EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6038 ISSN 1307-5543 – ejpam.com Published by New York Business Global Almost Contra-(τ1, τ2)p-continuity for Functions Prapart Pue-on1, 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 almost contra-(τ1, τ2)p-continuous functions. Moreover, some characterizations and several prop- erties concerning almost contra-(τ1, τ2)p-continuous functions are established. 2020 Mathematics Subject Classifications: 54C08, 54E55 Key Words and Phrases: τ1τ2-open set, almost contra-(τ1, τ2)p-continuous function 1. Introduction In 1966, Dontchev [1] introduced the concepts of contra-continuity and strong S- closedness in topological spaces. Moreover, Dontchev [1] obtained very interesting and important results concerning contra-continuity, compactness, S-closedness and strong S- closedness. In 1999, Dontchev et al. [2] defined a new class of functions called regular set-connected functions. Furthermore, Dontchev and Noiri [3] introduced and studied the concept of RC-continuity between topological spaces which is weaker than contra- continuity. In [4], the present authors introduced and investigated a new class of functions called contra-super-continuous functions which lies between classes of RC-continuous func- tions and contra-continuous functions. In 2002, Jafari and Noiri [5] 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. In par- ticular, Jafari and Noiri [5] defined contra-preclosed graphs and investigated relations between contra-precontinuity and contra-preclosed graphs. In 2004, Ekici [6] introduced and studied a new class of functions called almost contra-precontinuous functions which generalize classes of regular set-connected functions [2], contra-precontinuous functions [5], contra-continuous functions [1], almost s-continuous functions [7] and perfectly continu- ous functions [8]. Ekici [6] obtained basic properties and preservation theorems of almost ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6038 Email addresses: prapart.p@msu.ac.th (P. Pue-on), 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) P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6038 2 of 11 contra-precontinuous functions and relationships between almost contra-precontinuity and P -regular graphs. Noiri and Jafari [9] obtained the further characterizations and prop- erties of almost contra-precontinuous functions and showed that (s, p)-continuity due to Jafari [10] is equivalent to almost contra-precontinuity. In 2007, Al-Omari and Noorani [11] 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. On the other hand, the present authors introduced and studied the notions of 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], almost nearly (τ1, τ2)-continuous functions, almost weakly (τ1, τ2)-continuous func- tions [20] and almost contra-(Λ, sp)-continuous functions [21]. In this paper, we introduce the concept of almost contra-(τ1, τ2)p-continuous functions. We also investigate some characterizations of almost 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 [22] 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 [22] 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 [22] of A and is denoted by τ1τ2-Int(A). Lemma 1. [22] 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 subsetA of a bitopological space (X, τ1, τ2) is called (τ1, τ2)r-open [23] (resp. (τ1, τ2)s- open [24], (τ1, τ2)p-open [24], (τ1, τ2)β-open [24]) 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 P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6038 3 of 11 subset A of a bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-open [25] 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 called the (τ1, τ2)p-closure [26] (resp. (τ1, τ2)s-closure [24], α(τ1, τ2)-closure [27]) 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 [26] (resp. (τ1, τ2)s-interior [24], α(τ1, τ2)-interior [27]) of A and is denoted by (τ1, τ2)-pInt(A) (resp. (τ1, τ2)-sInt(A), α(τ1, τ2)-Int(A)). Lemma 2. 