EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6010 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and Lower Almost Contra-(τ1, τ2)-continuity Jeeranunt Khampakdee1, 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 introduces new classes of multifunctions between bitopological spaces, namely upper almost contra-(τ1, τ2)-continuous multifunctions and lower almost contra-(τ1, τ2)- continuous multifunctions. Furthermore, several characterizations of upper almost contra-(τ1, τ2)- continuous multifunctions and lower almost contra-(τ1, τ2)-continuous multifunctions are consid- ered. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: Upper almost contra-(τ1, τ2)-continuous multifunction, lower almost contra-(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. Stronger and weaker forms of open sets play an important role in the generalization of different forms of continuity. Using different forms of open sets, many authors have introduced and studied various types of continuity for functions and multifunctions. In [1], the present authors studied some properties of (Λ, sp)-open sets, r(Λ, sp)-open sets, s(Λ, sp)-open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets, β(Λ, sp)-open sets and b(Λ, sp)-open sets. Viriyapong and Boonpok [2] investigated some characterizations of (Λ, sp)-continuous functions by utilizing the no- tions of (Λ, sp)-open sets and (Λ, sp)-closed sets. 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 almost (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, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6010 Email addresses: jeeranunt.k@msu.ac.th (J. Khampakdee), 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) J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 2 of 19 ⋆-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 [5], [6], [7], [8], [9], [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. Chiangpradit 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] in- troduced and investigated the concept of faintly (τ1, τ2)-continuous functions. On the other hand, the present authors 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]. The notion of contra- continuity in topological spaces was introduced by Dontchev [28]. 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 functions called contra-precontinuous functions which is weaker than contra-continuous functions and studied several basic properties of contra-precontinuous functions. Ekici [31] intro- duced and studied a new class of functions called almost contra-precontinuous functions which generalize classes of regular set-connected functions [32], contra-precontinuous func- tions [30], contra-continuous functions [28], almost s-continuous functions [33] and per- fectly continuous functions [34]. In 2008, Ekici et al. [35] extended the notion of contra-continuous functions to the set- ting of multifunctions. Noiri and Popa [36] introduced the notion of weakly precontinuous multifunctions. Ekici et al. [37] introduced and studied two new concepts namely contra- precontinuous multifunctions and almost contra-precontinuous multifunctions which are containing the class of contra-continuous multifunctions [35] and contained in the class of weakly precontinuous multifunctions. Laprom et al. [38] introduced and investigated the notion of almost β(τ1, τ2)-continuous multifunctions. Moreover, some characterizations of (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 [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], [65], J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 3 of 19 [66] and [67], respectively. On the other hand, the present authors introduced and investi- gated the notions of rarely s-(τ1, τ2)p-continuous multifunctions [68], almost nearly (τ1, τ2)- continuous multifunctions [69], s-(τ1, τ2)-continuous multifunctions [70], quasi θ(τ1, τ2)- continuous multifunctions [71], almost nearly quasi (τ1, τ2)-continuous multifunctions [72], weakly s-(τ1, τ2)-continuous multifunctions [73], nearly (τ1, τ2)-continuous multifunctions [74] and almost quasi (τ1, τ2)-continuous multifunctions [75]. Ekici et al. [76] introduced and studied a new generalization of contra-continuous multifunctions called almost contra- continuous multifunctions. Boonpok and Khampakdee [77] 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 almost contra-(τ1, τ2)-continuous multifunctions and lower almost contra-(τ1, τ2)- continuous multifunctions. We also investigate some characterizations of upper almost contra-(τ1, τ2)-continuous multifunctions and lower almost 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 [78] 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 [78] 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 [78] of A and is denoted by τ1τ2-Int(A). Lemma 