EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6036 ISSN 1307-5543 – ejpam.com Published by New York Business Global Characterizations of Weakly Contra-(τ1, τ2)-continuous Functions Butsakorn Kong-ied1, Supunnee Sompong2, 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 Statistics, Faculty of Science and Technology, Sakon Nakhon Rajbhat University, Sakon Nakhon, 47000, Thailand Abstract. This paper presents a new class of functions called weakly contra-(τ1, τ2)-continuous functions. Moreover, several characterizations and some properties concerning weakly contra- (τ1, τ2)-continuous functions are established. 2020 Mathematics Subject Classifications: 54C08, 54E55 Key Words and Phrases: τ1τ2-open set, weakly contra-(τ1, τ2)-continuous function 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. In [1], the present authors investigated several 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 investigated the concept of g(m,n)-continuous functions. Duangphui et al. [4] intro- duced and studied the notion of (µ, µ′)(m,n)-continuous functions. Furthermore, several characterizations of almost (Λ, p)-continuous functions, strongly θ(Λ, p)-continuous func- tions, almost strongly θ(Λ, p)-continuous functions, θ(Λ, p)-continuous functions, weakly (Λ, b)-continuous functions, θ(⋆)-precontinuous functions, (Λ, p(⋆))-continuous functions, ⋆-continuous functions, θ-I -continuous functions, almost (g,m)-continuous functions, pairwise almost M -continuous functions, faintly (τ1, τ2)-continuous functions, δ(τ1, τ2)- continuous functions and almost nearly (τ1, τ2)-continuous functions were presented in [5], ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6036 Email addresses: butsakorn.k@msu.ac.th (B. Kong-ied), s−sompong@snru.ac.th (S. Sompong), chawalit.b@msu.ac.th (C. Boonpok) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6036 2 of 11 [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] and [18], respectively. In 1996, Dontchev [19] introduced the notion of contra-continuous functions. Jafari and Noiri [20] introduced and investigated the concept of contra-precontinuous functions. More- over, Jafari and Noiri [21] introduced and studied the notion of contra-α-continuous functions. In 1999, Dontchev and Noiri [22] introduced and investigated the concept of contra-semicontinuous functions. Caldas and Jafari [23] studied some properties of contra-β-continuous functions. In 2007, Baker [24] introduced and investigated the notion of weakly contra-continuous functions. Baker [25] introduced and studied the concept of weakly contra β-continuous functions. Noiri and Popa [26] introduced the notion of contra- m-continuous functions as functions from a set satisfying some minimal conditions into a topological space and investigated some characterizations and the relationships between contra-m-continuity and other related generalized forms of continuity. In 2011, Noiri and Popa [27] introduced a new class of functions called weakly contra-m-continuous functions as functions from a set satisfying some minimal conditions into a topological space and obtained some characterizations and several properties of such functions. It turns out that the weak contra-m-continuity is a unified form of several modifications of weak contra- continuity due to Baker [24]. On the other hand, the present authors introduced and stud- ied the concepts of (τ1, τ2)-continuous functions [28], almost (τ1, τ2)-continuous functions [29], weakly (τ1, τ2)-continuous functions [30], quasi θ(τ1, τ2)-continuous functions [31], al- most quasi (τ1, τ2)-continuous functions [32], weakly quasi (τ1, τ2)-continuous functions [33], almost weakly (τ1, τ2)-continuous functions [34] and almost contra-(Λ, sp)-continuous functions [35]. In this paper, we introduce the concept of weakly contra-(τ1, τ2)-continuous functions. We also investigate some characterizations of weakly contra-(τ1, τ2)-continuous functions. 