EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6014 ISSN 1307-5543 – ejpam.com Published by New York Business Global θ(τ1, τ2)-continuity for Functions Montri Thongmoon1, 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 introduces a new class of functions between bitopological spaces, namely θ(τ1, τ2)-continuous functions. Moreover, several characterizations and some properties concerning θ(τ1, τ2)-continuous functions are investigated. 2020 Mathematics Subject Classifications: 54C08; 54E55 Key Words and Phrases: (τ1, τ2)θ-open set, θ(τ1, τ2)-continuous function 1. Introduction Stronger and weaker forms of open sets in topological spaces such as semi-open sets, preopen sets, α-open sets, β-open sets, δ-open sets and θ-open sets play an important role in the research of generalizations of continuity. By using these sets many authors introduced and investigated various types of continuity. The notions of (Λ, sp)-open sets, s(Λ, sp)-open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets and β(Λ, sp)-open sets were stud- ied in [1]. Viriyapong and Boonpok [2] investigated several characterizations of (Λ, sp)- continuous functions by utilizing the notions of (Λ, sp)-open sets and (Λ, sp)-closed sets. Dungthaisong et al. [3] introduced and studied the concept of g(m,n)-continuous func- tions. Duangphui et al. [4] introduced and investigated the notion of almost (µ, µ′)(m,n)- continuous functions. Moreover, some characterizations of almost (Λ, p)-continuous func- tions, almost strongly 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 and pairwise weaklyM -continuous functions were presented in [5], [6], [7], [8], [9], [10], [11], [12] and [13], respectively. The concept of θ-continuous functions was introduced by Fomin [14]. Noiri [15] studied some properties of θ-continuous functions. Furthermore, the present author [16] investigated ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6014 Email addresses: montri.t@msu.ac.th (M. Thongmoon), 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. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6014 2 of 13 several characterizations of θ-continuous functions. Arya and Bhamini [17] introduced the notion of θ-semi-continuous functions. Jafari and Noiri [18] investigated several character- izations of θ-semi-continuous functions. Noiri [19] introduced and investigated the concept of θ-precontinuous functions. Baker [20] introduced and studied the notion of weakly θ- precontinuous functions. Noiri and Popa [21] introduced the concept of θ-M -continuous functions as functions from a set satisfying some minimal conditions into a set satisfying some minimal conditions and investigated some characterizations and several properties of θ-M -continuous functions. In particular, Noiri and Popa [21] defined and studied the notion of strongly θ-M -closed graphs. Noiri and Popa [22] introduced the concept of θ- m-continuous functions as functions from a set satisfying some minimal conditions into a topological space and obtained several characterizations of such functions. Long and Her- rington [23] investigated some characterizations of strongly θ-continuous functions. Jafari and Noiri [24] introduced and studied the notion of strongly θ-semi-continuous functions. Noiri [25] introduced and investigated the concept of strongly θ-precontinuous functions. Pue-on and Boonpok [26] introduced and studied the concept of θ(Λ, p)-continuous func- tions. Quite recently, Thongmoon and Boonpok [27] introduced and investigated the notion of strongly θ(Λ, p)-continuous functions. On the other hand, the present authors introduced and studied the concepts of (τ1, τ2)-continuous functions [28], almost (τ1, τ2)- continuous functions [29], weakly (τ1, τ2)-continuous functions [30] and quasi θ(τ1, τ2)- continuous functions [31]. In this paper, we introduce the concept of θ(τ1, τ2)-continuous functions. We also investigate several characterizations of θ(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 [32] 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 [32] 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 [32] of A and is denoted by τ1τ2-Int(A). Lemma 1. [32] 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). M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6014 3 of 13 (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 [33] (resp. (τ1, τ2)s-open [34], (τ1, τ2)p-open [34], (τ1, τ2)β-open [34]) 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). A subset A of a bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-open [35] 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). A point x ∈ X is called a (τ1, τ2)θ-cluster point [33] of A if τ1τ2-Cl(U) ∩ A ̸= ∅ for every τ1τ2-open set U containing x. The set of all (τ1, τ2)θ-cluster points of A is called the (τ1, τ2)θ-closure [33] of A and is denoted by (τ1, τ2)θ-Cl(A). A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)θ-closed [33] if A = (τ1, τ2)θ-Cl(A). The complement of a (τ1, τ2)θ-closed set is said to be (τ1, τ2)θ-open. The union of all (τ1, τ2)θ-open sets contained in A is called the (τ1, τ2)θ-interior [33] of A and is denoted by (τ1, τ2)θ-Int(A). Lemma 2. [33] For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: (1) If A is τ2τ2-open in X, then τ1τ2-Cl(A) = (τ1, τ2)θ-Cl(A). (2) (τ1, τ2)θ-Cl(A) is τ1τ2-closed in X. 3. θ(τ1, τ2)-continuous functions In this section, we introduce the concept of θ(τ1, τ2)-continuous functions. Further- more, several characterizations of θ(τ1, τ2)-continuous functions are discussed. Definition 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be θ(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(τ1τ2-Cl(U)) ⊆ σ1σ2-Cl(V ). A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be θ(τ1, τ2)-continuous if f is θ(τ1, τ2)-continuous at each point x of X. Theorem 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is θ(τ1, τ2)-continuous at x ∈ X if and only if for each σ1σ2-open set V of Y containing f(x), x ∈ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))). Proof. Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Since f is θ(τ1, τ2)-continuous at x, there exists a τ1τ2-open set U of X containing x such that f(τ1τ2-Cl(U)) ⊆ σ1σ2-Cl(V ). Then, we have x ∈ U ⊆ τ1τ2-Cl(U) ⊆ f−1(σ1σ2-Cl(V )) and hence x ∈ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))). Conversely, let V be any σ1σ2-open set of Y containing f(x). Then, by the hypothesis we have x ∈ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))). There exists a τ1τ2-open set U of X such that x ∈ U ⊆ τ1τ2-Cl(U) ⊆ f−1(σ1σ2-Cl(V )); hence f(τ1τ2-Cl(U)) ⊆ σ1σ2-Cl(V ). This shows that f is θ(τ1, τ2)-continuous at x. M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6014 4 of 13 Theorem 2. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is θ(τ1, τ2)-continuous if and only if f−1(V ) ⊆ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y . Proof. Let V be any σ1σ2-open set of Y and x ∈ f−1(V ). Then, f(x) ∈ V . Since f is θ(τ1, τ2)-continuous at x, by Theorem 1 we have x ∈ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))) and hence f−1(V ) ⊆ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))). Conversely, let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Then, we have x ∈ f−1(V ) ⊆ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))) and hence