1. [78] 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 subsetA of a bitopological space (X, τ1, τ2) is called (τ1, τ2)r-open [79] (resp. (τ1, τ2)s- open [39], (τ1, τ2)p-open [39], (τ1, τ2)β-open [39]) 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). Let A be a subset of a bitopological space (X, τ1, τ2). The set ∩{V | V is (τ1, τ2)r-open and A ⊆ V } is called the (τ1, τ2)r-kernel of A and is denoted by (τ1, τ2)r-ker(A). J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 4 of 19 Lemma 2. For subsets A,B of a bitopological space (X, τ1, τ2), the following properties hold: (1) A ⊆ (τ1, τ2)r-ker(A). (2) If A ⊆ B, then (τ1, τ2)r-ker(A) ⊆ (τ1, τ2)r-ker(B). (3) If A is (τ1, τ2)r-open, then (τ1, τ2)r-ker(A) = A. (4) x ∈ (τ1, τ2)r-ker(A) if and only if A ∩ K ̸= ∅ for every (τ1, τ2)r-closed set K con- taining x. A subset A of a bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-open [80] if A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)-open set is called 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 [65] (resp. (τ1, τ2)s-closure [39], α(τ1, τ2)-closure [66]) 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 [65] (resp. (τ1, τ2)s-interior [39], α(τ1, τ2)-interior [66]) of A and is denoted by (τ1, τ2)-pInt(A) (resp. (τ1, τ2)-sInt(A), α(τ1, τ2)-Int(A)). Lemma 3. For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: (1) (τ1, τ2)-pCl(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∪A [65]. (2) (τ1, τ2)-pInt(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∩A [26]. (3) (τ1, τ2)-sCl(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∪A [39]. (4) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A [62]. For a subset A of a bitopological space (X, τ1, τ2), a point x ∈ X is called a s(τ1, τ2)θ- cluster point [81] of A if τ1τ2-Cl(U) ∩ A ̸= ∅ for every (τ1, τ2)s-open set U containing x. The set of all s(τ1, τ2)θ-cluster points of A is called the s(τ1, τ2)θ-closure [81] of A and is denoted by s(τ1, τ2)θ-Cl(A). A subset A of a bitopological space (X, τ1, τ2) is called s(τ1, τ2)θ-closed [81] if s(τ1, τ2)θ-Cl(A) = A. The complement of a s(τ1, τ2)θ-closed set is said to be s(τ1, τ2)θ-open [81]. The union of all s(τ1, τ2)θ-open sets of X contained in A is called the s(τ1, τ2)θ-interior [81] of A and is denoted by s(τ1, τ2)θ-Int(A). 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). J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 5 of 19 3. Upper and lower almost contra-(τ1, τ2)-continuous multifunctions In this section, we introduce the concepts of upper almost contra-(τ1, τ2)-continuous multifunctions and lower almost contra-(τ1, τ2)-continuous multifunctions. Moreover, some characterizations of upper almost contra-(τ1, τ2)-continuous multifunctions and lower al- most contra-(τ1, τ2)-continuous multifunctions are considered. Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper almost contra-(τ1, τ2)-continuous at a point x ∈ X if for each (σ1, σ2)r-closed set K of Y with 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 said to be upper almost contra-(τ1, τ2)- continuous if F is upper almost 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 almost contra-(τ1, τ2)-continuous; (2) F+(K) is τ1τ2-open in X for every (σ1, σ2)r-closed set K of Y ; (3) F−(V ) is τ1τ2-closed in X for every (σ1, σ2)r-open set V of Y ; (4) F−(σ1σ2-Int(σ1σ2-Cl(V ))) is τ1τ2-closed in X for every σ1σ2-open set V of Y ; (5) F+(σ1σ2-Cl(σ1σ2-Int(K))) is τ1τ2-open in X for every σ1σ2-closed set K of Y ; (6) for each x ∈ X and each (σ1, σ2)s-open set V of Y with F (x) ⊆ V , there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ); (7) F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(V ))) for every (σ1, σ2)s-open set V of Y . Proof. (1) ⇒ (2): Let K be any (σ1, σ2)r-closed set of Y and x ∈ F+(K). Since F is is upper almost contra-(τ1, τ2)-continuous, there exists a τ1τ2-open set U of X containing x such that U ⊆ F+(K). It follows that F+(K) is τ1τ2-open in X. (2) ⇒ (1): The proof is obvious. (2) ⇔ (3): It follows from the fact that F+(Y −B) = X − F−(B) for every subset B of Y . (3) ⇔ (4): Let V be any σ1σ2-open set of Y . Since σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r- open in Y , by (3) we have F−(σ1σ2-Int(σ1σ2-Cl(V ))) is τ1τ2-closed in X. The converse is obvious. (4) ⇔ (5): It follows from the fact that F+(Y −B) = X − F−(B) for every subset B of Y . (5) ⇔ (2): It is similar to that of (3) ⇔ (4). (6) ⇒ (7): Let V be any (σ1, σ2)s-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V . By (6), there exists a τ1τ2-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). Thus, U ⊆ F+(σ1σ2-Cl(V )) and hence x ∈ τ1τ2-Int(F +(σ1σ2-Cl(V ))). This implies that F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(V ))). J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 6 of 19 (7) ⇒ (2): Let K be any (σ1, σ2)r-closed set of Y . Since K is (σ1, σ2)s-open in Y , by (7) we have F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(V ))) = τ1τ2-Int(F +(V )). Thus, F+(K) is τ1τ2-open in X. (2) ⇒ (6): Let x ∈ X and V be any (σ1, σ2)s-open set of Y with F (x) ⊆ V . Since σ1σ2-Cl(V ) is (σ1, σ2)r-closed in Y , by (2) we have F+(σ1σ2-Cl(V )) is τ1τ2-open in X. 