EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6574 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Weak Forms of Open and Closed Functions between Bitopological Spaces 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 presents new classes of open and closed functions defined between bitopo- logical spaces, called weakly θ(τ1, τ2)b-open functions and weakly θ(τ1, τ2)b-closed functions. More- over, several characterizations and some properties concerning weakly θ(τ1, τ2)b-open functions and weakly θ(τ1, τ2)b-closed functions are established. 2020 Mathematics Subject Classifications: 54C10, 54E55 Key Words and Phrases: Weakly θ(τ1, τ2)b-open function, weakly θ(τ1, τ2)b-closed function 1. Introduction Topology is concerned with all questions directly or indirectly related to openness and closedness. Semi-open sets, preopen sets, α-open sets, β-open sets, b-open sets, δ-open sets, θ-open sets and b-θ-open sets play an important role in the researches of generaliza- tions of open functions and closed functions. By using these sets, many authors introduced and studied various types of open functions and closed functions. In 1983, Rose [1] intro- duced and studied the notions of weakly open functions and almost open functions. In 1987, Rose and Janković [2] investigated some of the fundamental properties of weakly closed functions. In 2006, Caldas et al. [3] introduced and studied the concepts of θ- preopen functions and θ-preclosed functions by using the notions of pre-θ-interior and pre-θ-closure. Moreover, Caldas et al. [4] introduced and investigated the concepts of weakly semi-θ-open functions and weakly semi-θ-closed functions. In 2009, Noiri et al. [5] introduced and studied two new classes of functions called weakly b-θ-open functions and weakly b-θ-open functions by utilizing the notions of b-θ-open sets and the b-θ-closure op- erator. Weak b-θ-openness (resp. b-θ-closedness) is a generalization of both θ-preopenness ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6574 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 (3) (2025), 6574 2 of 10 and weak semi-θ-openness (resp. θ-preclosedness and weak semi-θ-closedness). Quite re- cently, Klanarong and Boonpok [6] studied the notions of weakly s(Λ, p)-open functions and weakly s(Λ, p)-closed functions by utilizing s(Λ, p)-open sets and the s(Λ, p)-closure operator. On the other hand, the present authors introduced and studied the concepts of θp(Λ, p)-open functions [7], θp(Λ, p)-closed functions [7], semi-(I ,J )-open functions [8], semi-(I ,J )-closed functions [8], weakly δ(Λ, p)-open functions [9], weakly δ(Λ, p)-closed functions [10], weakly θs(Λ, p)-open functions [11], weakly θs(Λ, p)-closed functions [11] and weakly b(Λ, p)-open functions [12]. In this paper, we introduce the notions of weakly θ(τ1, τ2)b-open functions and θ(τ1, τ2)b-closed functions. Furthermore, several characteri- zations of weakly θ(τ1, τ2)b-open functions and θ(τ1, τ2)b-closed functions are investigated. 