EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6039 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Contra-(τ1, τ2)p-continuous Functions Monchaya Chiangpradit1, 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 contra-(τ1, τ2)p-continuous func- tions. Furthermore, several characterizations and some properties concerning contra-(τ1, τ2)p- continuous functions are considered. 2020 Mathematics Subject Classifications: 54C08, 54E55 Key Words and Phrases: τ1τ2-open set, contra-(τ1, τ2)p-continuous function 1. Introduction The notions of contra-continuity and strong S-closedness in topological spaces were introduced by Dontchev [1]. Furthermore, Dontchev [1] obtained very interesting and important results concerning contra-continuity, compactness, S-closedness and strong S- closedness. Dontchev and Noiri [2] introduced and studied the concept of RC-continuity between topological spaces which is weaker than contra-continuity. Jafari and Noiri [3] 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. In 2002, Jafari and Noiri [4] 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. Moreover, the present authors [4] defined contra-preclosed graphs and investigated relations between contra-precontinuity and contra-preclosed graphs. In 2004, Ekici [5] introduced and studied a new class of func- tions called almost contra-precontinuous functions which generalize classes of regular set- connected functions [6], contra-precontinuous functions [4], contra-continuous functions [1], almost s-continuous functions [7] and perfectly continuous functions [8]. In 2007, Al- Omari and Noorani [9] introduced the concept of almost contra ω-continuous functions via the notion of ω-open sets and investigated several characterizations of contra ω-continuous ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6039 Email addresses: monchaya.c@msu.ac.th (M. Chiangpradit), 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) M. Chiangpradit, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6039 2 of 11 functions and almost contra ω-continuous functions. Noiri and Popa [10] 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 rela- tionships 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 [11]. On the other hand, the present authors introduced and studied the notions of (τ1, τ2)-continuous functions [12], almost (τ1, τ2)-continuous functions [13], weakly (τ1, τ2)-continuous functions [14], quasi θ(τ1, τ2)-continuous func- tions [15], δ(τ1, τ2)-continuous functions [16], almost quasi (τ1, τ2)-continuous functions [17], weakly quasi (τ1, τ2)-continuous functions [18], faintly (τ1, τ2)-continuous functions [19] and almost nearly (τ1, τ2)-continuous functions [20]. In this paper, we introduce the concept of contra-(τ1, τ2)p-continuous functions. We also investigate some characteriza- tions of contra-(τ1, τ2)p-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 [21] 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 [21] 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 [21] of A and is denoted by τ1τ2-Int(A). Lemma 1. [21] Let A and B be subsets of a bitopological space (X, τ1, τ2). For the τ1τ2- closure, the following properties hold: (1) A ⊆ τ1τ2-Cl(A) and τ1τ2-Cl(τ1τ2-Cl(A)) = τ1τ2-Cl(A). (2) If A ⊆ B, then τ1τ2-Cl(A) ⊆ τ1τ2-Cl(B). (3) τ1τ2-Cl(A) is τ1τ2-closed. (4) A is τ1τ2-closed if and only if A = τ1τ2-Cl(A). (5) τ1τ2-Cl(X −A) = X − τ1τ2-Int(A). A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [22] (resp. (τ1, τ2)s-open [23], (τ1, τ2)p-open [23], (τ1, τ2)β-open [23]) 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 [24] if A ⊆ M. Chiangpradit, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6039 3 of 11 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 [21] of A and is denoted by τ1τ2-ker(A). Lemma 2. [21] 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 [25] 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 [25] 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 [25]; (2) (τ1, τ2)-pInt(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∩A [26]. 