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 ). Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower almost contra-(τ1, τ2)-continuous at a point x ∈ X if for each (σ1, σ2)r-closed set K of Y with 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 said to be lower almost contra-(τ1, τ2)- continuous if F is lower almost 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 almost contra-(τ1, τ2)-continuous; (2) F−(K) is τ1τ2-open in X for every (σ1, σ2)r-closed set K of Y ; (3) F+(V ) is τ1τ2-closed in X for every (σ1, σ2)r-open set V of Y ; (4) F+(σ1σ2-Int(σ1σ2-Cl(V ))) is τ1τ2-closed in X for every σ1σ2-open set V of Y ; (5) F−(σ1σ2-Cl(σ1σ2-Int(K))) is τ1τ2-open in X for every σ1σ2-closed set K of Y ; (6) for each x ∈ X and each (σ1, σ2)s-open set V of Y with F (x)∩V ̸= ∅, there exists a τ1τ2-open set U of X containing x such that F (z)∩ σ1σ2-Cl(V ) ̸= ∅ for each z ∈ U ; (7) F−(V ) ⊆ τ1τ2-Int(F −(σ1σ2-Cl(V ))) for every (σ1, σ2)s-open set V of Y . 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 contra-(τ1, τ2)-continuous; (2) F+(σ1σ2-Cl(V )) is τ1τ2-open in X for every (σ1, σ2)β-open set V of Y ; (3) F+(σ1σ2-Cl(V )) is τ1τ2-open in X for every (σ1, σ2)s-open set V of Y ; (4) F−(σ1σ2-Int(σ1σ2-Cl(V ))) is τ1τ2-closed in X for every (σ1, σ2)p-open set V of Y . J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 7 of 19 Proof. (1) ⇒ (2): Let V be any (σ1, σ2)β-open set of Y . Then, σ1σ2-Cl(V ) is (σ1, σ2)r- closed in Y , by Theorem 1 we have F+(σ1σ2-Cl(V )) is τ1τ2-open in X. (2) ⇒ (3): This is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open. (3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . Thus, Y − σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r-closed in Y and hence Y − σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)s-open in Y . By (3), X − F−(σ1σ2-Int(σ1σ2-Cl(V ))) = F+(Y − σ1σ2-Int(σ1σ2-Cl(V ))) = F+(σ1σ2-Cl(Y − σ1σ2-Int(σ1σ2-Cl(V )))) is τ1τ2-open in X. Thus, F−(σ1σ2-Int(σ1σ2-Cl(V ))) is τ1τ2-closed in X. (4) ⇒ (1): Let V be any (σ1, σ2)r-open set of Y . Then, V is (σ1, σ2)p-open in Y and by (4), F−(V ) = F−(σ1σ2-Int(σ1σ2-Cl(V ))) is τ1τ2-closed in X. By Theorem 1, F is upper almost almost contra-(τ1, τ2)-continuous. Theorem 4. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost contra-(τ1, τ2)-continuous; (2) F−(σ1σ2-Cl(V )) is τ1τ2-open in X for every (σ1, σ2)β-open set V of Y ; (3) F−(σ1σ2-Cl(V )) is τ1τ2-open in X for every (σ1, σ2)s-open set V of Y ; (4) F+(σ1σ2-Int(σ1σ2-Cl(V ))) is τ1τ2-closed in X for every (σ1, σ2)p-open set V of Y . Proof. The proof is similar to that of Theorem 3. Lemma 4. [81] For a bitopological space (X, τ1, τ2), the following properties hold: (1) α(τ1, τ2)-Cl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)β-open set V of X; (2) (τ1, τ2)-pCl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)s-open set V of X; (3) (τ1, τ2)-sCl(V ) = τ1τ2-Int(τ1τ2-Cl(V )) for every (τ1, τ2)p-open set V of X. Corollary 1. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost contra-(τ1, τ2)-continuous; (2) F+(α(σ1, σ2)-Cl(V )) is τ1τ2-open in X for every (σ1, σ2)β-open set V of Y ; (3) F+((σ1, σ2)-pCl(V )) is τ1τ2-open in X for every (σ1, σ2)s-open set V of Y ; (4) F−((σ1, σ2)-sCl(V )) is τ1, τ2-closed in X for every (σ1, σ2)p-open set V of Y . Corollary 2. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 8 of 19 (1) F is lower almost contra-(τ1, τ2)-continuous; (2) F−(α(σ1, σ2)-Cl(V )) is τ1τ2-open in X for every (σ1, σ2)β-open set V of Y ; (3) F−((σ1, σ2)-pCl(V )) is τ1τ2-open in X for every (σ1, σ2)s-open set V of Y ; (4) F+((σ1, σ2)-sCl(V )) is τ1τ2-closed in X for every (σ1, σ2)p-open set V of Y . Theorem 5. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost contra-(τ1, τ2)-continuous; (2) τ1τ2-Cl(F −(V )) ⊆ F−(σ1σ2-Int(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (3) τ1τ2-Cl(F −(V )) ⊆ F−((σ1, σ2)-sCl(V )) for every σ1σ2-open set V of Y . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y . Then, σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r-open in Y . Thus by (1), F−(σ1σ2-Int(σ1σ2-Cl(V ))) is τ1τ2-closed in X. Since V ⊆ σ1σ2-Int(σ1σ2-Cl(V )), we have F−(V ) ⊆ F−(σ1σ2-Int(σ1σ2-Cl(V ))) and hence τ1τ2-Cl(F −(V )) ⊆ F−(σ1σ2-Int(σ1σ2-Cl(V ))). (2) ⇒ (1): Let V be any (σ1, σ2)r-open set of Y . By (2), we have τ1τ2-Cl(F −(V )) ⊆ F−(σ1σ2-Int(σ1σ2-Cl(V ))) = F−(V ) and hence F−(V ) is τ1τ2-closed in X, by Theorem 1 we have F is upper almost contra- (τ1, τ2)-continuous. (2) ⇔ (3): It follows from Lemma 4. Theorem 6. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost contra-(τ1, τ2)-continuous; (2) τ1τ2-Cl(F +(V )) ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (3) τ1τ2-Cl(F +(V )) ⊆ F+((σ1, σ2)-sCl(V )) for every σ1σ2-open set V of Y . Proof. The proof is similar to that of Theorem 5. Theorem 7. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost contra-(τ1, τ2)-continuous; (2) F−(V ) is τ1τ2-open in X for every s(σ1, σ2)θ-open set V of Y ; (3) F+(K) is τ1τ2-closed in X for every s(σ1, σ2)θ-closed set K of Y ; J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 9 of 19 (4) τ1τ2-Cl(F +(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+((σ1, σ2)-sCl(B)) for every subset B of Y ; (5) τ1τ2-Cl(F +(B)) ⊆ F+(s(σ1, σ2)θ-Cl(B)) for every subset B of Y ; (6) F (τ1τ2-Cl(A)) ⊆ s(σ1, σ2)θ-Cl(F (A)) for every subset A of X. Proof. (1) ⇒ (2): Let V be any s(σ1, σ2)θ-open set of Y . There exists a family of (σ1, σ2)r-closed sets {Kγ | γ ∈ ∇} such that V = ∪{Kγ | γ ∈ ∇}. It follows from Theorem 2 that F−(V ) = ∪{F−(Kγ) | γ ∈ ∇} is τ1τ2-open in X. (2) ⇒ (3): The proof is obvious. (3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Int(σ1σ2-Cl(B)) is (σ1, σ2)r-open and hence σ1σ2-Int(σ1σ2-Cl(B)) is s(σ1, σ2)θ-closed in Y . By (3), F+(σ1σ2-Int(σ1σ2-Cl(B))) is τ1τ2-closed in X. Thus, τ1τ2-Cl(F +(σ1σ2-Int(σ1σ2-Cl(B)))) = F+(σ1σ2-Int(σ1σ2-Cl(B))) ⊆ F+((σ1, σ2)-sCl(B)). (4) ⇒ (5): Let B be any subset of Y . For any (σ1, σ2)r-open set V of Y with B ⊆ V , by (4) and Lemma 4 we have τ1τ2-Cl(F +(B)) ⊆ τ1τ2-Cl(F +(V )) = τ1τ2-Cl(F +(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+((σ1, σ2)-sCl(V )) = F+(V ). Thus, τ1τ2-Cl(F +(B)) ⊆ F+(∩{V | V is (σ1, σ2)r-open in Y and B ⊆ V }) = F+(s(σ1, σ2)θ-Cl(B)). (5) ⇒ (1): Let V be any (σ1, σ2)s-open set of Y . By (5), we have X − τ1τ2-Int(F −(σ1σ2-Cl(V ))) = τ1τ2-Cl(F +(Y − σ1σ2-Cl(V ))) ⊆ F+((σ1, σ2)-sCl(Y − σ1σ2-Cl(V ))) = F+(Y − σ1σ2-Cl(V )) = X − F−(σ1σ2-Cl(V )) and hence F−(V ) ⊆ F−(σ1σ2-Cl(V )) ⊆ τ1τ2-Int(F −(σ1σ2-Cl(V ))). By Theorem 2, F is lower almost contra-(τ1, τ2)-continuous. (5) ⇒ (6): Let A be any subset of X and B = F (A). Then, A ⊆ F+(B) and by (5), τ1τ2-Cl(A) ⊆ τ1τ2-Cl(F +(B)) ⊆ F+(s(σ1, σ2)θ-Cl(B)). Thus, F (τ1τ2-Cl(A)) ⊆ F (F+(s(σ1, σ2)θ-Cl(B))) ⊆ s(σ1, σ2)θ-Cl(B) J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 10 of 19 = s(σ1, σ2)θ-Cl(F (A)). (6) ⇒ (5): Let B be any subset of Y . By (6), we have F (τ1τ2-Cl(F +(B))) ⊆ s(σ1, σ2)θ-Cl(F (F+(B))) ⊆ s(σ1, σ2)θ-Cl(B) and hence τ1τ2-Cl(F +(B)) ⊆ F+(s(σ1, σ2)θ-Cl(B)). Theorem 8. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost contra-(τ1, τ2)-continuous; (2) τ1τ2-Cl(F −(σ1σ2-Int(K))) ⊆ F−(K) for every (σ1, σ2)s-closed set K of Y ; (3) τ1τ2-Cl(F −(σ1σ2-Int((σ1, σ2)-sCl(B)))) ⊆ F−((σ1, σ2)-sCl(B)) for every subset B of Y ; (4) F+((σ1, σ2)-sInt(B)) ⊆ τ1τ2-Int(F +(σ1σ2-Cl((σ1, σ2)-sInt(B)))) for every subset B of Y . Proof. (1) ⇒ (2): LetK be any (σ1, σ2)s-closed set of Y . Then, Y −K is (σ1, σ2)s-open in Y . By Theorem 1, F+(Y −K) ⊆ τ1τ2)-Int(F +(Y − σ1σ2-Int(K))). Thus, X − F−(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). (2) ⇒ (3): Let B be any subset of Y . Then, (σ1, σ2)-sCl(B)) is (σ1, σ2)s-closed in Y , by (2) we have τ1τ2-Cl(F −(σ1σ2-Int((σ1, σ2)-sCl(B)))) ⊆ F−((σ1, σ2)-sCl(B)). (3) ⇒ (4): Let B be any subset of Y . By (3), we have X − F+((σ1, σ2)-sInt(B)) = F−((σ1, σ2)-sCl(Y −B)) ⊇ τ1τ2-Cl(F −(σ1σ2-Int((σ1, σ2)-sCl(Y −B)))) = τ1τ2-Cl(F −(σ1σ2-Int(Y − (σ1, σ2)-sInt(B)))) = τ1τ2-Cl(F −(Y − σ1σ2-Cl((σ1, σ2)-sInt(B)))) = τ1τ2-Cl(X − F+(σ1σ2-Cl((σ1, σ2)-sInt(B)))) = X − τ1τ2-Int(F +(σ1σ2-Cl((σ1, σ2)-sInt(B)))) and hence F+((σ1, σ2)-sInt(B)) ⊆ τ1τ2-Int(F +(σ1σ2-Cl((σ1, σ2)-sInt(B)))). (4) ⇒ (1): Let V be any (σ1, σ2)s-open set of Y . Then, V = (σ1, σ2)-sInt(V ) and by (4), F+(V ) ⊆ τ1τ2-Int(F +(σ1σ2-Cl(V ))). By Theorem 5, F is upper almost contra- (τ1, τ2)-continuous. J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 11 of 19 Theorem 9. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost contra-(τ1, τ2)-continuous; (2) τ1τ2-Cl(F +(σ1σ2-Int(K))) ⊆ F+(K) for every (σ1, σ2)s-closed set K of Y ; (3) τ1τ2-Cl(F +(σ1σ2-Int((σ1, σ2)-sCl(B)))) ⊆ F+((σ1, σ2)-sCl(B)) for every subset B of Y ; (4) F−((σ1, σ2)-sInt(B)) ⊆ τ1τ2-Int(F −(σ1σ2-Cl((σ1, σ2)-sInt(B)))) for every subset B of Y . Proof. The proof is similar to that of Theorem 8. Theorem 10. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction. If τ1τ2-Cl(F −(B)) ⊆ F−((σ1, σ2)r-ker(B)) for every subset B of Y , then F is upper almost contra-(τ1, τ2)- continuous. Proof. Suppose that τ1τ2-Cl(F −(B)) ⊆ F−((σ1, σ2)r-ker(B)) for every subset B of Y . Let V be any (σ1, σ2)r-open set of Y . By Lemma 2, we have τ1τ2-Cl(F −(V )) ⊆ F−((σ1, σ2)r-ker(V )) = F−(V ) and hence F−(V ) is τ1τ2-closed in X. By Theorem 1, F is upper almost contra-(τ1, τ2)- continuous. Theorem 11. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction. If F (τ1τ2-Cl(A)) ⊆ (σ1, σ2)r-ker(F (A)) for every subset A of X, then F is lower almost contra-(τ1, τ2)- continuous. Proof. Let V be any (σ1, σ2)r-open set of Y . Then, we have F (τ1τ2-Cl(F +(V ))) ⊆ (σ1, σ2)r-ker(V ) and hence τ1τ2-Cl(F +(V )) ⊆ F+((σ1, σ2)r-ker(V )). Thus by Lemma 2, τ1τ2-Cl(F +(V )) ⊆ F+((σ1, σ2)r-ker(V )) = F+(V ) and so F+(V ) is τ1τ2-closed in X. By Theorem 2, F is lower contra-(τ1, τ2)-continuous. Theorem 12. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction. If τ1τ2-Cl(F +(B)) ⊆ F+((σ1, σ2)r-ker(B)) for every subset B of Y , then F is lower almost contra-(τ1, τ2)- continuous. J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 12 of 19 Proof. Let V be any (σ1, σ2)r-open set of Y . Then, τ1τ2-Cl(F +(V )) ⊆ F+((σ1, σ2)r-ker(V )) and by Lemma 2, τ1τ2-Cl(F +(V )) ⊆ F+((σ1, σ2)r-ker(V )) = F+(V ). This implies that F+(V ) is τ1τ2-closed in X. By Theorem 2, F is lower almost contra-(τ1, τ2)-continuous. Definition 3. [60] 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 13. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an upper almost contra-(τ1, τ2)-continuous multifunction, 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 (σ1, σ2)r-closed set Y containing F (x). Since F is upper almost contra-(τ1, τ2)- continuous, there exists a τ1τ2-open set U ofX containing x such that U ⊆ F+(σ1σ2-Cl(V )); hence F (U) ⊆ σ1σ2-Cl(V ). Thus, F is upper weakly (τ1, τ2)-continuous. Definition 4. [60] 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 14. If F : (X, τ1, τ2) → (Y, σ1, σ2) is a lower almost contra-(τ1, τ2)-continuous multifunction, then F is lower weakly (τ1, τ2)-continuous. Proof. It is similar to that of Theorem 13. Definition 5. [82] 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 with 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 15. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an upper contra-(τ1, τ2)-continuous mul- tifunction, then F is upper almost contra-(τ1, τ2)-continuous. Proof. Let x ∈ X and K be any (σ1, σ2)r-closed set of Y with x ∈ F+(K). Then, K is σ1σ2-closed in Y . 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 is upper almost contra-(τ1, τ2)- continuous. The converse of Theorem 15 is not true in general as shown in the following example. J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 13 of 19 Example 1. Let X = {a, b, c, d} with topologies τ1 = {∅, {a}, {c}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, c, d}, {a, b, d}, X} and τ2 = {∅, {a}, {c}, {a, b}, {a, c}, {a, b, c}, {a, c, d}, X}. Let Y = {1, 2, 3, 4} with topolo- gies σ1 = {∅, {1}, {3}, {1, 2}, {1, 3}, {1, 2, 3}, {1, 3, 4}, Y } and σ2 = {∅, {1}, {3}, {1, 2}, {1, 3}, {1, 4}, {1, 2, 3}, {1, 3, 4}, {1, 2, 4}, Y }. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is defined as follows: F (a) = {4}, F (b) = {3}, F (c) = {1} and F (d) = {2}. Then, F is upper almost (τ1, τ2)-continuous but F is not upper contra-(τ1, τ2)-continuous. Definition 6. [82] A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be 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 with U ⊆ F−(K). A multi- function 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 16. If F : (X, τ1, τ2) → (Y, σ1, σ2) is a lower contra-(τ1, τ2)-continuous multi- function, then F is lower almost contra-(τ1, τ2)-continuous. Proof. It is similar to that of Theorem 15. Definition 7. [58] 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. [58] 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 ; (5) F+(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F +(B)) for every subset B of Y . Lemma 6. [58] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 14 of 19 (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 17. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an upper (τ1, τ2)-continuous multifunction and G : (Y, σ1, σ2) → (Z, ρ1, ρ2) is an upper almost contra-(σ1, σ2)-continuous multifunc- tion, then G ◦ F : (X, τ1, τ2) → (Z, ρ1, ρ2) is upper almost contra-(τ1, τ2)-continuous. Proof. Let K be any (ρ1, ρ2)r-closed set of Z. Since G is upper almost 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 almost contra-(τ1, τ2)-continuous. Theorem 18. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an lower (τ1, τ2)-continuous multifunction and G : (Y, σ1, σ2) → (Z, ρ1, ρ2) is an lower almost contra-(σ1, σ2)-continuous multifunc- tion, then G ◦ F : (X, τ1, τ2) → (Z, ρ1, ρ2) is lower almost contra-(τ1, τ2)-continuous. Proof. It is similar to that of Theorem 17. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), a multifunction ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is defined in [78] as follows: ClF⊛(x) = σ1σ2-Cl(F (x)) for each x ∈ X. Definition 8. [78] 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 7. [78] 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. [82] 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 . J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 15 of 19 Lemma 9. [78] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), ClF − ⊛ (V ) = F−(V ) for each σ1σ2-open set V of Y . Lemma 10. [82] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), ClF + ⊛ (K) = F+(K) for each σ1σ2-closed set K of Y . Theorem 19. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper almost contra-(τ1, τ2)- continuous if and only if ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is upper almost contra-(τ1, τ2)- continuous. Proof. Suppose that F is upper almost contra-(τ1, τ2)-continuous. Let K be any (σ1, σ2)r-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 almost contra-(τ1, τ2)-continuous. Conversely, suppose that ClF⊛ is upper almost contra-(τ1, τ2)-continuous. Let K be any (σ1, σ2)r-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 almost contra-(τ1, τ2)-continuous. Theorem 20. 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 almost contra- (τ1, τ2)-continuous if and only if ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is lower almost contra- (τ1, τ2)-continuous. Proof. Suppose that F is lower almost contra-(τ1, τ2)-continuous. Let K be any (σ1, σ2)r-closed set of Y . By Lemma 7, Lemma 8 and Theorem 2, ClF− ⊛ (K) = F−(K) is τ1τ2-open in X. This shows that ClF⊛ is lower almost contra-(τ1, τ2)-continuous. Conversely, suppose that ClF⊛ is lower almost contra-(τ1, τ2)-continuous. Let K be any (σ1, σ2)r-closed set of Y . By Lemma 7, Lemma 8 and Theorem 2, F−(K) = ClF− ⊛ (K) is τ1τ2-open in X. This shows that F is lower almost contra-(τ1, τ2)-continuous. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] C. Boonpok and J. Khampakdee. (Λ, sp)-open sets in topological spaces. European Journal of Pure and Applied Mathematics, 15(2):572–588, 2022. [2] C. Viriyapong and C. Boonpok. (Λ, sp)-continuous functions. WSEAS Transactions on Mathematics, 21:380–385, 2022. [3] T. Dungthaisong, C. Boonpok, and C. Viriyapong. Generalized closed sets in bigeneralized topological spaces. International Journal of Mathematical Analysis, 5(24):1175–1184, 2011. [4] T. Duangphui, C. Boonpok, and C. Viriyapong. Continuous functions on bigeneral- ized topological spaces. International Journal of Mathematical Analysis, 5(24):1165– 1174, 2011. J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 16 of 19 [5] N. Srisarakham and C. Boonpok. Almost (Λ, p)-continuous functions. International Journal of Mathematics and Computer Science, 18(2):255–259, 2023. [6] M. Thongmoon and C. Boonpok. Strongly θ(Λ, p)-continuous functions. International Journal of Mathematics and Computer Science, 19(2):475–479, 2024. [7] C. Boonpok and J. Khampakdee. Almost strong θ(Λ, p)-continuity for functions. European Journal of Pure and Applied Mathematics, 17(1):300–309, 2024. [8] P. Pue-on and C. Boonpok. θ(Λ, p)-continuity for functions. International Journal of Mathematics and Computer Science, 19(2):491–495, 2024. [9] C. Boonpok and N. Srisarakham. Weak forms of (Λ, b)-open sets and weak (Λ, b)- continuity. European Journal of Pure and Applied Mathematics, 16(1):29–43, 2023. [10] C. Boonpok. θ(⋆)-precontinuity. Mathematica, 65(1):31–42, 2023. [11] C. Boonpok. On some closed sets and low separation axioms via topological ideals. European Journal of Pure and Applied Mathematics, 15(3):1023–1046, 2022. [12] C. Boonpok. On some spaces via topological ideals. Open Mathematics, 21:20230118, 2023. [13] C. Boonpok. On characterizations of ⋆-hyperconnected ideal topological spaces. Jour- nal of Mathematics, 2020:9387601, 2020. [14] C. Boonpok. Almost (g,m)-continuous functions. International Journal of Mathe- matical Analysis, 4(40):1957–1964, 2010. [15] C. Boonpok. M -continuous functions in biminimal structure spaces. Far East Journal of Mathematical Sciences, 43(1):41–58, 2010. [16] C. Boonpok and N. Srisarakham. (τ1, τ2)-continuity for functions. Asia Pacific Jour- nal of Mathematics, 11:21, 2024. [17] C. Boonpok and P. Pue-on. Characterizations of almost (τ1, τ2)-continuous functions. International Journal of Analysis and Applications, 22:33, 2024. [18] C. Boonpok and C. Klanarong. On weakly (τ1, τ2)-continuous functions. European Journal of Pure and Applied Mathematics, 17(1):416–425, 2024. [19] P. Pue-on, S. Sompong, and C. Boonpok. Slightly (τ1, τ2)s-continuous functions. International Journal of Mathematics and Computer Science, 20(1):217–221, 2025. [20] B. Kong-ied, S. Sompong, and C. Boonpok. Almost quasi (τ1, τ2)-continuous func- tions. Asia