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 [13] if A = τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is called τ1τ2-open. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection of all τ1τ2-closed sets of X containing A is called the τ1τ2-closure [13] 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 [13] of A and is denoted by τ1τ2-Int(A). Lemma 1. [13] 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 [14] (resp. (τ1, τ2)s-open [15], (τ1, τ2)p-open [15], (τ1, τ2)β-open [15]) 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 [16] 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 intersection of all (τ1, τ2)p-closed (resp. (τ1, τ2)s-closed, α(τ1, τ2)-closed) sets of X containing A is N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6574 3 of 10 called the (τ1, τ2)p-closure [17] (resp. (τ1, τ2)s-closure [15], α(τ1, τ2)-closure [18]) of A and is denoted by (τ1, τ2)-pCl(A) (resp. (τ1, τ2)-sCl(A), α(τ1, τ2)-Cl(A)). The union of all (τ1, τ2)p-open (resp. (τ1, τ2)s-open, α(τ1, τ2)-open) sets of X contained in A is called the (τ1, τ2)p-interior [17] (resp. (τ1, τ2)s-interior [15], α(τ1, τ2)-interior [18]) of A and is denoted by (τ1, τ2)-pInt(A) (resp. (τ1, τ2)-sInt(A), α(τ1, τ2)-Int(A)). Lemma 2. For subsets A and B of a bitopological space (X, τ1, τ2), the following properties hold: (1) α(τ1, τ2)-Int(A) = τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))) ∩A; (2) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A [19]; (3) (τ1, τ2)-pInt(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∩A [20]. For a subset A of a bitopological space (X, τ1, τ2), a point x ∈ X is called (τ1, τ2)θ- cluster point [14] 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 [14] 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 [14] 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 [14] of A and is denoted by (τ1, τ2)θ-Int(A). Lemma 3. [14] 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. A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)b-open if A ⊆ τ1τ2-Int(τ1τ2-Cl(A)) ∪ τ1τ2-Cl(τ1τ2-Int(A)). The complement of a (τ1, τ2)b-open set is called (τ1, τ2)b-closed. Let A be a subset of a bitopological space (X, τ1, τ2). The union of all (τ1, τ2)b-open sets of X contained in A is called the (τ1, τ2)b-interior of A and is denoted by (τ1, τ2)-bInt(A). The intersection of all (τ1, τ2)b-closed sets of X containing A is called the (τ1, τ2)b-closure of A and is denoted by (τ1, τ2)-bCl(A). Lemma 4. For subsets A and B of a bitopological space (X, τ1, τ2), the following properties hold: (1) (τ1, τ2)-bInt(A) = (τ1, τ2)-sInt(A) ∪ (τ1, τ2)-pInt(A); (2) (τ1, τ2)-bCl(A) = (τ1, τ2)-sCl(A) ∩ (τ1, τ2)-pCl(A); (3) (τ1, τ2)-bCl(X −A) = X − (τ1, τ2)-bInt(A); (4) x ∈ (τ1, τ2)-bInt(A) if and only if A ∩ U ̸= ∅ for every (τ1, τ2)b-open set U of X containing x; N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6574 4 of 10 (5) A is (τ1, τ2)b-closed if and only if A = (τ1, τ2)-bCl(A); (6) (τ1, τ2)-pInt((τ1, τ2)-bCl(A)) = (τ1, τ2)-bCl((τ1, τ2)-pInt(A)). For a subset A of a bitopological space (X, τ1, τ2), a point x ∈ X