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 [36] 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 [36] 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 [36] of A and is denoted by τ1τ2-Int(A). Lemma 1. [36] 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. B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6036 3 of 11 (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 [37] (resp. (τ1, τ2)s- open [38], (τ1, τ2)p-open [38], (τ1, τ2)β-open [38]) 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 said to be (τ1, τ2)r-closed (resp. (τ1, τ2)s-closed, (τ1, τ2)p-closed, (τ1, τ2)β-closed). A subset A of a bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-open [39] if A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)-open set is said to be α(τ1, τ2)-closed. Let A be a subset of a bitopological space (X, τ1, τ2). The set ∩{G | A ⊆ G and G is τ1τ2-open} is called the τ1τ2-kernel [36] of A and is denoted by τ1τ2-ker(A). Lemma 2. [36] 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. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection of all (τ1, τ2)p- closed sets of X containing A is called the (τ1, τ2)p-closure [40] of A and is denoted by (τ1, τ2)-pCl(A). The union of all (τ1, τ2)p-open sets of X contained in A is called the (τ1, τ2)p-interior [40] of A and is denoted by (τ1, τ2)-pInt(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 [40]; (2) (τ1, τ2)-pInt(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∩A [34]. 3. Characterizations of weakly contra-(τ1, τ2)-continuous functions In this section, we introduce the concept of weakly contra-(τ1, τ2)-continuous functions. Furthermore, several characterizations of weakly contra-(τ1, τ2)-continuous functions are discussed. Definition 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly contra-(τ1, τ2)- continuous if for each σ1σ2-open set V of Y and each σ1σ2-closed set K of Y such that K ⊆ V , τ1τ2-Cl(f −1(K)) ⊆ f−1(V ). B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6036 4 of 11 Definition 2. [41] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be contra-(τ1, τ2)- continuous if f−1(V ) is τ1τ2-closed in X for every σ1σ2-open set V of Y . Lemma 4. [41] For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is contra-(τ1, τ2)-continuous; (2) f−1(K) is τ1τ2-open in X for every σ1σ2-closed set K of Y ; (3) 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 f(U) ⊆ K; (4) f(τ1τ2-Cl(A)) ⊆ σ1σ2-ker(f(A)) for every subset A of X; (5) τ1τ2-Cl(f −1(B)) ⊆ f−1(σ1σ2-ker(B)) for every subset B of Y . Theorem 1. If a function f : (X, τ1, τ2) → (Y, σ1, σ2) is contra-(τ1, τ2)-continuous, then f is weakly contra-(τ1, τ2)-continuous. Proof. Let V be any σ1σ2-open set of Y and K be any σ1σ2-closed set of Y such that K ⊆ V . Since f is contra-(τ1, τ2)-continuous, by Lemma 4 we have f−1(V ) is τ1τ2-closed in X and hence τ1τ2-Cl(f −1(K)) ⊆ τ1τ2-Cl(f −1(V )) = f−1(V ). This shows that f is weakly