Pacific Journal of Mathematics, 11:64, 2024. [21] M. Chiangpradit, S. Sompong, and C. Boonpok. Weakly quasi (τ1, τ2)-continuous functions. International Journal of Analysis and Applications, 22:125, 2024. [22] M. Thongmoon, S. Sompong, and C. Boonpok. Rarely (τ1, τ2)-continuous functions. International Journal of Mathematics and Computer Science, 20(1):423–427, 2025. [23] N. Srisarakham, A. Sama-Ae, and C. Boonpok. Characterizations of faintly (τ1, τ2)-continuous functions. European Journal of Pure and Applied Mathematics, 17(4):2753–2762, 2024. [24] C. Prachanpol, C. Boonpok, and C. Viriyapong. δ(τ1, τ2)-continuous functions. Eu- ropean Journal of Pure and Applied Mathematics, 17(4):3730–3742, 2024. [25] N. Srisarakham, S. Sompong, and C. Boonpok. Quasi θ(τ1, τ2)-continuous functions. European Journal of Pure and Applied Mathematics, 18(1):5722, 2025. [26] J. Khampakdee, S. Sompong, and C. Boonpok. Almost weakly (τ1, τ2)-continuous J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 17 of 19 functions. European Journal of Pure and Applied Mathematics, 18(1):5721, 2025. [27] B. Kong-ied, A. Sama-Ae, and C. Boonpok. Almost nearly (τ1, τ2)-continuous func- tions. International Journal of Analysis and Applications, 23:14, 2025. [28] J. Dontchev. Contra-continuous functions and strongly S-closed spaces. International Journal of Mathematics and Mathematical Sciences, 19:303–310, 1966. [29] J. Dontchev and T. Noiri. Contra-semicontinuous functions. Mathematica Pannonica, 10:159–168, 1999. [30] S. Jafari and T. Noiri. On contra-precontinuous functions. Bulletin of the Malaysian Mathematical Sciences Society, 25:115–128, 2002. [31] E. Ekici. Almost contra-precontinuous functions. Bulletin of the Malaysian Mathe- matical Sciences Society, 27:53–65, 2004. [32] J. Dontchev, M. Ganster, and I. Reilly. More on almost s-continuity. Indian Journal of Mathematics, 41:139–146, 1999. [33] T. Noiri, B. Ahmad, and M. Khan. Almost s-continuous functions. Kyungpook Mathematical Journal, 35:311–322, 1995. [34] T. Noiri. Super-continuity and some strong forms of continuity. Indian Journal of Pure and Applied Mathematics, 15:241–250, 1984. [35] E. Ekici, S. Jafari, and T. Noiri. On upper and lower contra-continuous multifunc- tions. Analele Ştiinţifice ale Universităţii Al. I. Cuza din Iaşi Matematică, 54(1):75– 85, 2008. [36] T. Noiri and V. Popa. Almost weakly continuous multifunctions. Demonstratio Mathematica, 26:363–380, 1993. [37] E. Ekici, S. Jafari, and V. Popa. On contra-precontinuous and almost contra- precontinuous multifunctions. Journal of Advanced Research in Pure Mathematics, 2(1):11–25, 2010. [38] K. Laprom, C. Boonpok, and C. Viriyapong. β(τ1, τ2)-continuous multifunctions on bitopological spaces. Journal of Mathematics, 2020:4020971, 2020. [39] C. Boonpok. (τ1, τ2)δ-semicontinuous multifunctions. Heliyon, 6:e05367, 2020. [40] C. Boonpok and C. Viriyapong. Upper and lower almost weak (τ1, τ2)-continuity. European Journal of Pure and Applied Mathematics, 14(4):1212–1225, 2021. [41] C. Viriyapong and C. Boonpok. Weak quasi (Λ, sp)-continuity for multifunctions. International Journal of Mathematics and Computer Science, 17(3):1201–1209, 2022. [42] C. Boonpok. On continuous multifunctions in ideal topological spaces. Lobachevskii Journal of Mathematics, 40(1):24–35, 2019. [43] C. Boonpok. Upper and lower β(⋆)-continuity. Heliyon, 7:e05986, 2021. [44] C. Boonpok and J. Khampakdee. Upper and lower α-⋆-continuity. European Journal of Pure and Applied Mathematics, 17(1):201–211, 2024. [45] C. Boonpok and N. Srisarakham. Almost α-⋆-continuity for multifunctions. Interna- tional Journal of Analysis and Applications, 21:107, 2023. [46] C. Boonpok. Weak quasi continuity for multifunctions in ideal topological spaces. Advances in Mathematics: Scientific Journal, 9(1):339–355, 2020. [47] C. Boonpok and P. Pue-on. Upper and lower weakly α-⋆-continuous multifunctions. International Journal of Analysis and Applications, 21:90, 2023. J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 18 of 19 [48] C. Boonpok and P. Pue-on. Upper and lower sβ(⋆)-continuous multifunctions. Eu- ropean Journal of Pure and Applied Mathematics, 16(3):1634–1646, 2023. [49] C. Boonpok and J. Khampakdee. Upper and lower weak sβ(⋆)-continuity. European Journal of Pure and Applied Mathematics, 16(4):2544–2556, 2023. [50] C. Boonpok. θ(⋆)-quasi continuity for multifunctions. WSEAS Transactions on Math- ematics, 21:245–251, 2022. [51] C. Boonpok and P. Pue-on. Continuity for multifunctions in ideal topological spaces. WSEAS Transactions on Mathematics, 19:624–631, 2020. [52] C. Boonpok and P. Pue-on. Upper and lower weakly (Λ, sp)-continuous multifunc- tions. European Journal of Pure and Applied Mathematics, 16(2):1047–1058, 2023. [53] J. Khampakdee