were presented in [5], [6], [7], [8], [9], ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6008 Email addresses: nongluk.h@msu.ac.th (N. Viriyapong), areeyuth.s@psu.ac.th (A. Sama-Ae), chawalit.b@msu.ac.th (C. Boonpok) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6008 2 of 15 [10], [11], [12], [13], [14], [15], [16], [17], [18] and [19], respectively. Kong-ied at al. [20] introduced and studied the concept of almost quasi (τ1, τ2)-continuous functions. Chi- angpradit et al. [21] introduced and investigated the notion of weakly quasi (τ1, τ2)- continuous functions. Thongmoon et al. [22] introduced and studied the notion of rarely (τ1, τ2)-continuous functions. Srisarakham et al. [23] introduced and investigated the concept of faintly (τ1, τ2)-continuous functions. On the other hand, the present au- thors introduced and studied the notions of δ(τ1, τ2)-continuous functions [24], quasi θ(τ1, τ2)-continuous functions [25], almost weakly (τ1, τ2)-continuous functions [26] and almost nearly (τ1, τ2)-continuous functions [27]. In 1966, Dontchev [28] introduced the notion of contra-continuity in topological spaces. Dontchev and Noiri [29] introduced and studied the concept of RC-continuity between topological spaces which is weaker than contra-continuity. Jafari and Noiri [30] introduced a new class of function called contra- precontinuous functions which is weaker than contra-continuous functions and studied several basic properties of contra-precontinuous functions. Ekici [31] introduced and stud- ied a new class of functions called almost contra-precontinuous functions which gener- alize classes of regular set-connected functions [32], contra-precontinuous functions [30], contra-continuous functions [28], almost s-continuous functions [33] and perfectly contin- uous functions [34]. In 2008, Ekici et al. [35] extended the notion of contra-continuous functions to the setting of multifunctions. Noiri and Popa [36] introduced the notion of weakly precon- 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, s-(τ1, τ2)p- continuous multifunctions, slightly α(τ1, τ2)-continuous multifunctions and slightly (τ1, τ2)p- continuous multifunctions were established in [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64] and [65], respectively. On the other hand, the present authors introduced and investi- gated the notions of rarely s-(τ1, τ2)p-continuous multifunctions [66], almost nearly (τ1, τ2)- continuous multifunctions [67], s-(τ1, τ2)-continuous multifunctions [68], quasi θ(τ1, τ2)- continuous multifunctions [69], almost nearly quasi (τ1, τ2)-continuous multifunctions [70], weakly s-(τ1, τ2)-continuous multifunctions [71], nearly (τ1, τ2)-continuous multifunctions [72] and almost quasi (τ1, τ2)-continuous multifunctions [73]. Ekici et al. [74] introduced and studied two new concepts namely contra-precontinuous multifunctions and almost N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6008 3 of 15 contra-precontinuous multifunctions which are containing the class of contra-continuous multifunctions [35] and contained in the class of weakly precontinuous multifunctions. Ekici et al. [75] introduced and studied a new generalization of contra-continuous multi- functions called almost contra-continuous multifunctions. Recently, the present authors [76] introduced and investigated the notions of upper almost contra-(Λ, sp)-continuous multifunctions and lower almost contra-(Λ, sp)-continuous multifunctions. In this paper, we introduce the concepts of upper contra-(τ1, τ2)-continuous multifunctions and lower contra-(τ1, τ2)-continuous multifunctions. We also investigate several characterizations of upper contra-(τ1, τ2)-continuous multifunctions and lower contra-(τ1, τ2)-continuous mul- tifunctions. 