x ∈ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))). By Theorem 1, f is θ(τ1, τ2)-continuous. Theorem 3. For a function (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equiva- lent: (1) f is θ(τ1, τ2)-continuous; (2) (τ1, τ2)θ-Cl(f −1(B)) ⊆ f−1(σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) (τ1, τ2)θ-Cl(f −1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (4) f−1(V ) ⊆ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (5) f((τ1, τ2)θ-Cl(A)) ⊆ (σ1, σ2)θ-Cl(f(A)) for every subset A of X; (6) (τ1, τ2)θ-Cl(f −1(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (7) (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; (8) (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(K))) ⊆ f−1(K) for every (σ1, σ2)r-closed set K of Y ; (9) (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(K))) ⊆ f−1(K) for every σ1σ2-closed set K of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . Suppose that x ̸∈ f−1(σ1, σ2)θ-Cl(B)). Then, we have x ∈ f−1(Y − (σ1, σ2)θ-Cl(B)) = f−1((σ1, σ2)θ-Int(Y − B)). Therefore, f(x) ∈ (σ1, σ2)θ-Int(Y −B). There exists a σ1σ2-open set V of Y such that f(x) ∈ V ⊆ σ1σ2-Cl(V ) ⊆ Y −B. Since f is θ(τ1, τ2)-continuous, there exists a τ1τ2-open set U of X containing x such that f(τ1τ2-Cl(U)) ⊆ σ1σ2-Cl(V ); hence τ1τ2-Cl(U) ⊆ f−1(σ1σ2-Cl(V )) ⊆ f−1(Y −B) = X − f−1(B). Thus, τ1τ2-Cl(U) ∩ f−1(B) = ∅ and so x ̸∈ (τ1, τ2)θ-Cl(f −1(B)). M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6014 5 of 13 (2) ⇒ (3): This is obvious since σ1σ2-Cl(V ) = (σ1, σ2)θ-Cl(V ) for every σ1σ2-open set V of Y . (3) ⇒ (4): Let V be any σ1σ2-open set of Y . Thus by (3), we have X − (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))) = (τ1, τ2)θ-Cl(X − f−1(σ1σ2-Cl(V ))) = (τ1, τ2)θ-Cl(f −1(Y − σ1σ2-Cl(V ))) ⊆ f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V ))) ⊆ f−1(σ1σ2-Cl(Y − V )) = f−1(Y − V ) = X − f−1(V ) and hence f−1(V ) ⊆ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))). (4) ⇒ (1): It follows from Theorem 2. (2) ⇒ (5): Let A be any subset of X. By (2), we have (τ1, τ2)θ-Cl(A) ⊆ (τ1, τ2)θ-Cl(f −1(f(A))) ⊆ f−1((σ1, σ2)θ-Cl(f(A))). Thus, f((τ1, τ2)θ-Cl(A)) ⊆ (σ1, σ2)θ-Cl(f(A)). (5) ⇒ (2): Let B be any subset of Y . Then by (5), we have f((τ1, τ2)θ-Cl(f −1(B))) ⊆ (σ1, σ2)θ-Cl(f(f −1(B))) ⊆ (σ1, σ2)θ-Cl(B) and hence (τ1, τ2)θ-Cl(f −1(B)) ⊆ f−1((σ1, σ2)θ-Cl(B)). (3) ⇒ (6): Let B be any subset of Y . Since (σ1, σ2)θ-Cl(B) is σ1σ2-closed in Y , by (3) we have (τ1, τ2)θ-Cl(f −1(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ f−1(σ1σ2-Cl(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)). (6) ⇒ (7): This is obvious since σ1σ2-Cl(V ) = (σ1, σ2)θ-Cl(V ) for every σ1σ2-open set V of Y . (7) ⇒ (8): Let K be any (σ1, σ2)r-closed set of Y . Thus by (7), we have (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(K))) = (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))))) ⊆ f−1(σ1σ2-Cl(σ1σ2-Int(K))) = f−1(K). (8) ⇒ (9): Let K be any σ1σ2-closed set of Y . Since σ1σ2-Cl(σ1σ2-Int(K)) is (σ1, σ2)r- closed in Y , by (8) we have (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(K))) = (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))))) ⊆ f−1(K). M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6014 6 of 13 (9) ⇒ (4): Let V be any σ1σ2-open set of Y . Then, Y − V is σ1σ2-closed in Y and by (9), (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(Y − V ))) ⊆ f−1(Y − V ) = X − f−1(V ). Moreover, we have (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(Y − V ))) = (τ1, τ2)θ-Cl(f −1(Y − σ1σ2-Cl(V ))) = (τ1, τ2)θ-Cl(X − f−1(σ1σ2-Cl(V ))) = X − (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))). Thus, f−1(V ) ⊆ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))). Theorem 4. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is θ(τ1, τ2)-continuous if and only if (τ1, τ2)θ-Cl(f −1(V )) ⊆ f−1((σ1, σ2)θ-Cl(V )) for every σ1σ2-open set V of Y . Proof. This is an immediate consequence of Theorem 3. Theorem 5. For a function (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equiva- lent: (1) f is θ(τ1, τ2)-continuous; (2) (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y . Proof. (1) ⇒ (2): Let V be any (σ1, σ2)β-open set of Y . Then, V ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))) and σ1σ2-Cl(V ) = σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))). Since σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))) is (σ1, σ2)r-closed in Y , by Theorem 3 we have (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )). (2) ⇒ (3): The proof is obvious. (3) ⇒ (1): Let V be any (σ1, σ2)β-open set of Y . Since σ1σ2-Cl(V ) is (σ1, σ2)s-open in Y , by (3) we have (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )). By Theorem 3, f is θ(τ1, τ2)-continuous. Theorem 6. For a function (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equiva- lent: (1) f is θ(τ1, τ2)-continuous; (2) (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6014 7 of 13 (3) (τ1, τ2)θ-Cl(f −1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (4) f−1(V ) ⊆ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y . Proof. (1) ⇒ (2): It follows from Theorem 5. (2) ⇒ (3): Let V be any (σ1, σ2)p-open set of Y . Then by (2), we have (τ1, τ2)θ-Cl(f −1(V )) ⊆ (τ1, τ2)θ-Cl(f −1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )). (3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . By (3), we have X − (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))) = (τ1, τ2)θ-Cl(X − f−1(σ1σ2-Cl(V ))) = (τ1, τ2)θ-Cl(f −1(Y − σ1σ2-Cl(V ))) ⊆ f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V ))) = X − f−1(σ1σ2-Int(σ1σ2-Cl(V ))) ⊆ X − f−1(V ) and hence f−1(V ) ⊆ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))). (4) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, V is (σ1, σ2)p-open in Y , by (4) we have f−1(V ) ⊆ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))). By Theorem 3, f is θ(τ1, τ2)-continuous. Definition 2. [36] A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-regular if for each τ1τ2-closed set F and each x ̸∈ F , there exist disjoint τ1τ2-open sets U and V such that x ∈ U and F ⊆ V . Lemma 3. [37] A bitopological space (X, τ1, τ2) is (τ1, τ2)-regular if and only if for each x ∈ X and each τ1τ2-open set U containing x, there exists a τ1τ2-open set V such that x ∈ V ⊆ τ1τ2-Cl(V ) ⊆ U . Lemma 4. [37] Let (X, τ1, τ2) be a (τ1, τ2)-regular space. Then, the following properties hold: (1) τ1τ2-Cl(A) = (τ1, τ2)θ-Cl(A) for every subset A of X. (2) Every τ1τ2-open set is (τ1, τ2)θ-open. 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. Definition 4. [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 τ1τ2-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. M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6014 8 of 13 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 7. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)- regular, the following properties are equivalent: (1) f is (τ1, τ2)-continuous; (2) f is θ(τ1, τ2)-continuous; (3) f is weakly (τ1, τ2)-continuous. Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing f(x). Thus by Lemma 5, f−1(V ) is τ1τ2-open in X. Since σ1σ2-Cl(V ) is σ1σ2-closed, by Lemma 5 we have f−1(σ1σ2-Cl(V )) is τ1τ2-closed. Put U = f−1(V ). Then, U is a τ1τ2-open set U of X such that x ∈ U ⊆ f−1(σ1σ2-Cl(V )) = τ1τ2-Cl(f −1(σ1σ2-Cl(V ))). This implies that τ1τ2-Cl(U) ⊆ τ1τ2-Cl(f −1(σ1σ2-Cl(V ))) = f−1(σ1σ2-Cl(V )). Thus, f(τ1τ2-Cl(U)) ⊆ σ1σ2-Cl(V ). This shows that f is θ(τ1, τ2)-continuous. (2) ⇒ (3): The proof is obvious. (3) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Since (Y, σ1, σ2) is (σ1, σ2)-regular, by Lemma 3 there exists a σ1σ2-open set W of Y such that f(x) ∈ W ⊆ σ1σ2-Cl(W ) ⊆ V . Since f is weakly (τ1, τ2)-continuous, there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(W ) ⊆ V . Thus, f is (τ1, τ2)-continuous. Theorem 8. Let (X, τ1, τ2) be (τ1, τ2)-regular. Then a function f : (X, τ1, τ2) → (Y, σ1, σ2) is θ(τ1, τ2)-continuous if and only if f is weakly (τ1, τ2)-continuous. Proof. We prove only the sufficiency. Suppose that f is weakly (τ1, τ2)-continuous. Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Then, there exists a τ1τ2-open set W of X containing x such that f(W ) ⊆ σ1σ2-Cl(V ). Since (X, τ1, τ2) is (τ1, τ2)-regular, by Lemma 3 there exists a τ1τ2-open set U of X such that x ∈ U ⊆ τ1τ2-Cl(U) ⊆ W . Thus, f(τ1τ2-Cl(U)) ⊆ σ1σ2-Cl(V ). This shows that f is θ(τ1, τ2)-continuous. M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6014 9 of 13 4. Some results on θ(τ1, τ2)-continuity Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-T2 [38] 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. Definition 5. [39] 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 9. Let (X, τ1, τ2) be a bitopological space. If for any distinct points x and x′ in X, there exists a function f : (X, τ1, τ2) → (Y, σ1, σ2) such that (1) (Y, σ1, σ2) is σ1σ2-Urysohn, (2) f(x) ̸= f(x′), and (3) f is θ(τ1, τ2)-continuous at x and x′, then (X, τ1, τ2) is τ1τ2-Urysohn. Proof. Let x, x′ be any distinct points of X. Then, by the hypothesis there exists a function f : (X, τ1, τ2) → (Y, σ1, σ2) which satisfies three conditions. Now 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 V ′ of Y containing y and y′, respectively, such that σ1σ2-Cl(V )∩ σ1σ2-Cl(V ′) = ∅. Since f is θ(τ1, τ2)-continuous at x and x′, there exist τ1τ2-open sets U and U ′ of X containing x and x′, respectively, such that f(τ1τ2-Cl(U)) ⊆ σ1σ2-Cl(V ) and f(τ1τ2-Cl(U ′)) ⊆ σ1σ2-Cl(V ′). This implies that τ1τ2-Cl(U) ∩ τ1τ2-Cl(U ′) = ∅. Thus, (X, τ1, τ2) is τ1τ2-Urysohn. Definition 6. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to have a strong θ(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-open set V of Y containing y such that [τ1τ2-Cl(U)× σ1σ2-Cl(V )] ∩G(f) = ∅. Lemma 6. A function f : (X, τ1, τ2) → (Y, σ1, σ2) has a strong θ(τ1, τ2)s-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-open set V of Y containing y such that f(τ1τ2-Cl(U)) ∩ σ1σ2-Cl(V ) = ∅. Theorem 10. If f : (X, τ1, τ2) → (Y, σ1, σ2) is θ(τ1, τ2)-continuous and (Y, σ1, σ2) is σ1σ2-Urysohn, then G(f) is strong θ(τ1, τ2)-closed. Proof. Suppose that (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 θ(τ1, τ2)-continuous, there exists a τ1τ2-open set U of X containing x such that f(τ1τ2-Cl(U)) ⊆ σ1σ2-Cl(W ). This implies that f(τ1τ2-Cl(U)) ∩ σ1σ2-Cl(V ) = ∅ and by Lemma 6, G(f) is strong θ(τ1, τ2)-closed. M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6014 10 of 13 Theorem 11. If f : (X, τ1, τ2) → (Y, σ1, σ2) is an injective θ(τ1, τ2)-continuous function with a strong θ(τ1, τ2)-closed graph, then (X, τ1, τ2) is σ1σ2-Urysohn. 