and C. Boonpok. Upper and lower α(Λ, sp)-continuous multifunc- tions. WSEAS Transactions on Mathematics, 21:684–690, 2022. [54] C. Boonpok and J. Khampakdee. On almost α(Λ, sp)-continuous multifunctions. European Journal of Pure and Applied Mathematics, 15(2):626–634, 2022. [55] C. Boonpok and M. Thongmoon. Weak α(Λ, sp)-continuity for multifunctions. Eu- ropean Journal of Pure and Applied Mathematics, 16(1):465–478, 2023. [56] M. Thongmoon and C. Boonpok. Upper and lower almost β(Λ, sp)-continuous mul- tifunctions. WSEAS Transactions on Mathematics, 21:844–853, 2022. [57] C. Boonpok and J. Khampakdee. Slight (Λ, sp)-continuity and Λsp-extremally dis- connectedness. European Journal of Pure and Applied Mathematics, 15(3):1180–1188, 2022. [58] P. Pue-on, S. Sompong, and C. Boonpok. Upper and lower (τ1, τ2)-continuous mul- functions. International Journal of Mathematics and Computer Science, 19(4):1305– 1310, 2024. [59] C. Klanarong, S. Sompong, and C. Boonpok. Upper and lower almost (τ1, τ2)- continuous multifunctions. European Journal of Pure and Applied Mathematics, 17(2):1244–1253, 2024. [60] M. Thongmoon, S. Sompong, and C. Boonpok. Upper and lower weak (τ1, τ2)- continuity. European Journal of Pure and Applied Mathematics, 17(3):1705–1716, 2024. [61] P. Pue-on, S. Sompong, and C. Boonpok. Weakly quasi (τ1, τ2)-continuous multifunc- tions. European Journal of Pure and Applied Mathematics, 17(3):1553–1564, 2024. [62] P. Pue-on, S. Sompong, and C. Boonpok. Almost quasi (τ1, τ2)-continuity for multi- functions. International Journal of Analysis and Applications, 22:97, 2024. [63] J. Khampakdee, S. Sompong, and C. Boonpok. c-(τ1, τ2)-continuity for multifunc- tions. European Journal of Pure and Applied Mathematics, 17(3):2289–2299, 2024. [64] P. Pue-on, A. Sama-Ae, and C. Boonpok. c-quasi (τ1, τ2)-continuous multifunctions. European Journal of Pure and Applied Mathematics, 17(4):3242–3253, 2024. [65] N. Viriyapong, S. Sompong, and C. Boonpok. Upper and lower s-(τ1, τ2)p-continuous multifunctions. European Journal of Pure and Applied Mathematics, 17(3):2210–2220, 2024. [66] C. Viriyapong, S. Sompong, and C. Boonpok. Upper and lower slight α(τ1, τ2)- continuity. European Journal of Pure and Applied Mathematics, 17(3):2142–2154, J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6010 19 of 19 2024. [67] N. Viriyapong, S. Sompong, and C. Boonpok. Slightly (τ1, τ2)p-continuous multifunc- tions. International Journal of Analysis and Applications, 22:152, 2024. [68] B. Kong-ied, S. Sompong, and C. Boonpok. Rarely s-(τ1, τ2)p-continuous multifunc- tions. European Journal of Pure and Applied Mathematics, 18(1):5649, 2025. [69] N. Chutiman, A. Sama-Ae, and C. Boonpok. Almost near (τ1, τ2)-continuity for multifunctions. European Journal of Pure and Applied Mathematics, 18(1):5650, 2025. [70] M. Chiangpradit, A. Sama-Ae, and C. Boonpok. s-(τ1, τ2)-continuity for multifunc- tions. European Journal of Pure and Applied Mathematics, 18(1):5634, 2025. [71] P. Pue-on, A. Sama-Ae, and C. Boonpok. Quasi θ(τ1, τ2)-continuity for multifunc- tions. European Journal of Pure and Applied Mathematics, 18(1):5717, 2025. [72] J. Khampakdee, A. Sama-Ae, and C. Boonpok. Almost nearly quasi (τ1, τ2)- continuous multifunctions. European Journal of Pure and Applied Mathematics, 18(1):5720, 2025. [73] P. Pue-on, A. Sama-Ae, and C. Boonpok. Upper and lower weakly s-(τ1, τ2)- continuous multifunctions. European Journal of Pure and Applied Mathematics, 18(1):5718, 2025. [74] M. Thongmoon, A. Sama-Ae, and C. Boonpok. Upper and lower near (τ1, τ2)- continuity. European Journal of Pure and Applied Mathematics, 18(1):5633, 2025. [75] M. Chiangpradit, S. Sompong, and C. Boonpok. Upper and lower almost quasi (τ1, τ2)-continuity. Asia Pacific Journal of Mathematics, 12:12, 2025. [76] E. Ekici, S. Jafari, and V. Popa. On almost contra-continuous multifunctions. Lobachevskii Journal of Mathematics, 30(2):124–131, 2009. [77] C. Boonpok and J. Khampakdee. Upper and lower almost contra-(Λ, sp)-continuity. European Journal of Pure and Applied Mathematics, 16(1):156–168, 2023. [78] C. Boonpok, C. Viriyapong, and M. Thongmoon. On upper and lower (τ1, τ2)- precontinuous multifunctions. Journal of Mathematics and Computer Science, 18:282–293, 2018. [79] C. Viriyapong and C. Boonpok. (τ1, τ2)α-continuity for multifunctions. Journal of Mathematics, 2020:6285763, 2020. [80] N. Viriyapong, S. Sompong, and C. Boonpok. (τ1, τ2)-extremal disconnectedness in bitopological spaces. International Journal of Mathematics and Computer Science, 19(3):855–860, 2024. [81] P. Pue-on, S. Sompong, and C. Boonpok. Almost contra-(τ1, τ2)p-continuity for func- tions. (accepted). [82] N. Viriyapong, A. Sama-Ae, and C. Boonpok. Upper and lower contra-(τ1, τ2)- continuity. (accepted).