contra-(τ1, τ2)-continuous. Definition 3. [28] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is called (τ1, τ2)-continuous at a point x ∈ X if for 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) ⊆ V . A function f : (X, τ1, τ2) → (Y, σ1, σ2) is called (τ1, τ2)-continuous if f has this property at each point of X. Lemma 5. [28] For a function (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equiv- alent: (1) f is (τ1, τ2)-continuous; (2) f−1(V ) is τ1τ2-open in X for every σ1σ2-open set V of Y ; (3) f(τ1τ2-Cl(A)) ⊆ σ1σ2-Cl(f(A)) for every subset A of X; (4) τ1τ2-Cl(f −1(B)) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ; (5) f−1(σ1σ2-Int(B)) ⊆ τ1τ2-Int(f −1(B)) for every subset B of Y ; (6) f−1(K) is τ1τ2-closed in X for every σ1σ2-closed set K of Y . Theorem 2. If a function f : (X, τ1, τ2) → (Y, σ1, σ2) is (τ1, τ2)-continuous, then f is weakly contra-(τ1, τ2)-continuous. B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6036 5 of 11 Proof. Let V be any σ1σ2-open set of Y and K be any σ1σ2-closed set of Y such that K ⊆ V . Since f is (τ1, τ2)-continuous, by Lemma 5 we have f−1(K) is τ1τ2-closed inX and so τ1τ2-Cl(f −1(K)) = f−1(K) ⊆ f−1(V ). Thus, f is weakly contra-(τ1, τ2)-continuous. Definition 4. [42] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be slightly (τ1, τ2)- continuous if for each x ∈ X and each σ1σ2-clopen set V of Y containing f(x), there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ V . Lemma 6. [42] For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is slightly (τ1, τ2)-continuous; (2) f−1(V ) is τ1τ2-open in X for each σ1σ2-clopen set V of Y ; (3) f−1(V ) is τ1τ2-closed in X for each σ1σ2-clopen set V of Y ; (4) f−1(V ) is τ1τ2-clopen in X for each σ1σ2-clopen set V of Y . Theorem 3. If a function f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly contra-(τ1, τ2)-continuous, then f is slightly (τ1, τ2)-continuous. Proof. Let V be any σ1σ2-clopen set of Y . If we put K = V , then by the weak contra- (τ1, τ2)-continuity we have τ1τ2-Cl(f −1(V )) ⊆ f−1(V ) and hence f−1(V ) is τ1τ2-closed in X. It follows from Lemma 6 that f is slightly (τ1, τ2)-continuous. Definition 5. [30] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly (τ1, τ2)- continuous at a point x ∈ X if for 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 ). A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly (τ1, τ2)-continuous if f has this property at each point of X. Lemma 7. [30] For a function (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equiv- alent: (1) f is weakly (τ1, τ2)-continuous; (2) f−1(V ) ⊆ τ1τ2-Int(f −1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (3) τ1τ2-Cl(f −1(σ1σ2-Int(K))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(f −1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ; (5) f−1(σ1σ2-Int(B)) ⊆ τ1τ2-Int(f −1(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ; (6) τ1τ2-Cl(f −1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y . Definition 6. [39] A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-extremally discon- nected if the τ1τ2-closure of every τ1τ2-open set U of X is τ1τ2-open. B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6036 6 of 11 Theorem 4. If f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly contra-(τ1, τ2)-continuous and (Y, σ1, σ2) is (σ1, σ2)-extremally disconnected, then f is weakly (τ1, τ2)-continuous. Proof. Let V be any σ1σ2-open set of Y . Since (Y, σ1, σ2) is (σ1, σ2)-extremally dis- connected, σ1σ2-Cl(V ) is σ1σ2-open. Since f weakly contra-(τ1, τ2)-continuous, τ1τ2-Cl(f −1(V )) ⊆ τ1τ2-Cl(f −1(σ1σ2-Cl(V ))) ⊆ f−1(σ1σ2-Cl(V )). Thus, τ1τ2-Cl(f −1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y . It follows from Lemma 7 that f is weakly (τ1, τ2)-continuous. Recall that a subset A of a bitopological space (X, τ1, τ2) is said to be generalized (τ1, τ2)-closed (briefly, g-(τ1, τ2)-closed) [43] if τ1τ2-Cl(A) ⊆ U whenever A ⊆ U and U is τ1τ2-open. Definition 7. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be g-(τ1, τ2)-continuous if f−1(K) is g-(τ1, τ2)-closed in X for every σ1σ2-closed set K of Y . Definition 8. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be approximately (τ1, τ2)- continuous if τ1τ2-Cl(K) ⊆ f−1(V ) whenever V is σ1σ2-open in Y and K is g-(τ1, τ2)- closed in X such that K ⊆ f−1(V ). Theorem 5. If f : (X, τ1, τ2) → (Y, σ1, σ2) is g-(τ1, τ2)-continuous and approximately (τ1, τ2)-continuous, then f is weakly contra-(τ1, τ2)-continuous. Proof. Let V be any σ1σ2-open set of Y and K be any σ1σ2-closed set of Y such that K ⊆ V . Since f is g-(τ1, τ2)-continuous, f−1(K) is g-(τ1, τ2)-closed in X. Since f−1(K) ⊆ f−1(V ) and f is approximately (τ1, τ2)-continuous, τ1τ2-Cl(K) ⊆ f−1(V ). This shows that f is weakly contra-(τ1, τ2)-continuous. Theorem 6. If f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly contra-(τ1, τ2)-continuous and f(K) is σ1σ2-closed in Y for every g-(τ1, τ2)-closed set K of X, then f is approximately (τ1, τ2)- continuous. Proof. Let V be any σ1σ2-open set of Y and K be any g-(τ1, τ2)-closed set of X such that K ⊆ f−1(V ). Then, f(K) is σ1σ2-closed and f(K) ⊆ V . Since f is weakly contra- (τ1, τ2)-continuous, τ1τ2-Cl(f −1(f(K))) ⊆ f−1(V ) and hence τ1τ2-Cl(K) ⊆ f−1(V ). This shows that f is approximately (τ1, τ2)-continuous. Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-compact [36] if every cover of X by τ1τ2-open sets of X has a finite subcover. Definition 9. [41] 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. Definition 10. A bitopological space (X, τ1, τ2) is called a C -(τ1, τ2)-space if for every τ1τ2-open set U of X and each x ∈ U , there exists a τ1τ2-closed set F of X such that x ∈ F ⊆ U . B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6036 7 of 11 Theorem 7. Let f : (X, τ1, τ2) → (Y, σ1, σ2) be a weakly contra-(τ1, τ2)-continuous func- tion and (Y, σ1, σ2) be a C -(σ1, σ2)-space. If (X, τ1, τ2) is strongly S-τ1τ2-closed, then f(X) is σ1σ2-compact. Proof. Suppose that (X, τ1, τ2) is strongly S-τ1τ2-closed. Let {Vγ ∈ ∇} be any cover of f(X) by σ1σ2-open sets of Y . For each x ∈ X, there exists γ(x) ∈ ∇ such that f(x) ∈ Vγ(x). Since (Y, σ1, σ2) be a C -(σ1, σ2)-space, there exists a σ1σ2-closed set Fγ(x) of Y such that f(x) ∈ Fγ(x) ⊆ Vγ(x). Since f is weakly contra-(τ1, τ2)-continuous, τ1τ2-Cl(f −1(Fγ(x))) ⊆ f−1(Vγ(x)). The family {τ1τ2-Cl(f−1(Fγ(x))) | x ∈ X} is a τ1τ2- closed cover of X. Since (X, τ1, τ2) is strongly S-τ1τ2-closed, there exists a finite number of pints, say, x1, x2, x3, ..., xn in X such that X = ∪{τ1τ2-Cl(f−1(Fγ(xk))) | xk ∈ X; 1 ≤ k ≤ n}. Thus, f(X) = ∪{f(τ1τ2-Cl(f−1(Fγ(xk)))) | xk ∈ X; 1 ≤ k ≤ n} ⊆ ∪{Vγ(xk) | xk ∈ X; 1 ≤ k ≤ n}. This shows that f(X) is σ1σ2-compact. Definition 11. [44] A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-T1 if for any pair of distinct points x, y in X, there exist τ1τ2-open sets U and V such that x ∈ U , y ̸∈ U and y ∈ V , x ̸∈ V . Lemma 8. [44] For a bitopological space (X, τ1, τ2), the following properties are equivalent: (1) (X, τ1, τ2) is (τ1, τ2)-T1; (2) for each x ∈ X, the singleton {x} is τ1τ2-closed in X; (3) for each x ∈ X, the singleton {x} is a Λ(τ1,τ2)-set. Theorem 8. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a weakly contra-(τ1, τ2)-continuous injection and (Y, σ1, σ2) is (σ1, σ2)-T1, (X, τ1, τ2) is (τ1, τ2)-T1. Proof. Let x and x′ be any distinct points of X. Since f is injective, f(x) ̸= f(x′). Moreover, since (Y, σ1, σ2) is (σ1, σ2)-T1, there exists a σ1σ2-open set V of Y such that f(x) ∈ V and f(x′) ̸∈ V . By Lemma 8, {f(x)} is σ1σ2-closed in Y . Since f is weakly contra-(τ1, τ2)-continuous, τ1τ2-Cl(f −1({f(x)})) ⊆ f−1(V ). Since x′ ̸∈ f−1(V ), we have x′ ̸∈ τ1τ2-Cl(f −1({f(x)})). Then by Lemma 1, τ1τ2-Cl(f −1({f(x)})) is τ1τ2-closed and hence X − τ1τ2-Cl(f −1({f(x)})) is a τ1τ2-open set of X containing x′ but not x. This shows that (X, τ1, τ2) is (τ1, τ2)-T1. Definition 12. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to have a contra-C -closed graph if for each (x, y) ∈ (X × Y )−G(f), there exist a τ1τ2-closed set F of X containing x and a σ1σ2-closed set F ′ of Y containing y such that (F × F ′) ∩G(f) = ∅. B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6036 8 of 11 Theorem 9. If f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly contra-(τ1, τ2)-continuous and (Y, σ1, σ2) is (σ1, σ2)-T1, then G(f) is contra-C -closed. Proof. Let (x, y) ∈ (X × Y ) − G(f). Then, y ̸= f(x). Since (Y, σ1, σ2) is (σ1, σ2)- T1, there exists a σ1σ2-open set V of Y such that y ̸∈ V and f(x) ̸∈ V . By Lemma 8, we have {f(x)} is σ1σ2-closed in Y . Since f is weakly contra-(τ1, τ2)-continuous, τ1τ2-Cl(f −1({f(x)})) ⊆ f−1(V ) and hence (x, y) ∈ τ1τ2-Cl(f −1({f(x)}))× (Y − V ) ⊆ (X × Y )−G(f). This shows that G(f) is contra-C -closed. Definition 13. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to have a contra-CR-closed graph if for each (x, y) ∈ (X × Y )−G(f), there exist a τ1τ2-closed set F of X containing x and a (σ1, σ2)r-closed set F ′ of Y containing y such that (F × F ′) ∩G(f) = ∅. Definition 14. [45] A bitopological space (X, τ1, τ2) is said to be τ1τ2-Urysohn if for each pair of distinct points x and y in X, there exist τ1τ2-open sets U and V such that x ∈ U , y ∈ V and τ1τ2-Cl(U) ∩ τ1τ2-Cl(V ) = ∅. Theorem 10. If f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly contra-(τ1, τ2)-continuous and (Y, σ1, σ2) is σ1σ2-Urysohn, then G(f) is contra-CR-closed. Proof. Let (x, y) ∈ (X × Y ) − G(f). Then, y ̸= f(x). Since (Y, σ1, σ2) is σ1σ2- Urysohn, there exist σ1σ2-open sets V and W of Y containing y and f(x), respectively, such that σ1σ2-Cl(V ) ∩ σ1σ2-Cl(W ) = ∅; hence σ1σ2-Cl(V ) ⊆ Y − σ1σ2-Cl(W ). Since f is weakly contra-(τ1, τ2)-continuous, τ1τ2-Cl(f −1(σ1σ2-Cl(V ))) ⊆ f−1(Y − σ1σ2-Cl(W )). Thus, (x, y) ∈ τ1τ2-Cl(f −1(σ1σ2-Cl(V )))×σ1σ2-Cl(W ) ⊆ (X×Y )−G(f) and hence G(f) is contra-CR-closed. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] C. Viriyapong and C. Boonpok. (Λ, sp)-continuous functions. WSEAS Transactions on Mathematics, 21:380–385, 2022. [2] C. Boonpok and J. Khampakdee. (Λ, sp)-open sets in topological spaces. European Journal of Pure and Applied Mathematics, 15(2):572–588, 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. B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. 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