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 strong θ(τ1, τ2)-closed, by Lemma 6 there exist a τ1τ2-open set U of X containing x and a σ1σ2-open set V of Y containing f(y) such that such that f(τ1τ2-Cl(U)) ∩ σ1σ2-Cl(V ) = ∅. Since f is θ(τ1, τ2)-continuous, there exists a τ1τ2-open set W of X containing y such that f(τ1τ2-Cl(W )) ⊆ σ1σ2-Cl(V ). Thus, f(τ1τ2-Cl(U)) ∩ f(τ1τ2-Cl(W )) = ∅ and hence τ1τ2-Cl(U) ∩ τ1τ2-Cl(W ) = ∅. This shows that (X, τ1, τ2) is σ1σ2-Urysohn. Recall that a bitopological space (X, τ1, τ2) is said to be quasi (τ1, τ2)-H -closed [40] if every τ1τ2-open cover {Uγ | γ ∈ Γ}, there exists a finite subset Γ0 of Γ such that X = ∪{τ1τ2-Cl(Uγ) | γ ∈ Γ0}. A subset K of a bitopological space (X, τ1, τ2) is said to be quasi (τ1, τ2)-H -closed relative to X if for any cover {Vγ | γ ∈ Γ} by τ1τ2-open sets of X, there exists a finite subset Γ0 of Γ such that K ⊆ ∪{τ1τ2-Cl(Vγ) | γ ∈ Γ0}. Theorem 12. If f : (X, τ1, τ2) → (Y, σ1, σ2) is θ(τ1, τ2)-continuous and K is quasi (τ1, τ2)- H -closed relative to X, then f(K) is quasi (σ1, σ2)-H -closed relative to Y . Proof. Let {Vγ | γ ∈ Γ} be a cover of f(K) by σ1σ2-open sets of Y . For each k ∈ K, there exists γ(k) ∈ Γ such that f(k) ∈ Vγ(k). Since f is θ(τ1, τ2)-continuous, there exists a τ1τ2-open set Uk of X containing k such that f(τ1τ2-Cl(Uk)) ⊆ σ1σ2-Cl(Vγ(k)). Since {Uk | k ∈ K} is a cover of K by τ1τ2-open sets in X, there exists a finite subset K0 of K such that K ⊆ ∪{τ1τ2-Cl(Uk) | k ∈ K0}. Thus, f(K) ⊆ ∪{f(τ1τ2-Cl(Uk)) | k ∈ K0} ⊆ ∪{σ1σ2-Cl(Vγ(k)) | k ∈ K0}. This shows that f(K) is quasi (σ1, σ2)-H -closed relative to Y . Corollary 1. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a θ(τ1, τ2)-continuous surjection and (X, τ1, τ2) is quasi (τ1, τ2)-H -closed, then (Y, σ1, σ2) is quasi (σ1, σ2)-H -closed. Definition 7. Let A be a subset of a bitopological space (X, τ1, τ2). The (τ1, τ2)θ-frontier of A, (τ1, τ2)θ-fr(A), is defined by (τ1, τ2)θ-fr(A) = (τ1, τ2)θ-Cl(A) ∩ (τ1, τ2)θ-Cl(X −A). Theorem 13. The set of all points x ∈ X at which a function f : (X, τ1, τ2) → (Y, σ1, σ2) is not θ(τ1, τ2)-continuous is identical with the union of the (τ1, τ2)θ-frontier of the inverse images of the σ1σ2-closure of σ1σ2-open sets containing f(x). Proof. Suppose that f is not θ(τ1, τ2)-continuous. Then, there exists a σ1σ2-open set V of Y containing f(x) such that f(τ1τ2-Cl(U)) is not contained in σ1σ2-Cl(V ) for every τ1τ2-open set U of X containing x. Then, τ1τ2-Cl(U) ∩ (X − f−1(σ1σ2-Cl(V ))) ̸= ∅ for every τ1τ2-open set U of X containing x. Thus, x ∈ (τ1, τ2)θ-Cl(X − f−1(σ1σ2-Cl(V ))). On the other hand, we have x ∈ f−1(V ) ⊆ (τ1, τ2)θ-Cl(f −1(σ1σ2-Cl(V ))) and hence x ∈ (τ1, τ2)θ-fr(f −1(σ1σ2-Cl(V ))). M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6014 11 of 13 Conversely, suppose that f is θ(τ1, τ2)-continuous at x ∈ X. Let V be any σ1σ2-open set of Y containing f(x). Then by Theorem 2 we have x ∈ f−1(V ) ⊆ (τ1, τ2)θ-Int(f −1(σ1σ2-Cl(V ))). Thus, x ̸∈ (τ1, τ2)θ-fr(f −1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y containing f(x). This completes the proof. 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. [5] N. Srisarakham and C. Boonpok. Almost (Λ, p)-continuous functions. International Journal of Mathematics and Computer Science, 18(2):255–259, 2023. [6] 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. [7] 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. [8] C. Boonpok. θ(⋆)-precontinuity. Mathematica, 65(1):31–42, 2023. [9] 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. [10] C. Boonpok. On some spaces via topological ideals. Open Mathematics, 21:20230118, 2023. [11] C. Boonpok. On characterizations of ⋆-hyperconnected ideal topological spaces. Jour- nal of Mathematics, 2020:9387601, 2020. [12] C. Boonpok. Almost (g,m)-continuous functions. International Journal of Mathe- matical Analysis, 4(40):1957–1964, 2010. [13] C. Boonpok. M -continuous functions in biminimal structure spaces. Far East Journal of Mathematical Sciences, 43(1):41–58, 2010. [14] S. Fomin. Extensions of topological spaces. Doklady Akademii Nauk SSSR, 32:114– 116, 1941. M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6014 12 of 13 [15] T. Noiri. Properties of θ-continuous functions. Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, Series (8), 58:887–891, 1975. [16] V. Popa. Characterizations of θ-continuous functions. Studii şi Cercetări Ştiinţifice. Seria Matematică, 32:113–119, 1980. [17] S. P. Arya and M. P. Bhamini. Some weaker forms of semi-continuous functions. Ganita, 33:124–134, 1982. [18] S. Jafari and T. Noiri. Properties of θ-semi-continuous functions. Journal of Institute of Mathematics and Computer Sciences, Mathematics Series, 13:123–128, 2000. [19] T. Noiri. On θ-precontinuous functions. International Journal of Mathematics and Mathematical Sciences, 28:285–292, 2001. [20] C. W. Baker. Weakly θ-precontinuous functions. Acta Mathematica Hungarica, 100:343–351, 2003. [21] T. Noiri and V. Popa. A unified theory of θ-continuity for functions. Rendiconti del Circolo Matematico di Palermo Series 2, 52:163–188, 2003. [22] T. Noiri and V. Popa. On θ-m-continuous functions. Libertas Mathematica, 26:1–13, 2006. [23] P. E. Long and L. L. Herrington. Strongly θ-continuous functions. Journal of the Korean Mathematical Society, 18:21–28, 1981. [24] S. Jafari and T. Noiri. Strongly θ-semi-continuous functions. Indian Journal of Pure and Applied Mathematics, 29:1195–1201, 1998. [25] T. Noiri. Strongly θ-precontinuous functions. Acta Mathematica Hungarica, 90:307– 316, 2001. [26] P. Pue-on and C. Boonpok. θ(Λ, p)-continuity for functions. International Journal of Mathematics and Computer Science, 19(2):491–495, 2024. [27] M. Thongmoon and C. Boonpok. Strongly θ(Λ, p)-continuous functions. International Journal of Mathematics and Computer Science, 19(2):475–479, 2024. [28] C. Boonpok and N. Srisarakham. (τ1, τ2)-continuity for functions. Asia Pacific Jour- nal of Mathematics, 11:21, 2024. [29] C. Boonpok and P. Pue-on. Characterizations of almost (τ1, τ2)-continuous functions. International Journal of Analysis and Applications, 22:33, 2024. [30] C. Boonpok and C. Klanarong. On weakly (τ1, τ2)-continuous functions. European Journal of Pure and Applied Mathematics, 17(1):416–425, 2024. [31] N. Srisarakham, S. Sompong, and C. Boonpok. Quasi θ(τ1, τ2)-continuous functions. European Journal of Pure and Applied Mathematics, 18(1):5722, 2025. [32] 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. [33] C. Viriyapong and C. Boonpok. (τ1, τ2)α-continuity for multifunctions. Journal of Mathematics, 2020:6285763, 2020. [34] C. Boonpok. (τ1, τ2)δ-semicontinuous multifunctions. Heliyon, 6:e05367, 2020. [35] N. Viriyapong, S. Sompong, and C. Boonpok. (τ1, τ2)-extremal disconnectedness in bitopological spaces. International Journal of Mathematics and Computer Science, M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6014 13 of 13 19(3):855–860, 2024. [36] M. Chiangpradit, S. Sompong, and C. Boonpok. On characterizations of (τ1, τ2)- regular spaces. International Journal of Mathematics and Computer Science, 19(4):1229–1334, 2024. [37] C. Klanarong, S. Sompong, and C. Boonpok. (τ1, τ2)-continuity and (τ1, τ2)θ-closed sets. International Journal of Mathematics and Computer Science, 19(4):1299–1304, 2024. [38] N. Chutiman, S. Sompong, and C. Boonpok. On some separation axioms in bitopo- logical spaces. Asia Pacific Journal of Mathematics, 11:41, 2024. [39] P. Pue-on, A. Sama-Ae, and C. Boonpok. Characterizations of quasi θ(τ1, τ2)- continuous multifunctions. International Journal of Analysis and Applications, 23:59, 2025. [40] 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.