EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6037 ISSN 1307-5543 – ejpam.com Published by New York Business Global Characterizations of Contra-(τ1, τ2)-continuous Functions Nipaporn Chutiman1, 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 a new class of functions between bitopological spaces, namely contra-(τ1, τ2)-continuous functions. Furthermore, several characterizations and some properties concerning contra-(τ1, τ2)-continuous functions are investigated. 2020 Mathematics Subject Classifications: 54C08, 54E55 Key Words and Phrases: τ1τ2-open set, contra-(τ1, τ2)-continuous function 1. Introduction The field of the mathematical science which goes under the name of topology is con- cerned with all questions directly or indirectly related to 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. Moreover, several characterizations of almost (Λ, p)-continuous func- tions, strongly θ(Λ, p)-continuous functions, almost strongly θ(Λ, p)-continuous functions, θ(Λ, p)-continuous functions, weakly (Λ, b)-continuous functions, θ(⋆)-precontinuous func- tions, (Λ, p(⋆))-continuous functions, ⋆-continuous functions, θ-I -continuous functions, al- most (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], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] and [18], respectively. The notions of contra-continuity and strong S-closedness in topological spaces were introduced by Dontchev [19]. Dontchev [19] obtained very interesting and important results concerning contra-continuity, compactness, S-closedness and strong S- closedness. Dontchev and Noiri [20] introduced and studied the concept of RC-continuity ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6037 Email addresses: nipaporn.c@msu.ac.th (N. Chutiman), 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. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6037 2 of 12 between topological spaces which is weaker than contra-continuity. Jafari and Noiri [21] introduced and investigated a new class of functions called contra-super-continuous func- tions which lies between classes of RC-continuous functions and contra-continuous func- tions. Jafari and Noiri [22] 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. Furthermore, the present authors [22] de- fined contra-preclosed graphs and investigated relations between contra-precontinuity and contra-preclosed graphs. Ekici [23] introduced and studied a new class of functions called almost contra-precontinuous functions which generalize classes of regular set-connected functions [24], contra-precontinuous functions [22], contra-continuous functions [19], al- most s-continuous functions [25] and perfectly continuous functions [26]. Al-Omari and Noorani [27] introduced the concept of almost contra ω-continuous functions via the notion of ω-open sets and investigated several characterizations of contra ω-continuous functions and almost contra ω-continuous functions. Noiri and Popa [28] introduced the 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. It turns out that the contra m-continuity is a unified form of several modifications of weak contra-continuity due to Baker [29]. On the other hand, the present authors introduced and studied the no- tions of (τ1, τ2)-continuous functions [30], almost (τ1, τ2)-continuous functions [31], weakly (τ1, τ2)-continuous functions [32], quasi θ(τ1, τ2)-continuous functions [33], almost quasi (τ1, τ2)-continuous functions [34], weakly quasi (τ1, τ2)-continuous functions [35], almost weakly (τ1, τ2)-continuous functions [36] and almost contra-(Λ, sp)-continuous functions [37]. In this paper, we introduce the concept of contra-(τ1, τ2)-continuous functions. We also investigate some characterizations of 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 [38] 