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 [77] 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 [77] 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 [77] of A and is denoted by τ1τ2-Int(A). Lemma 1. [77] 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 [78] 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 [79] (resp. (τ1, τ2)s-open [37], (τ1, τ2)p-open [37], (τ1, τ2)β-open [37]) 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, (τ1, τ2)β- closed, α(τ1, τ2)-closed). Let A be a subset of a bitopological space (X, τ1, τ2). The set ∩{G | A ⊆ G and G is τ1τ2-open} is called the τ1τ2-kernel [77] of A and is denoted by τ1τ2-ker(A). N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6008 4 of 15 Lemma 2. [77] For subsets A,B of a bitopological space (X, τ1, τ2), the following properties hold: (1) A ⊆ τ1τ2-ker(A). (2) If A ⊆ B, then τ1τ2-ker(A) ⊆ τ1τ2-ker(B). (3) If A is τ1τ2-open, then τ1τ2-ker(A) = A. (4) x ∈ τ1τ2-ker(A) if and only if A ∩H ̸= ∅ for every τ1τ2-closed set H containing 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 contra-(τ1, τ2)-continuous multifunctions In this section, we introduce the concepts of upper contra-(τ1, τ2)-continuous multi- functions and lower contra-(τ1, τ2)-continuous multifunctions. Furthermore, several char- acterizations of upper contra-(τ1, τ2)-continuous multifunctions and lower contra-(τ1, τ2)- continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called upper contra-(τ1, τ2)- continuous at a point x ∈ X if for each σ1σ2-closed set K of Y such that x ∈ F+(K), there exists a τ1τ2-open set U of X containing x such that U ⊆ F+(K). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called upper contra-(τ1, τ2)-continuous if F is upper contra- (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 contra-(τ1, τ2)-continuous; (2) F+(K) is τ1τ2-open in X for every σ1σ2-closed set K of Y ; (3) F−(V ) is τ1τ2-closed in X for every σ1σ2-open set V of Y ; (4) for each x ∈ X and each σ1σ2-closed set K of Y containing F (x), there exists a τ1τ2-open set U of X containing x such that if y ∈ U , then F (y) ⊆ K. Proof. (1) ⇔ (2): Let K be any σ1σ2-closed set of Y and x ∈ F+(K). Since F is upper contra-(τ1, τ2)-continuous, there exists a τ1τ2-open set U of X containing x such that U ⊆ F+(K). Thus, F+(K) is τ1τ2-open in X. The converse of the proof is similar. (2) ⇔ (3): This follows from the fact that F+(Y −B) = X − F−(B) for every subset B of Y . (1) ⇔ (4): Obvious. N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6008 5 of 15 Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called lower contra-(τ1, τ2)- continuous at a point x ∈ X if for each σ1σ2-closed set K of Y such that x ∈ F−(K), there exists a τ1τ2-open set U of X containing x such that U ⊆ F−(K). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called lower contra-(τ1, τ2)-continuous if F is lower contra- (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 contra-(τ1, τ2)-continuous; (2) F−(K) is τ1τ2-open in X for every σ1σ2-closed set K of Y ; (3) F+(V ) is τ1τ2-closed in X for every σ1σ2-open set V of Y ; (4) for each x ∈ X and each σ1σ2-closed set K of Y such that F (x)∩K ̸= ∅, there exists a τ1τ2-open set U of X containing x such that if y ∈ U , then F (y) ∩K ̸= ∅. Proof. The proof is similar to that of Theorem 1. Theorem 3. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction. If τ1τ2-Cl(F −(B)) ⊆ F−(σ1σ2-ker(B)) for every subset B of Y , then F is upper contra-(τ1, τ2)-continuous. Proof. Suppose that τ1τ2-Cl(F −(B)) ⊆ F−(σ1σ2-ker(B)) for every subset B of Y . Let V be any σ1σ2-open set of Y . By Lemma 2, we have τ1τ2-Cl(F −(V )) ⊆ F−(σ1σ2-ker(V )) = F−(V ) and hence F−(V ) is τ1τ2-closed inX. By Theorem 1, F is upper contra-(τ1, τ2)-continuous. Theorem 4. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction. If F (τ1τ2-Cl(A)) ⊆ σ1σ2-ker(F (A)) for every subset A of X, then F is lower contra-(τ1, τ2)-continuous. Proof. Let V be any σ1σ2-open set of Y . Then, F (τ1τ2-Cl(F +(V ))) ⊆ σ1σ2-ker(V ) and hence τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-ker(V )). By Lemma 2, we have τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-ker(V )) = F+(V ) and hence F+(V ) is τ1τ2-closed in X. By Theorem 2, F is lower contra-(τ1, τ2)-continuous. Theorem 5. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction. If τ1τ2-Cl(F +(B)) ⊆ F+(σ1σ2-ker(B)) for every subset B of Y , then F is lower contra-(τ1, τ2)-continuous. Proof. Let V be any σ1σ2-open set of Y . Then, τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-ker(V )) and by Lemma 2, τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-ker(V )) = F+(V ). This implies that F+(V ) is τ1τ2-closed in X. By Theorem 2, F is lower contra-(τ1, τ2)-continuous. N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6008 6 of 15 Definition 3. [6] A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper weakly (τ1, τ2)-continuous if for each x ∈ X and each σ1σ2-open set V of Y containing F (x), there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). Theorem 6. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an upper contra-(τ1, τ2)-continuous multi- function, then F is upper weakly (τ1, τ2)-continuous. Proof. Let x ∈ X and V be any σ1σ2-open set of Y containing F (x). Then, σ1σ2-Cl(V ) is a σ1σ2-closed set Y containing F (x). Since F is upper contra-(τ1, τ2)-continuous, by Theorem 1 there exists a τ1τ2-open set U ofX containing x such that U ⊆ F+(σ1σ2-Cl(V )); hence F (U) ⊆ σ1σ2-Cl(V ). This shows that F is upper weakly (τ1, τ2)-continuous. The converse of Theorem 6 is not true in general as shown in the following example. Example 1. Let X = {a, b, c, d} with topologies τ1 = {∅, {a}, {a, b}, {a, b, c}, X} and τ2 = {∅, {a}, {a, b}, {a, b, c}, {a, b, d}, X}. Let Y = {1, 2, 3, 4} with topologies σ1 = {∅, {1}, {1, 2}, {1, 2, 3}, {1, 2, 4}, Y } and σ2 = {∅, {1}, {1, 2}, {1, 2, 3}, Y }. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is defined as follows: F (a) = {1, 2}, F (b) = {2}, F (c) = {1, 2} and F (d) = {4}. Then, F is upper weakly (τ1, τ2)-continuous but F is not upper contra-(τ1, τ2)-continuous. Definition 4. [6] A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower weakly (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 σ1σ2-Cl(V )∩F (z) ̸= ∅ for each z ∈ U . Theorem 7. If F : (X, τ1, τ2) → (Y, σ1, σ2) is a lower contra-(τ1, τ2)-continuous multi- function, then F is lower weakly (τ1, τ2)-continuous. Proof. The proof is similar to that of Theorem 6. Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-connected [77] if X cannot be written as the union of two nonempty disjoint τ1τ2-open sets. Lemma 3. [6] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper weakly (τ1, τ2)-continuous; (2) F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (3) τ1τ2-Cl(F −(σ1σ2-Int(K))) ⊆ F−(K) for every 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 ; (5) F+(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ; N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6008 7 of 15 (6) τ1τ2-Cl(F −(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (7) τ1τ2-Cl(F −(V )) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (8) τ1τ2-Cl(F −(σ1σ2-Int(K))) ⊆ F−(K) for every (σ1, σ2)r-closed set K of Y . Lemma 4. [6] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower weakly (τ1, τ2)-continuous; (2) F−(V ) ⊆ τ1τ2-Int(F −(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (3) τ1τ2-Cl(F +(σ1σ2-Int(K))) ⊆ F+(K) for every 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 ; (5) F−(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F −(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ; (6) τ1τ2-Cl(F +(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (7) τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (8) τ1τ2-Cl(F +(σ1σ2-Int(K))) ⊆ F+(K) for every (σ1, σ2)r-closed set K of Y . Theorem 8. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an upper or lower contra-(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 σ1σ2-connected. Proof. Suppose that (Y, σ1, σ2) is not σ1σ2-connected. There exist nonempty σ1σ2- open sets U and V of Y 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 show that F+(U) and F+(V ) are τ1τ2-open in X. (i) Let F be upper contra- (τ1, τ2)-continuous, by Theorem 6 we have F is upper weakly (τ1, τ2)-continuous. By Lemma 3, 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. Consequently, this shows that (X, τ1, τ2) is not τ1τ2-connected. N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6008 8 of 15 (ii) Let F be lower contra-(τ1, τ2)-continuous, by Theorem 7 we have F is lower weakly (τ1, τ2)-continuous. By Lemma 4, τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-Cl(V )) = F+(V ) since V is σ1σ2-clopen. Therefore, F+(V ) = τ1τ2-Cl(F +(V )) and so F+(V ) is τ1τ2-closed in X. Thus, we have 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. Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-compact [77] if every cover of X by τ1τ2-open sets of X has a finite subcover. Definition 5. [80] A bitopological space (X, τ1, τ2) is said to be strongly S-τ1τ2-closed if every cover of X by τ1τ2-closed sets of X has a finite subcover. Theorem 9. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a surjective multifunction and F (x) is strongly S-σ1σ2-closed for each x ∈ X. If F is upper contra-(τ1, τ2)-continuous and (X, τ1, τ2) is τ1τ2-compact, then (Y, σ1, σ2) is strongly S-σ1σ2-closed. Proof. Suppose that (X, τ1, τ2) is τ1τ2-compact. Let {Vγ | γ ∈ ∇} be any cover of Y by σ1σ2-closed sets of Y . Since F (x) is strongly S-σ1σ2-closed for each x ∈ X, there exists a finite subset ∇(x) of ∇ such that F (x) ⊆ ∪{Vγ | γ ∈ ∇(x)}. Put V (x) = ∪{Vγ | γ ∈ ∇(x)}. Then, V (x) is σ1σ2-closed in Y and F (x) ⊆ V (x). Since F is upper contra-(τ1, τ2)-continuous, there exists a τ1τ2-open set U(x) of X containing x such that F (U(x)) ⊆ V (x). The family {U(x) | x ∈ X} is a τ1τ2-open cover of X. Since (X, τ1, τ2) is τ1τ2-compact, there exists a finite number of pints, say, x1, x2, x3, ..., xn in X such that X = ∪{U(xk) | xk ∈ X; 1 ≤ k ≤ n}. Thus, Y = F (X) = ∪{F (U(xk)) | xk ∈ X; 1 ≤ k ≤ n} ⊆ ∪{Vγ(xk) | xk ∈ X; 1 ≤ k ≤ n}. This shows that (Y, σ1, σ2) is strongly S-σ1σ2-closed. Definition 6. [56] A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be: (1) upper (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 (U) ⊆ V ; (2) 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 5. [56] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper (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 ; N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6008 9 of 15 (5) F+(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F +(B)) for every subset B of Y . Theorem 10. If F : (X, τ1, τ2) → (Y, σ1, σ2) is upper (τ1, τ2)-continuous and G : (Y, σ1, σ2) → (Z, ρ1, ρ2) is upper contra-(σ1, σ2)-continuous, then G ◦ F : (X, τ1, τ2) → (Z, ρ1, ρ2) is upper contra- (τ1, τ2)-continuous. Proof. Let K be any ρ1ρ2-closed set of Z. Since G is upper contra-(σ1, σ2)-continuous, by Theorem 1 we have F+(K) is σ1σ2-open in Y . Since F is upper (τ1, τ2)-continuous, by Lemma 5 we have (G ◦ F )+(K) = F+(G+(K)) is τ1τ2-open in X. Thus by Theorem 1, G ◦ F is upper contra-(τ1, τ2)-continuous. Lemma 6. [56] 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 11. If F : (X, τ1, τ2) → (Y, σ1, σ2) is lower (τ1, τ2)-continuous and G : (Y, σ1, σ2) → (Z, ρ1, ρ2) is lower contra-(σ1, σ2)-continuous, then G ◦ F : (X, τ1, τ2) → (Z, ρ1, ρ2) is lower contra- (τ1, τ2)-continuous. Proof. The proof is similar to that of Theorem 10. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), a multifunction ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is defined in [77] as follows: ClF⊛(x) = σ1σ2-Cl(F (x)) for each x ∈ X. Definition 7. [77] 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; N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6008 10 of 15 (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 7. [77] If F : (X, τ1, τ2) → (Y, σ1, σ2) is a multifunction such that F (x) is σ1σ2- regular and σ1σ2-paracompact for each x ∈ X, then ClF+ ⊛ (V ) = F+(V ) for each σ1σ2-open set V of Y . Lemma 8. If F : (X, τ1, τ2) → (Y, σ1, σ2) is a multifunction such that F (x) is σ1σ2-regular and σ1σ2-paracompact for each x ∈ X, then ClF− ⊛ (K) = F−(K) for each σ1σ2-closed set K of Y . Proof. It follows from Lemma 7. Lemma 9. [77] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), ClF − ⊛ (V ) = F−(V ) for each σ1σ2-open set V of Y . Lemma 10. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), ClF+ ⊛ (K) = F+(K) for each σ1σ2-closed set K of Y . Proof. It follows from Lemma 9. Theorem 12. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper contra-(τ1, τ2)- continuous if and only if ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is upper contra-(τ1, τ2)-continuous. Proof. Suppose that F is upper contra-(τ1, τ2)-continuous. Let K be any σ1σ2-closed set of Y . It follows from Lemma 9, Lemma 10 and Theorem 1, ClF+ ⊛ (K) = F+(K) is τ1τ2-open in X. Thus, ClF⊛ is upper contra-(τ1, τ2)-continuous. Conversely, suppose that ClF⊛ is upper contra-(τ1, τ2)-continuous. Let K be any σ1σ2- closed set of Y . By Lemma 9, Lemma 10 and Theorem 1, F+(K) = ClF+ ⊛ (K) is τ1τ2-open in X. Thus, F is upper contra-(τ1, τ2)-continuous. Theorem 13. 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 contra-(τ1, τ2)-continuous if and only if ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is lower contra-(τ1, τ2)-continuous. Proof. Suppose that F is lower contra-(τ1, τ2)-continuous. Let K be any σ1σ2-closed set of Y . It follows from Lemma 7, Lemma 8 and Theorem 2 that ClF− ⊛ (K) = F−(K) is τ1τ2-open in X. This shows that ClF⊛ is lower contra-(τ1, τ2)-continuous. Conversely, suppose that ClF⊛ is lower contra-(τ1, τ2)-continuous. Let K be any σ1σ2- closed set of Y . 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