3. Contra-(τ1, τ2)p-continuous functions In this section, we introduce the concept of contra-(τ1, τ2)p-continuous functions. More- over, some characterizations of contra-(τ1, τ2)p-continuous functions are discussed. Definition 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be contra-(τ1, τ2)p- continuous if for each x ∈ X and for each σ1σ2-closed set F of Y containing f(x), there exists a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ F . Theorem 1. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is contra-(τ1, τ2)p-continuous; (2) f−1(F ) is (τ1, τ2)p-open in X for every σ1σ2-closed set F of Y ; (3) f−1(V ) is (τ1, τ2)p-closed in X for every σ1σ2-open set V of Y ; (4) f((τ1, τ2)-pCl(A)) ⊆ σ1σ2-ker(f(A)) for every subset A of X; M. Chiangpradit, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6039 4 of 11 (5) (τ1, τ2)-pCl(f −1(B)) ⊆ f−1(σ1σ2-ker(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let F be any σ1σ2-closed set of Y and x ∈ f−1(F ). Then, f(x) ∈ F . Since f is contra-(τ1, τ2)p-continuous, there exists a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ F . Thus, U ⊆ f−1(F ) and hence x ∈ U ⊆ f−1(F ). Therefore, x ∈ (τ1, τ2)-pInt(f −1(F )). This implies that f−1(F ) ⊆ (τ1, τ2)-pInt(f −1(F )). Thus, f−1(F ) is (τ1, τ2)p-open in X. (2) ⇔ (3): Let V be any σ1σ2-open set of Y . Then, Y − V is σ1σ2-closed in Y . By (2), we have f−1(Y − V ) = X − f−1(V ) is (τ1, τ2)p-open in X and hence f−1(V ) is (τ1, τ2)p-closed in X. The converse can be shown easily. (2) ⇒ (4): Let A be any subset of X. Suppose that y ̸∈ σ1σ2-ker(f(A)). Then by Lemma 2, there exists a σ1σ2-closed set K of Y containing y such that f(A) ∩ K = ∅. Thus, A ∩ f−1(K) = ∅ and hence (τ1, τ2)-pCl(A) ∩ f−1(K) = ∅. Therefore, f((τ1, τ2)-pCl(A)) ∩K = ∅ and y ̸∈ f((τ1, τ2)-pCl(A)). This shows that f((τ1, τ2)-pCl(A)) ⊆ σ1σ2-ker(f(A)). (4) ⇒ (5): Let B be any subset of Y . By (4) and Lemma 2, we have f((τ1, τ2)-pCl(f −1(B))) ⊆ σ1σ2-ker(f(f −1(B))) ⊆ σ1σ2-ker(B) and hence (τ1, τ2)-pCl(f −1(B)) ⊆ f−1(σ1σ2-ker(B)). (5) ⇒ (3): Let V be any σ1σ2-open set of Y . Then by (5) and Lemma 2, we have (τ1, τ2)-pCl(f −1(V )) ⊆ f−1(σ1σ2-ker(V )) = f−1(V ) and hence f−1(V ) is (τ1, τ2)p-closed in X. (2) ⇒ (1): Let F be any σ1σ2-closed set of Y containing f(x). By (2), f−1(F ) is (τ1, τ2)p-open in X. Then we have, x ∈ (τ1, τ2)-pInt(f −1(F )) and therefore there exists a (τ1, τ2)p-open set U of X such that x ∈ U ⊆ f−1(F ); hence f(U) ⊆ F . This shows that f is contra-(τ1, τ2)p-continuous. Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-regular [27] if for each τ1τ2-closed set F and each point x ∈ X − F , there exist disjoint τ1τ2-open sets U and V such that x ∈ U and F ⊆ V . Definition 2. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be (τ1, τ2)p-continuous if for each x ∈ X and for each σ1σ2-open set V of Y containing f(x), there exists a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ V . Theorem 2. If a function f : (X, τ1, τ2) → (Y, σ1, σ2) is contra-(τ1, τ2)p-continuous and (Y, σ1, σ2) is (σ1, σ2)-regular, then f is (τ1, τ2)p-continuous. Proof. Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Since (Y, σ1, σ2) is (σ1, σ2)-regular, there exists a σ1σ2-open set W of Y containing f(x) such that σ1σ2-Cl(W ) ⊆ V. M. Chiangpradit, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6039 5 of 11 Since f is contra-(τ1, τ2)p-continuous, there exists a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(W ). Thus, f(U) ⊆ σ1σ2-Cl(W ) ⊆ V and hence f is (τ1, τ2)p- continuous. Definition 3. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost (τ1, τ2)p- continuous if for each x ∈ X and for each σ1σ2-open set V of Y containing f(x), there exists a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). Definition 4. A functions f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be (τ1, τ2)p-open if f(U) is (σ1, σ2)p-open in Y for every (τ1, τ2)p-open set U of X. Theorem 3. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a (τ1, τ2)p-open contra-(τ1, τ2)p-continuous function, then f is almost (τ1, τ2)p-continuous. Proof. Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Since f is contra-(τ1, τ2)p-continuous, there exists a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). Since f is (τ1, τ2)p-open, f(U) is σ1σ2-open in Y . Therefore, f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(f(U))) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). This shows that f is almost (τ1, τ2)p-continuous. Definition 5. [26] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost weakly (τ1, τ2)-continuous if for each x ∈ X and each σ1σ2-open set V of Y containing f(x), x ∈ τ1τ2-Int(τ1τ2-Cl(f −1(σ1σ2-Cl(V )))). Lemma 4. [26] For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost weakly (τ1, τ2)-continuous; (2) f−1(V ) ⊆ τ1τ2-Int(τ1τ2-Cl(f −1(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ; (3) τ1τ2-Cl(τ1τ2-Int(f −1(V ))) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (4) (τ1, τ2)-pCl(f −1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (5) f−1(V ) ⊆ (τ1, τ2)-pInt(f −1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (6) for each x ∈ X and each σ1σ2-open set V of Y containing f(x), there exists a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). Theorem 4. If a function f : (X, τ1, τ2) → (Y, σ1, σ2) is contra-(τ1, τ2)p-continuous, then f is almost weakly (τ1, τ2)-continuous. Proof. Let V be any σ1σ2-open set of Y . Since f is contra-(τ1, τ2)p-continuous and σ1σ2-Cl(V ) is σ1σ2-closed in Y , by Theorem 1 we have f−1(σ1σ2-Cl(V )) is (τ1, τ2)p-open in X. Thus, f−1(V ) ⊆ f−1(σ1σ2-Cl(V )) ⊆ τ1τ2-Int(τ1τ2-Cl(f −1(σ1σ2-Cl(V ))). By Lemma 4(2), f is almost weakly (τ1, τ2)-continuous. M. Chiangpradit, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6039 6 of 11 The (τ1, τ2)p-frontier [25] of a subset A of a bitopological space (X, τ1, τ2), denoted by (τ1, τ2)-pfr(A), is defined by (τ1, τ2)-pfr(A) = (τ1, τ2)-pCl(A) ∩ (τ1, τ2)-pCl(X −A) = (τ1, τ2)-pCl(A)− (τ1, τ2)-pInt(A). Theorem 5. The set of all points x of X at which a function f : (X, τ1, τ2) → (Y, σ1, σ2) is not contra-(τ1, τ2)p-continuous is identical with the union of the (τ1, τ2)p-frontier of the inverse images of σ1σ2-closed sets of Y containing f(x). Proof. Suppose that f is not contra-(τ1, τ2)p-continuous at x ∈ X. Then, there exists a σ1σ2-closed set F of Y containing f(x) such that f(U) ∩ (Y − F ) ̸= ∅ for every (τ1, τ2)p-open set U of X containing x. This implies that U ∩ f−1(Y −F ) ̸= ∅. Therefore, x ∈ (τ1, τ2)-pCl(f −1(Y − F )) = (τ1, τ2)-pCl(X − f−1(F )). On the other hand, we have x ∈ f−1(F ) ⊆ (τ1, τ2)-pCl(f −1(F )) and hence x ∈ (τ1, τ2)-pfr(f −1(F )). Conversely, suppose that x ∈ (τ1, τ2)-pfr(f −1(F )) for some σ1σ2-closed set F of Y containing f(x). Now, we assume that f is contra-(τ1, τ2)p-continuous at x. Then, there exists a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ F . Thus, U ⊆ f−1(F ) and hence x ∈ (τ1, τ2)-pInt(f −1(F )) ⊆ X − (τ1, τ2)-pfr(f −1(F )). This is a contradiction. This means that f is not contra-(τ1, τ2)p-continuous. Definition 6. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is called contra-α(τ1, τ2)-continuous if f−1(V ) is α(τ1, τ2)-closed in X for each σ1σ2-open set V of Y . Definition 7. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is called contra-(τ1, τ2)s-continuous if f−1(V ) is (τ1, τ2)s-closed in X for each σ1σ2-open set V of Y . Lemma 5. For a subset A of a bitopological space (X, τ1, τ2), the following properties are equivalent: (1) A is α(τ1, τ2)-open; (2) A is (τ1, τ2)p-open and (τ1, τ2)s-open. Proof. (1) ⇒ (2): Let A be α(τ1, τ2)-open. Then, A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). Therefore, A ⊆ τ1τ2-Int(τ1τ2-Cl(A)) and A ⊆ τ1τ2-Cl(τ1τ2-Int(A)). This shows that A is (τ1, τ2)p-open and (τ1, τ2)s-open. (2) ⇒ (1): Let A be (τ1, τ2)p-open and (τ1, τ2)s-open. Then, A ⊆ τ1τ2-Int(τ1τ2-Cl(A)) and A ⊆ τ1τ2-Cl(τ1τ2-Int(A)). Thus, A ⊆ τ1τ2-Int(τ1τ2-Cl(A)) ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))) and hence A is α(τ1, τ2)-open. Theorem 6. For a f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is contra-α(τ1, τ2)-continuous; M. Chiangpradit, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6039 7 of 11 (2) f is contra-(τ1, τ2)p-continuous and contra-(τ1, τ2)s-continuous. Proof. This is an immediate consequence of Lemma 5. Definition 8. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be RC-(τ1, τ2)-continuous if f−1(V ) is (τ1, τ2)r-closed in X for each σ1σ2-open set V of Y . Definition 9. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be (τ1, τ2)s-continuous if for x ∈ X and each σ1σ2-open set V of Y containing f(x), there exists a (τ1, τ2)s-open set U of X containing x such that f(U) ⊆ V . Definition 10. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be (τ1, τ2)β-continuous if for x ∈ X and each σ1σ2-open set V of Y containing f(x), there exists a (τ1, τ2)β-open set U of X containing x such that f(U) ⊆ V . Lemma 6. For a subset A of a bitopological space (X, τ1, τ2), the following properties are equivalent: (1) A is (τ1, τ2)r-closed; (2) A is (τ1, τ2)p-closed and (τ1, τ2)s-open; (3) A is α(τ1, τ2)-closed and (τ1, τ2)β-open. Proof. (1) ⇒ (2): Let A be (τ1, τ2)r-closed. Then, we have A = τ1τ2-Cl(τ1τ2-Int(A)). Thus, A is (τ1, τ2)p-closed and (τ1, τ2)s-open. (2) ⇒ (3): Let A be (τ1, τ2)p-closed and (τ1, τ2)s-open. Then, τ1τ2-Cl(τ1τ2-Int(A)) ⊆ A and A ⊆ τ1τ2-Cl(τ1τ2-Int(A)). Thus, τ1τ2-Cl(τ1τ2-Int(A)) = τ1τ2-Cl(A) and hence τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A))) = τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A)))) = τ1τ2-Cl(τ1τ2-Int(A)) ⊆ A. This shows that A is α(τ1, τ2)-closed. It is obvious that A is (τ1, τ2)β-open. (3) ⇒ (1): Let A be α(τ1, τ2)-closed and (τ1, τ2)β-open. Then, we have τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A))) ⊆ A and A ⊆ τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A))). Thus, A = τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A))) and hence τ1τ2-Cl(τ1τ2-Int(A)) = τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A))))) = τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A))) = A. Therefore, A is (τ1, τ2)r-closed. As a consequence of Lemma 6, we have the following result: Theorem 7. For a f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: M. Chiangpradit, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6039 8 of 11 (1) f is RC-(τ1, τ2)-continuous; (2) f is contra-(τ1, τ2)p-continuous and (τ1, τ2)s-continuous; (3) f is contra-α(τ1, τ2)-continuous and (τ1, τ2)β-continuous. Definition 11. [28] 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 ) = ∅. Definition 12. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to have a contra-(τ1, τ2)p- closed graph if for each (x, y) ∈ (X × Y )−G(f), there exists a (τ1, τ2)p-open set U of X containing x and a σ1σ2-closed set K of Y containing y such that (U ×K) ∩G(f) = ∅. Lemma 7. A function f : (X, τ1, τ2) → (Y, σ1, σ2) has a contra-(τ1, τ2)p-closed graph if and only if for each (x, y) ∈ (X × Y ) − G(f), there exists a (τ1, τ2)p-open set U of X containing x and a σ1σ2-closed set K of Y containing y such that f(U) ∩K = ∅. Theorem 8. If a function f : (X, τ1, τ2) → (Y, σ1, σ2) is contra-(τ1, τ2)p-continuous and (Y, σ1, σ2) is σ1σ2-Urysohn, then G(f) is contra-(τ1, τ2)p-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)p-continuous, there exists a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(W ). Thus, f(U) ∩ σ1σ2-Cl(V ) = ∅ and hence by Lemma 7, G(f) is contra-(τ1, τ2)p-closed. Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-connected [21] if X cannot be written as the union of two nonempty disjoint τ1τ2-open sets. Definition 13. A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)p-connected if X cannot be written as the union of two nonempty disjoint (τ1, τ2)p-open sets. Theorem 9. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a contra-(τ1, τ2)p-continuous surjection and (X, τ1, τ2) is (τ1, τ2)p-connected, then (Y, σ1, σ2) is σ1σ2-connected. Proof. Suppose that (Y, σ1, σ2) is not σ1σ2-connected. Then, there exist nonempty σ1σ2-open sets V and W such that Y = V ∪ W . Therefore, V and W are σ1σ2-clopen in Y . Since f is contra-(τ1, τ2)p-continuous, f −1(V ) and f−1(W ) are (τ1, τ2)p-open in X. Moreover, f−1(V ) and f−1(W ) are nonempty disjoint and X = f−1(V ) ∪ f−1(W ). This shows that (X, τ1, τ2) is not (τ1, τ2)p-connected. This is a contradiction. This means that (Y, σ1, σ2) is σ1σ2-connected. Definition 14. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is called perfectly (τ1, τ2)-continuous if f−1(V ) is τ1τ2-clopen in X for each σ1σ2-open set V of Y . Definition 15. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is called α(τ1, τ2)-continuous if f−1(V ) is α(τ1, τ2)-open in X for each σ1σ2-open set V of Y . M. Chiangpradit, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6039 9 of 11 Theorem 10. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is perfectly (τ1, τ2)-continuous if and only if f is contra-(τ1, τ2)p-continuous and α(τ1, τ2)-continuous. Proof. This is obvious. Conversely, let V be any σ1σ2-open set of Y . Since f is contra-(τ1, τ2)p-continuous and α(τ1, τ2)-continuous, f −1(V ) is (τ1, τ2)p-closed and α(τ1, τ2)-open in X. Therefore, we have τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(f −1(V )))) ⊆ τ1τ2-Cl(τ1τ2-Int(f −1(V ))) ⊆ f−1(V ) ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(f −1(V )))) ⊆ τ1τ2-Cl(τ1τ2-Int(f −1(V ))). This implies that f−1(V ) is τ1τ2-clopen in X. Thus, f is perfectly (τ1, τ2)-continuous. Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-compact [21] if every cover of X by τ1τ2-open sets has a finite subcover. Definition 16. A bitopological space (X, τ1, τ2) is said to be mildly τ1τ2-compact if every τ1τ2-clopen cover of X has a finite subcover. Theorem 11. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a perfectly (τ1, τ2)-continuous surjection and (X, τ1, τ2) is mildly τ1τ2-compact, then (Y, σ1, σ2) is σ1σ2-compact. Proof. Let {Vγ | γ ∈ ∇} be any σ1σ2-open cover of Y . Since f is perfectly (τ1, τ2)- continuous, we have {f−1(Vγ) | γ ∈ ∇} is a τ1τ2-clopen cover of X. Since (X, τ1, τ2) is mildly τ1τ2-compact, there exists a finite subset ∇0 of ∇ such that X = ∪{f−1(Vγ) | γ ∈ ∇0}. Since f is surjective, Y = ∪{Vγ | γ ∈ ∇0} and (Y, σ1, σ2) is σ1σ2-compact. Definition 17. A bitopological space (X, τ1, τ2) is said to be: (1) (τ1, τ2)p-irreducible if every pair of nonempty (τ1, τ2)p-closed sets of X has a nonempty intersection; (2) τ1τ2-hyperconnected if τ1τ2-Cl(V ) = X for every nonempty τ1τ2-open set V of X. Theorem 12. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a contra-(τ1, τ2)p-continuous surjection and (X, τ1, τ2) is (τ1, τ2)p-irreducible, then (Y, σ1, σ2) is σ1σ2-hyperconnected. Proof. Suppose that (Y, σ1, σ2) is not σ1σ2-hyperconnected. 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