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 [38] 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 [38] of A and is denoted by τ1τ2-Int(A). Lemma 1. [38] 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). N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6037 3 of 12 (3) τ1τ2-Cl(A) is τ1τ2-closed. (4) A is τ1τ2-closed if and only if A = τ1τ2-Cl(A). (5) τ1τ2-Cl(X −A) = X − τ1τ2-Int(A). A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [39] (resp. (τ1, τ2)s-open [40], (τ1, τ2)p-open [40], (τ1, τ2)β-open [40]) 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 [41] 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 [38] of A and is denoted by τ1τ2-ker(A). Lemma 2. [38] 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 [42] 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 [42] 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 [42]; (2) (τ1, τ2)-pInt(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∩A [36]. 3. Characterizations of contra-(τ1, τ2)-continuous functions In this section, we introduce the concept of contra-(τ1, τ2)-continuous functions. Fur- thermore, some characterizations of contra-(τ1, τ2)-continuous functions are discussed. Definition 1. 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 . N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6037 4 of 12 Theorem 1. 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 . Proof. (1) ⇒ (2): The proof is obvious. (2) ⇒ (3): Let x ∈ X and K be any σ1σ2-closed set of Y containing f(x). By (2), f−1(K) is τ1τ2-open in X. Then, we have x ∈ τ1τ2-Int(f −1(K)) and therefore there exists a τ1τ2-open set U of X containing x such that U ⊆ f−1(K). Thus, f(U) ⊆ K. (3) ⇒ (4): Let A be any subset of X. Let x ∈ τ1τ2-Cl(A) and K be any σ1σ2-closed set of Y containing f(x). Then by (3), there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ K; hence U ⊆ f−1(K). Since x ∈ τ1τ2-Cl(A), U ∩ A ̸= ∅ and so ∅ ≠ f(U ∩ A) ⊆ f(U) ∩ f(A) ⊆ K ∩ f(A). By Lemma 2, we have f(x) ∈ σ1σ2-ker(f(A)) and hence f(τ1τ2-Cl(A)) ⊆ σ1σ2-ker(f(A)). (4) ⇒ (5): Let B be any subset of Y . By (4) and Lemma 2, we have f(τ1τ2-Cl(f −1(B))) ⊆ σ1σ2-ker(f(f −1(B))) ⊆ σ1σ2-ker(B) and hence τ1τ2-Cl(f −1(B)) ⊆ f−1(σ1σ2-ker(B)). (5) ⇒ (1): Let V be any σ1σ2-open set of Y . Then by (5) and Lemma 2, we have τ1τ2-Cl(f −1(V )) ⊆ f−1(σ1σ2-ker(V )) = f−1(V ) and so f−1(V ) is τ1τ2-closed in X. This shows that f is contra-(τ1, τ2)-continuous. Definition 2. [32] 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. Theorem 2. If f : (X, τ1, τ2) → (Y, σ1, σ2) is contra-(τ1, τ2)-continuous, then f is 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 of Y containing f(x). Since f is contra-(τ1, τ2)-continuous, by Theorem 1 there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). This shows that f is weakly (τ1, τ2)-continuous. N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6037 5 of 12 Definition 3. [31] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost (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-Int(σ1σ2-Cl(V )). A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost (τ1, τ2)-continuous if f has this property at each point of X. Definition 4. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is called almost (τ1, τ2)-open if f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(f(U))) for every τ1τ2-open set U of X. Theorem 3. If f : (X, τ1, τ2) → (Y, σ1, σ2) is contra-(τ1, τ2)-continuous and almost (τ1, τ2)-open, then f is almost (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 of Y containing f(x). Since f is contra-(τ1, τ2)-continuous, by Theorem 1 there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). Since f is almost (τ1, τ2)-open, we have f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(f(U))) ⊆ σ1σ2-Int(σ1σ2-Cl(V )) and hence f is almost (τ1, τ2)-continuous. Recall that a bitopological space (X, τ1, τ2) is said to be almost (τ1, τ2)-regular [43] if for each (τ1, τ2)r-closed set F and each x ̸∈ F , there exist disjoint τ1τ2-open sets U and V such that x ∈ U and F ⊆ V . Lemma 4. [43] A bitopological space (X, τ1, τ2) is almost (τ1, τ2)-regular if and only if for each x ∈ X and each (τ1, τ2)r-open set U with x ∈ U , there exists a τ1τ2-open set V such that x ∈ V ⊆ τ1τ2-Cl(V ) ⊆ U . Theorem 4. If f : (X, τ1, τ2) → (Y, σ1, σ2) is contra-(τ1, τ2)-continuous and (Y, σ1, σ2) is almost (σ1, σ2)-regular, then f is almost (τ1, τ2)-continuous. Proof. Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Then, we have σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r-open in Y . Since (Y, σ1, σ2) is almost (σ1, σ2)-regular, by Lemma 4 there exists a σ1σ2-open set W of Y such that f(x) ∈ W ⊆ σ1σ2-Cl(W ) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). Since f is contra-(τ1, τ2)-continuous and σ1σ2-Cl(W ) is σ1σ2-closed in Y , by Theorem 1 there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(W ) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). Thus, f is almost (τ1, τ2)-continuous. Lemma 5. [31] For a function (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equiv- alent: (1) f is almost (τ1, τ2)-continuous at x ∈ X; N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6037 6 of 12 (2) x ∈ τ1τ2-Int(f −1(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y containing f(x); (3) x ∈ τ1τ2-Int(f −1(V )) for every (σ1, σ2)r-open set V of Y containing f(x); (4) for each (σ1, σ2)r-open set V of Y containing f(x), there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ V . Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-extremally disconnected [41] if the τ1τ2-closure of every τ1τ2-open set U of X is τ1τ2-open. Lemma 6. [41] For a bitopological space (X, τ1, τ2), the following properties are equivalent: (1) (X, τ1, τ2) is (τ1, τ2)-extremally disconnected; (2) every (τ1, τ2)r-open set of X is τ1τ2-closed; (3) every (τ1, τ2)r-closed set of X is τ1τ2-open. Theorem 5. If f : (X, τ1, τ2) → (Y, σ1, σ2) is contra-(τ1, τ2)-continuous and (Y, σ1, σ2) is (σ1, σ2)-extremally disconnected, then f is almost (τ1, τ2)-continuous. Proof. Let x ∈ X and V be any (σ1, σ2)r-open set of Y containing f(x). Since (Y, σ1, σ2) is (σ1, σ2)-extremally disconnected, by Lemma 6 we have V is σ1σ2-clopen. Since f is contra-(τ1, τ2)-continuous, by Theorem 1 there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ V . Thus by Lemma 5, f is almost (τ1, τ2)-continuous. Definition 5. [30] 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 7. [30] 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 . Definition 6. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to satisfy the (τ1, τ2)- interiority condition if τ1τ2-Int(f −1(σ1σ2-Cl(V ))) ⊆ f−1(V ) for each σ1σ2-open set V of Y . N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6037 7 of 12 Theorem 6. If f : (X, τ1, τ2) → (Y, σ1, σ2) is contra-(τ1, τ2)-continuous and satisfy the (τ1, τ2)-interiority condition, then f is (τ1, τ2)-continuous. Proof. Let V be any σ1σ2-open set of Y . Since f is contra-(τ1, τ2)-continuous, by Theorem 1 we have f−1(V ) ⊆ f−1(σ1σ2-Cl(V )) = τ1τ2-Int(f −1(σ1σ2-Cl(V ))) = τ1τ2-Int(τ1τ2-Int(f −1(σ1σ2-Cl(V )))) ⊆ τ1τ2-Int(f −1(V )) ⊆ f−1(V ) and hence f−1(V ) is τ1τ2-open in X. By Lemma 7, f is (τ1, τ2)-continuous. Definition 7. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to have a contra (τ1, τ2)- closed graph if for each (x, y) ∈ (X × Y ) − G(f), there exist a τ1τ2-open set U of X containing x and a σ1σ2-closed set F of Y containing y such that (U × F ) ∩G(f) = ∅. Lemma 8. A function f : (X, τ1, τ2) → (Y, σ1, σ2) has a contra (τ1, τ2)-closed graph if and only if for each (x, y) ∈ (X × Y )−G(f), there exist a τ1τ2-open set U of X containing x and a σ1σ2-closed set F of Y containing y such that f(U) ∩ F = ∅. Definition 8. [44] 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 7. If f : (X, τ1, τ2) → (Y, σ1, σ2) is contra-(τ1, τ2)-continuous and (Y, σ1, σ2) is σ1σ2-Urysohn, then G(f) is contra (τ1, τ2)-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 ) = ∅. Since f is contra-(τ1, τ2)-continuous, by Theorem 1 there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(W ). This implies that f(U) ∩ σ1σ2-Cl(V ) = ∅ and by Lemma 8, G(f) is contra (τ1, τ2)-closed. Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-T1 [45] 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 . Theorem 8. If f : (X, τ1, τ2) → (Y, σ1, σ2) is contra-(τ1, τ2)-continuous and (Y, σ1, σ2) is (σ1, σ2)-T1, then G(f) is contra (τ1, τ2)-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 f(x) ∈ V and y ̸∈ V . Since f is (τ1, τ2)- continuous, there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ V . Thus, f(U) ∩ (Y − V ) = ∅ and Y − V is a σ1σ2-closed set of Y containing y. This shows that G(f) is contra (τ1, τ2)-closed. N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6037 8 of 12 Definition 9. [46] A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-T2 if for any pair of distinct points x, y in X, there exist disjoint τ1τ2-open sets U and V of X containing x and y, respectively. Theorem 9. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a contra-(τ1, τ2)-continuous injection with a contra (τ1, τ2)-closed graph, then (X, τ1, τ2) is (τ1, τ2)-T2. Proof. Let x and y be any distinct points of X. Since f is injective, f(x) ̸= f(y). Then, we have (x, f(y)) ∈ (X×Y )−G(f). Since G(f) is contra (τ1, τ2)-closed, by Lemma 8 there exist a τ1τ2-open set U of X containing x and a σ1σ2-closed set K of Y containing f(y) such that f(U) ∩K = ∅. Since f is contra-(τ1, τ2)-continuous, by Theorem 1 there exists a τ1τ2-open set U0 of X containing y such that f(U0) ⊆ K. Thus, f(U)∩ f(U0) = ∅ and hence U ∩ U0 = ∅. This shows that (X, τ1, τ2) is (τ1, τ2)-T2. Theorem 10. Let (X, τ1, τ2) be a bitopological space. If for each pair of distinct points x and x′ in X, there exists a function f of (X, τ1, τ2) into a σ1σ2-Urysohn space (Y, σ1, σ2) such that f(x) ̸= f(x′) and f is contra-(τ1, τ2)-continuous at x and x′, then (X, τ1, τ2) is (τ1, τ2)-T2. Proof. Let x and x′ be any distinct points of X. Then by the hypothesis, there exists a σ1σ2-Urysohn space (Y, σ1, σ2) and a function f : (X, τ1, τ2) → (Y, σ1, σ2) which satisfies the conditions of this theorem. Let y = f(x) and y′ = f(x′). Then, y ̸= y′. Since (Y, σ1, σ2) is σ1σ2-Urysohn, there exist σ1σ2-open sets V and W of Y containing y and y′, respectively, such that σ1σ2-Cl(V )∩σ1σ2-Cl(W ) = ∅. Since f is contra-(τ1, τ2)-continuous at x and x′, by Theorem 1 there exist τ1τ2-open sets U and U ′ of X containing x and x′, respectively, such that f(U) ⊆ σ1σ2-Cl(V ) and f(U ′) ⊆ σ1σ2-Cl(W ). This implies that U ∩ U ′ = ∅. Thus, (X, τ1, τ2) is (τ1, τ2)-T2. Corollary 1. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a contra-(τ1, τ2)-continuous injection and (Y, σ1, σ2) is σ1σ2-Urysohn, then (X, τ1, τ2) is (τ1, τ2)-T2. Proof. For each pair of distinct points x and x′ in X, f is a contra-(τ1, τ2)-continuous function of (X, τ1, τ2) into a σ1σ2-Urysohn space (Y, σ1, σ2) such that f(x) ̸= f(x′) because f is injective. Thus by Theorem 10, (X, τ1, τ2) is (τ1, τ2)-T2. Definition 10. A bitopological space (X, τ1, τ2) is said to be ultra-τ1τ2-Huasdorff if for each pair of distinct points x and y in X, there exist τ1τ2-clopen sets U and V of X containing x and y, respectively, such that U ∩ V = ∅. Theorem 11. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a contra-(τ1, τ2)-continuous injection and (Y, σ1, σ2) is ultra-σ1σ2-Hausdorff, then (X, τ1, τ2) is (τ1, τ2)-T2. Proof. Let x and y be any distinct points in X. Then, since f is injective, f(x) ̸= f(y). Moreover, since (Y, σ1, σ2) is ultra-σ1σ2-Hausdorff, there exist σ1σ2-clopen sets V and W of Y containing x and y, respectively, such that V ∩ W = ∅. Since f is contra-(τ1, τ2)- continuous, by Theorem 1 there exist τ1τ2-open sets U and G of X containing x and y, N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6037 9 of 12 respectively, such that f(U) ⊆ V and f(G) ⊆ W . Thus, U ∩G = ∅ and hence (X, τ1, τ2) is (τ1, τ2)-T2. Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-compact [38] if every cover of X by τ1τ2-open sets of X has a finite subcover. A bitopological space (X, τ1, τ2) is said to be quasi (τ1, τ2)-H -closed [47] if every τ1τ2-open cover {Uγ | γ ∈ ∇}, there exists a finite subset ∇0 of ∇ such that X = ∪{τ1τ2-Cl(Uγ) | γ ∈ ∇0}. Definition 11. 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 12. A bitopological space (X, τ1, τ2) is said to be S-τ1τ2-closed if every (τ1, τ2)s- open cover {Uγ | γ ∈ ∇}, there exists a finite subset ∇0 of ∇ such that X = ∪{τ1τ2-Cl(Uγ) | γ ∈ ∇0}. Theorem 12. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a contra-(τ1, τ2)-continuous surjection 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 . For each x ∈ X, there exists γ(x) ∈ ∇ such that f(x) ∈ Vγ(x). Since f is contra-(τ1, τ2)-continuous, by Theorem 1 there exists a τ1τ2-open set U(x) containing x such that f(U(x)) ⊆ Vγ(x). The family {U(x) | x ∈ X} is a cover of X by τ1τ2-open sets. 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. Corollary 2. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a contra-(τ1, τ2)-continuous surjection and (X, τ1, τ2) is τ1τ2-compact, then (Y, σ1, σ2) is S-σ1σ2-closed and hence quasi (σ1, σ2)-H - closed. Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-connected [38] if X cannot be written as the union of two nonempty disjoint τ1τ2-open sets. Theorem 13. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a contra-(τ1, τ2)-continuous surjection and (X, τ1, τ2) is τ1τ2-connected, then (Y, σ1, σ2) is σ1σ2-connected. Proof. Assume that (Y, σ1, σ2) is not σ1σ2-connected. Then, there exist σ1σ2-open sets V and W of Y such that V ∩W = ∅ and V ∪W = Y . Thus, we have f−1(V )∩f−1(W ) = ∅ and f−1(V )∪f−1(W ) = X. Since f is surjective, f−1(V ) ̸= ∅ and f−1(W ) ̸= ∅. Since f is contra-(τ1, τ2)-continuous and V,W are σ1σ2-clopen sets, by Theorem 1 we have f−1(V ) and f−1(W ) are τ1τ2-open in X. Therefore, (X, τ1, τ2) is not τ1τ2-connected. The τ1τ2-frontier [31] of a subset A of a bitopological space (X, τ1, τ2), denoted by τ1τ2-fr(A), is defined by τ1τ2-fr(A) = τ1τ2-Cl(A) ∩ τ1τ2-Cl(X −A) = τ1τ2-Cl(A)− τ1τ2-Int(A). N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6037 10 of 12 Theorem 14. The set of all points x ∈ X at which a function f : (X, τ1, τ2) → (Y, σ1, σ2) is not contra-(τ1, τ2)-continuous is identical with the union of the τ1τ2-frontier of the in- verse images of σ1σ2-closed sets of Y containing f(x). Proof. Suppose that f is not contra-(τ1, τ2)-continuous at x ∈ X. Then, there exists a σ1σ2-closed set K of Y containing f(x) such that f(U)∩ (Y −K) ̸= ∅ for every τ1τ2-open set U of X containing x. Thus, x ∈ τ1τ2-Cl(f −1(Y −K)) = τ1τ2-Cl(X − f−1(K)). On the other hand, we have x ∈ f−1(K) ⊆ τ1τ2-Cl(f −1(K)) and hence x ∈ τ1τ2-fr(f −1(K)). Conversely, suppose that f is contra-(τ1, τ2)-continuous at x ∈ X. Let K be any σ1σ2- closed set of Y containing f(x). By Theorem 1, x ∈ f−1(K) = τ1τ2-Int(f −1(K)). Thus, x ̸∈ τ1τ2-fr(f −1(K)) for every σ1σ2-closed set K of Y containing f(x). This completes the proof. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] C. Viriyapong and C. Boonpok. 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