is called a θ(τ1, τ2)b- cluster point of A if (τ1, τ2)-bCl(U)∩A ̸= ∅ for every (τ1, τ2)b-open set U of X containing x. The set of all θ(τ1, τ2)b-cluster points of A is called the θ(τ1, τ2)b-closure of A and is denoted by θ(τ1, τ2)b-Cl(A). If A = θ(τ1, τ2)b-Cl(A), then A is called θ(τ1, τ2)b-closed. The complement of a θ(τ1, τ2)b-closed set is called θ(τ1, τ2)b-open. The θ(τ1, τ2)b-interior of A is defined by the union of all θ(τ2, τ1)b-open sets of X contained in A and is denoted by θ(τ1, τ2)b-Int(A). Lemma 5. For subsets A and Cγ(γ ∈ ∇) of a bitopological space (X, τ1, τ2), the following properties hold: (1) If Cγ is θ(τ1, τ2)b-open for each γ ∈ ∇, then ∪γ∈∇Cγ is θ(τ1, τ2)b-open. (2) If A is (τ1, τ2)b-closed, then (τ1, τ2)-bInt(A) = θ(τ1, τ2)b-Int(A). (3) θ(τ1, τ2)b-Cl(A) is θ(τ1, τ2)b-closed. 3. Weakly θ(τ1, τ2)b-open functions In this section, we introduce the concept of weakly θ(τ1, τ2)b-open functions. Moreover, some characterizations of weakly θ(τ1, τ2)b-open functions are discussed. Definition 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly θ(τ1, τ2)b-open if f(U) ⊆ θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U))) for every τ1τ2-open set U of X. Theorem 1. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly θ(τ1, τ2)b-open; (2) f((τ1, τ2)θ-Int(A)) ⊆ θ(σ1, σ2)b-Int(f(A)) for every subset A of X; (3) (τ1, τ2)θ-Int(f −1(B)) ⊆ f−1(θ(σ1, σ2)b-Int(B)) for every subset B of Y ; (4) f−1(θ(σ1, σ2)b-Cl(B)) ⊆ (τ1, τ2)θ-Cl(f −1(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let A be any subset of X and x ∈ (τ1, τ2)θ-Int(A). Then, there exists a τ1τ2-open set U of X such that x ∈ U ⊆ τ1τ2-Cl(U) ⊆ A. Therefore, f(x) ∈ f(U) ⊆ f(τ1τ2-Cl(U)) ⊆ f(A). Since f is weakly θ(τ1, τ2)b-open, we have f(U) ⊆ θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U))) ⊆ θ(σ1, σ2)b-Int(f(A)). N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6574 5 of 10 This implies that f(x) ∈ θ(σ1, σ2)b-Int(f(A)). Thus, x ∈ f−1(θ(σ1, σ2)b-Int(f(A))) and hence (τ1, τ2)θ-Int(A) ⊆ f−1(θ(σ1, σ2)b-Int(f(A))). This shows that f((τ1, τ2)θ-Int(A)) ⊆ θ(σ1, σ2)b-Int(f(A)). (2) ⇒ (3): Let B be any subset of Y . Thus by (2), f((τ1, τ2)θ-Int(f −1(B))) ⊆ θ(σ1, σ2)b-Int(B) and so (τ1, τ2)θ-Int(f −1(B)) ⊆ f−1(θ(σ1, σ2)b-Int(B)). (3) ⇒ (4): Let B be any subset of Y . Using (3), we have X − (τ1, τ2)θ-Cl(f −1(B)) = (τ1, τ2)θ-Int(X − f−1(B)) = (τ1, τ2)θ-Int(f −1(Y −B)) ⊆ f−1(θ(σ1, σ2)b-Int(Y −B)) = f−1(Y − θ(σ1, σ2)b-Cl(B)) = X − f−1(θ(σ1, σ2)b-Cl(B)). Thus, f−1(θ(σ1, σ2)b-Cl(B)) ⊆ (τ1, τ2)θ-Cl(f −1(B)). (4) ⇒ (1): Let U be any τ1τ2-open set of X. By (4), f−1(θ(σ1, σ2)b-Cl(Y − f(τ1τ2-Cl(U)))) ⊆ (τ1, τ2)θ-Cl(f −1(Y − f(τ1τ2-Cl(U)))). Thus, f−1(Y − θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U)))) ⊆ (τ1, τ2)θ-Cl(X − f−1(f(τ1τ2-Cl(U)))) ⊆ (τ1, τ2)θ-Cl(X − τ1τ2-Cl(U)) and hence U ⊆ (τ1, τ2)θ-Int(τ1τ2-Cl(U)) ⊆ f−1(θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U)))). Therefore, f(U) ⊆ θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U))). This shows that f is weakly θ(τ1, τ2)b-open. Theorem 2. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly θ(τ1, τ2)b-open; (2) for each x ∈ X and each τ1τ2-open set U of X containing x, there exists a θ(σ1, σ2)b- open set V of Y containing f(x) such that V ⊆ f(τ1τ2-Cl(U)). Proof. (1) ⇒ (2): Let x ∈ X and U be any τ1τ2-open set of X containing x. Since f is weakly θ(τ1, τ2)b-open, f(x) ∈ f(U) ⊆ θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U))). Let V = θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U))). Then, V is θ(σ1, σ2)b-open in Y and f(x) ∈ V ⊆ f(τ1τ2-Cl(U)). N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6574 6 of 10 (2) ⇒ (1): Let U be any τ1τ2-open set of X and y ∈ f(U). It follows from (2) that V ⊆ f(τ1τ2-Cl(U)) for some θ(σ1, σ2)b-open set V of Y containing y. Thus, y ∈ V ⊆ θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U))) and hence f(U) ⊆ θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U))). This shows that f is weakly θ(τ1, τ2)b- open. Theorem 3. For a bijective function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly θ(τ1, τ2)b-open; (2) θ(σ1, σ2)b-Cl(f(τ1τ2-Int(K))) ⊆ f(K) for each τ1τ2-closed set K of X; (3) θ(σ1, σ2)b-Cl(f(U)) ⊆ f(τ1τ2-Cl(U)) for each τ1τ2-open set U of X. Proof. (1) ⇒ (2): Let K be any τ1τ2-closed set of X. Then, we have f(X −K) = Y − f(K) ⊆ θ(σ1, σ2)b-Int(f(τ1τ2-Cl(X −K))) and hence Y − f(K) ⊆ Y − θ(σ1, σ2)b-Cl(f(τ1τ2-Int(K))). Thus, θ(σ1, σ2)b-Cl(f(τ1τ2-Int(K))) ⊆ f(K). (2) ⇒ (3): Let U be any τ1τ2-open set of X. Since τ1τ2-Cl(U) is a τ1τ2-closed set and U ⊆ τ1τ2-Int(τ1τ2-Cl(U)), by (2) we have θ(σ1, σ2)b-Cl(f(U)) ⊆ θ(σ1, σ2)b-Cl(f(τ1τ2-Int(τ1τ2-Cl(U)))) ⊆ f(τ1τ2-Cl(U)). (3) ⇒ (1): Let U be any τ1τ2-open set of X. Using (3), we have Y − θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U))) = θ(σ1, σ2)b-Cl(Y − f(τ1τ2-Cl(U))) = θ(σ1, σ2)b-Cl(f(X − τ1τ2-Cl(U))) ⊆ f(τ1τ2-Cl(X − τ1τ2-Cl(U))) = f(X − τ1τ2-Int(τ1τ2-Cl(U))) ⊆ f(X − U) = Y − f(U) and so f(U) ⊆ θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U))). This shows that f is weakly θ(τ1, τ2)b-open. The proof of the following theorem is straightforward and thus is omitted. Theorem 4. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly θ(τ1, τ2)b-open; N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6574 7 of 10 (2) f(U) ⊆ θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U))) for each (τ1, τ2)p-open set U of X; (3) f(U) ⊆ θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U))) for each α(τ1, τ2)-open set U of X; (4) f(τ1τ2-Int(τ1τ2-Cl(U))) ⊆ θ(σ1, σ2)b-Int(f(τ1τ2-Cl(U))) for each τ1τ2-open set U of X; (5) f(τ1τ2-Int(K)) ⊆ θ(σ1, σ2)b-Int(f(K)) for each τ1τ2-closed set K of X. 4. Weakly θ(τ1, τ2)b-closed functions In this section, we introduce the concept of weakly θ(τ1, τ2)b-closed functions. Fur- thermore, some characterizations of weakly θ(τ1, τ2)b-closed functions are investigated. Definition 2. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly θ(τ1, τ2)b-closed if θ(σ1, σ2)b-Cl(f(τ1τ2-Int(K))) ⊆ f(K) for every τ1τ2-closed set K of X. Theorem 5. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly θ(τ1, τ2)b-closed; (2) θ(σ1, σ2)b-Cl(f(U)) ⊆ f(τ1τ2-Cl(U)) for every τ1τ2-open set U of X. Proof. (1) ⇒ (2): Let U be any τ1τ2-open set of X. Since τ1τ2-Cl(U) is a τ1τ2-closed set and U ⊆ τ1τ2-Int(τ1τ2-Cl(U)), we have θ(σ1, σ2)b-Cl(f(U)) ⊆ θ(σ1, σ2)b-Cl(f(τ1τ2-Int(τ1τ2-Cl(U)))) ⊆ f(τ1τ2-Cl(U)). (2) ⇒ (1): Let K be any τ1τ2-closed set of X. Then, we have θ(σ1, σ2)b-Cl(f(τ1τ2-Int(K))) ⊆ f(τ1τ2-Cl(τ1τ2-Int(K))) ⊆ f(τ1τ2-Cl(K)) = f(K) and hence f is weakly θ(τ1, τ2)b-closed. Corollary 1. A bijective function f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly θ(τ1, τ2)b-open if and only if f is weakly θ(τ1, τ2)b-closed. Proof. This is an immediate consequence of Theorem 3 and 5. The proof of the following theorem is straightforward and thus is omitted. Theorem 6. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6574 8 of 10 (1) f is weakly θ(τ1, τ2)b-closed; (2) θ(σ1, σ2)b-Cl(f(τ1τ2-Int(K))) ⊆ f(K) for every (τ1, τ2)p-closed set K of X; (3) θ(σ1, σ2)b-Cl(f(τ1τ2-Int(K))) ⊆ f(K) for every α(τ1, τ2)-closed set K of X; (4) θ(σ1, σ2)b-Cl(f(τ1τ2-Int(τ1τ2-Cl(A)))) ⊆ f(τ1τ2-Cl(A)) for every subset A of X; (5) θ(σ1, σ2)b-Cl(f(U)) ⊆ f(τ1τ2-Cl(U)) for every (τ1, τ2)p-open set U of X. Theorem 7. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly θ(τ1, τ2)b-closed; (2) θ(σ1, σ2)b-Cl(f(U)) ⊆ f(τ1τ2-Cl(U)) for every (τ1, τ2)r-open set U of X; (3) for each subset B of Y and each τ1τ2-open set U of X with f−1(B) ⊆ U , there exists a θ(σ1, σ2)b-open set V of Y such that B ⊆ V and f−1(V ) ⊆ τ1τ2-Cl(U); (4) for each point y ∈ Y and each τ1τ2-open set U of X with f−1(y) ⊆ U , there exists a θ(σ1, σ2)b-open set V of Y containing y and f−1(V ) ⊆ τ1τ2-Cl(U). Proof. (1) ⇒ (2): It follows from Theorem 5. (2) ⇒ (3): Let B be any subset of Y and U be any τ1τ2-open set ofX with f−1(B) ⊆ U . Then, we have f−1(B) ∩ τ1τ2-Cl(X − τ1τ2-Cl(U)) = ∅ and hence B ∩ f(τ1τ2-Cl(X − τ1τ2-Cl(U))) = ∅. Since X − τ1τ2-Cl(U) is (τ1, τ2)r-open, B ∩ θ(σ1, σ2)b-Cl(f(X − τ1τ2-Cl(U))) = ∅. Let V = Y − θ(σ1, σ2)b-Cl(f(X − τ1τ2-Cl(U))). Then, V is a θ(σ1, σ2)b-open set with B ⊆ V and f−1(V ) ⊆ X − f−1(θ(σ1, σ2)b-Cl(f(X − τ1τ2-Cl(U)))) ⊆ X − f−1(f(X − τ1τ2-Cl(U))) ⊆ τ1τ2-Cl(U). (3) ⇒ (4): The proof is obvious. (4) ⇒ (1): Let K be any τ1τ2-closed set of Y and y ∈ Y −f(K). Since f−1(y) ⊆ X−K, by (4) there exists a θ(σ1, σ2)b-open set V of Y such that y ∈ V and f−1(V ) ⊆ τ1τ2-Cl(X −K) = X − τ1τ2-Int(K). Thus, V ∩ f(τ1τ2-Int(K)) = ∅ and hence y ̸∈ θ(σ1, σ2)b-Cl(f(τ1τ2-Int(K))). Therefore, θ(σ1, σ2)b-Cl(f(τ1τ2-Int(K))) ⊆ f(K). This shows that f is weakly θ(τ1, τ2)b-closed. Theorem 8. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a bijective weakly θ(τ1, τ2)b-closed function, then for every subset B of Y and every τ1τ2-open set U of X with f−1(B) ⊆ U , there exists a θ(σ1, σ2)b-closed set K of Y such that B ⊆ K and f−1(K) ⊆ τ1τ2-Cl(U). N. Chutiman, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6574 9 of 10 Proof. Let B be any subset of Y and U be any τ1τ2-open set of X with f−1(B) ⊆ U . 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