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 [26]; (2) (τ1, τ2)-pInt(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∩A [20]; (3) (τ1, τ2)-sCl(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∪A [24]; (4) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A [28]. Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is called a s(τ1, τ2)θ-cluster point of A if τ1τ2-Cl(U)∩A ̸= ∅ for every (τ1, τ2)s-open set U containing x. The set of all s(τ1, τ2)θ-cluster points of A is called the s(τ1, τ2)θ-closure of A and is denoted by s(τ1, τ2)θ-Cl(A). A subset A of a bitopological space (X, τ1, τ2) is called s(τ1, τ2)θ-closed if s(τ1, τ2)θ-Cl(A) = A. The complement of a s(τ1, τ2)θ-closed set is said to be s(τ1, τ2)θ-open. The union of all s(τ1, τ2)θ-open sets of X contained in A is called the s(τ1, τ2)θ-interior of A and is denoted by s(τ1, τ2)θ-Int(A). 3. Almost contra-(τ1, τ2)p-continuous functions In this section, we introduce the concept of almost contra-(τ1, τ2)p-continuous func- tions. Moreover, some characterizations of almost contra-(τ1, τ2)p-continuous functions are discussed. Definition 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost contra-(τ1, τ2)p- continuous if for each x ∈ X and for each (σ1, σ2)r-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 almost contra-(τ1, τ2)p-continuous; (2) f−1(F ) is (τ1, τ2)p-open in X for every (σ1, σ2)r-closed set F of Y ; (3) f−1(V ) is (τ1, τ2)p-closed in X for every (σ1, σ2)r-open set V of Y ; P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6038 4 of 11 (4) f−1(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X for every σ1σ2-open set V of Y ; (5) f−1(σ1σ2-Cl(σ1σ2-Int(F ))) is (τ1, τ2)p-open in X for every σ1σ2-closed set F of Y . Proof. (1) ⇒ (2): Let F be any (σ1, σ2)r-closed set of Y and x ∈ f−1(F ). Then, f(x) ∈ F . By (1), there exists a (τ1, τ2)p-open set U ofX containing x such that f(U) ⊆ F . Thus, x ∈ U ⊆ f−1(F ) and hence x ∈ (τ1, τ2)-pInt(f −1(F )). This implies that f−1(F ) ⊆ (τ1, τ2)-pInt(f −1(F )). Therefore, f−1(F ) is (τ1, τ2)p-open in X. (2) ⇒ (3): The proof is obvious. (3) ⇒ (4): Let V be any σ1σ2-open set of Y . Then, we have σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r-open in Y . Thus by (3), f−1(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X. (4) ⇒ (5): Let F be any σ1σ2-closed set of Y . Then, Y − F is σ1σ2-open in Y . By (4), we have f−1(σ1σ2-Cl(σ1σ2-Int(Y −F ))) = Y −f−1(σ1σ2-Int(σ1σ2-Cl(F ))) is (τ1, τ2)p- closed in X. Thus, f−1(σ1σ2-Cl(σ1σ2-Int(F ))) is (τ1, τ2)p-open in X. (5) ⇒ (1): Let F be any (σ1, σ2)r-closed set of Y containing f(x). Since F is σ1σ2- closed in Y and by (5), f−1(F ) is (τ1, τ2)p-open in X. Let U = f−1(F ). Then, U is a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ F . This shows that f is almost contra-(τ1, τ2)p-continuous. Definition 2. A function 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 2. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a (τ1, τ2)p-open surjection and g : (Y, σ1, σ2) → (Z, ρ1, ρ2) is a function such that g◦f : (X, τ1, τ2) → (Z, ρ1, ρ2) is almost contra-(τ1, τ2)p-continuous, then g is almost contra-(σ1, σ2)p-continuous. Proof. Let F be any (ρ1, ρ2)r-closed set of Z. Since g ◦ f is almost contra-(τ1, τ2)p- continuous, by Theorem 1 we have (g◦f)−1(F ) = f−1(g−1(F )) is (τ1, τ2)p-open inX. Since f is (τ1, τ2)p-open surjective, f(f−1(g−1(F ))) = g−1(F ) is (σ1, σ2)p-open in Y . Thus, g is almost contra-(σ1, σ2)p-continuous. Definition 3. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be (τ1, τ2)p-closed if f(K) is (σ1, σ2)p-closed in Y for every (τ1, τ2)p-closed set K of X. Theorem 3. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a (τ1, τ2)p-closed surjection and g : (Y, σ1, σ2) → (Z, ρ1, ρ2) is a function such that g◦f : (X, τ1, τ2) → (Z, ρ1, ρ2) is almost contra-(τ1, τ2)p-continuous, then g is almost contra-(σ1, σ2)p-continuous. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6038 5 of 11 Proof. The proof is similar to that of Theorem 2. Definition 4. A bitopological space (X, τ1, τ2) is said to be weakly τ1τ2-Hausdorff if each element of X is an intersection of (τ1, τ2)r-closed sets. Definition 5. A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)p-T1 if for each pair of distinct points x and y of X, there exist (τ1, τ2)p-open sets U and V containing x and y, respectively, such that y ̸∈ U and x ̸∈ V . Theorem 4. If f : (X, τ1, τ2) → (Y, σ1, σ2) is an almost contra-(τ1, τ2)p-continuous injec- tion and (Y, σ1, σ2) is weakly σ1σ2-Hausdorff, then (X, τ1, τ2) is (τ1, τ2)p-T1. Proof. Suppose that (Y, σ1, σ2) is weakly σ1σ2-Hausdorff. For any distinct points x and y in X, there exist (σ1, σ2)r-closed sets H and K of Y such that f(x) ∈ H, f(y) ̸∈ H, f(y) ∈ K and f(x) ̸∈ K. Since f is almost contra-(τ1, τ2)p-continuous, f−1(H) and f−1(K) are (τ1, τ2)p-open sets of X such that x ∈ f−1(H), y ̸∈ f−1(H), y ∈ f−1(K) and x ̸∈ f−1(K). This shows that (X, τ1, τ2) is (τ1, τ2)p-T1. Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-connected [22] if X cannot be written as the union of two nonempty disjoint τ1τ2-open sets. Definition 6. 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 5. If f : (X, τ1, τ2) → (Y, σ1, σ2) is an almost contra-(τ1, τ2)p-continuous sur- jection 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 almost contra-(τ1, τ2)p-continuous, f −1(V ) and f−1(W ) are (τ1, τ2)p-open in X. Furthermore, 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. Thus, (Y, σ1, σ2) is σ1σ2-connected. Definition 7. A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)p-compact if every (τ1, τ2)p-open cover of X has a finite subcover. Definition 8. A bitopological space (X, τ1, τ2) is said to be S-τ1τ2-closed if every (τ1, τ2)r- closed cover of X has a finite subcover. Theorem 6. If f : (X, τ1, τ2) → (Y, σ1, σ2) is an almost contra-(τ1, τ2)p-continuous sur- jection and (X, τ1, τ2) is (τ1, τ2)p-compact, then (Y, σ1, σ2) is S-σ1σ2-closed. Proof. Let {Vγ | γ ∈ ∇} be any (σ1, σ2)r-closed cover of Y . Since f is almost contra- (τ1, τ2)p-continuous, we have {f−1(Vγ) | γ ∈ ∇} is a (τ1, τ2)p-open cover ofX and therefore there exists a finite subset ∇0 of ∇ such that X = ∪{f−1(Vγ) | γ ∈ ∇0}. Thus, we have Y = ∪{Vγ | γ ∈ ∇0} and hence (Y, σ1, σ2) is S-σ1σ2-closed. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6038 6 of 11 Definition 9. A bitopological space (X, τ1, τ2) is said to be P-τ1τ2-closed if every (τ1, τ2)p- closed cover of X has a finite subcover. Definition 10. A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-compact if every (τ1, τ2)r-open cover of X has a finite subcover. Theorem 7. If f : (X, τ1, τ2) → (Y, σ1, σ2) is an almost contra-(τ1, τ2)p-continuous sur- jection and (X, τ1, τ2) is P-τ1τ2-closed, then (Y, σ1, σ2) is (σ1, σ2)r-compact. Proof. Let {Vγ | γ ∈ ∇} be any (σ1, σ2)r-open cover of Y . Since f is almost contra- (τ1, τ2)p-continuous, we have {f−1(Vγ) | γ ∈ ∇} is a (τ1, τ2)p-closed cover of X. Since (X, τ1, τ2) is P-τ1τ2-closed, there exists a finite subset ∇0 of ∇ such that X = ∪{f−1(Vγ) | γ ∈ ∇0}. Thus, we have Y = ∪{Vγ | γ ∈ ∇0} and hence (Y, σ1, σ2) is (σ1, σ2)r-compact. Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is called a s(τ1, τ2)θ-cluster point of A if τ1τ2-Cl(U)∩A ̸= ∅ for every (τ1, τ2)s-open set U containing x. The set of all s(τ1, τ2)θ-cluster points of A is called the s(τ1, τ2)θ-closure of A and is denoted by s(τ1, τ2)θ-Cl(A). A subset A of a bitopological space (X, τ1, τ2) is called s(τ1, τ2)θ-closed if s(τ1, τ2)θ-Cl(A) = A. The complement of a s(τ1, τ2)θ-closed set is said to be s(τ1, τ2)θ-open. The union of all s(τ1, τ2)θ-open sets of X contained in A is called the s(τ1, τ2)θ-interior of A and is denoted by s(τ1, τ2)θ-Int(A). Definition 11. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be p(τ1, τ2)s-continuous if for each x ∈ X and for each (σ1, σ2)s-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 8. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is p(τ1, τ2)s-continuous; (2) f is almost contra-(τ1, τ2)p-continuous; (3) f−1(V ) is (τ1, τ2)p-open in X for each s(σ1, σ2)θ-open set V of Y ; (4) f−1(F ) is (τ1, τ2)p-closed in X for each s(σ1, σ2)θ-closed set F of Y . Proof. (1) ⇒ (2): Let F be any (σ1, σ2)r-closed set of Y and x ∈ f−1(F ). Then, f(x) ∈ F and F is (σ1, σ2)s-open. Since f is p(τ1, τ2)s-continuous, there exists a (τ1, τ2)p- open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(F ) = F . Therefore, we have x ∈ U ⊆ f−1(F ) which implies that x ∈ (τ1, τ2)-pInt(f −1(F )). Thus, f−1(F ) ⊆ (τ1, τ2)-pInt(f −1(F )) and hence f−1(F ) = (τ1, τ2)-pInt(f −1(F )). This shows that f−1(F ) is (τ1, τ2)p-open in X. It follows from Theorem 1 that f is almost contra-(τ1, τ2)p-continuous. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6038 7 of 11 (2) ⇒ (3): This follows from the fact that every s(σ1, σ2)θ-open set is the union of (σ1, σ2)r-closed sets. (3) ⇔ (4): This is obvious. (4) ⇒ (1): Let x ∈ X and V be any (σ1, σ2)s-open set of Y containing f(x). Since σ1σ2-Cl(V ) is (σ1, σ2)r-closed, we have σ1σ2-Cl(V ) is s(σ1, σ2)θ-open. Thus by (4), f−1(σ1σ2-Cl(V )) is (τ1, τ2)p-open in X. Now, put U = f−1(σ1σ2-Cl(V )). Then, U is a (τ1, τ2)p-open set of X containing x and f(U) ⊆ σ1σ2-Cl(V ). This shows that f is p(τ1, τ2)s-continuous. Theorem 9. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost contra-(τ1, τ2)p-continuous; (2) f((τ1, τ2)-pCl(A)) ⊆ s(σ1, σ2)θ-Cl(f(A)) for every subset A of X; (3) (τ1, τ2)-pCl(f −1(B)) ⊆ f−1(s(σ1, σ2)θ-Cl(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let A be any subset of X. Let x ∈ (τ1, τ2)-pCl(A) and V be any (σ1, σ2)s-open set of Y containing f(x). Since f is almost contra-(τ1, τ2)p-continuous, by Theorem 8 there exists a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). Since x ∈ (τ1, τ2)-pCl(A), we have U ∩A ̸= ∅ and hence ∅ ≠ f(U) ∩ f(A) ⊆ σ1σ2-Cl(V ) ∩ f(A). Therefore, f(x) ∈ s(σ1, σ2)θ-Cl(f(A)). Thus, f((τ1, τ2)-pCl(A)) ⊆ s(σ1, σ2)θ-Cl(f(A)). (2) ⇒ (3): Let B be any subset of Y . By (2), we have f((τ1, τ2)-pCl(f −1(B))) ⊆ s(σ1, σ2)θ-Cl(f(f −1(B))) ⊆ s(σ1, σ2)θ-Cl(B) and hence (τ1, τ2)-pCl(f −1(B)) ⊆ f−1(s(σ1, σ2)θ-Cl(B)). (3) ⇒ (1): Let V be any (σ1, σ2)s-open set of Y containing f(x). Since σ1σ2-Cl(V ) ∩ (Y − σ1σ2-Cl(V )) = ∅, we have f(x) ̸∈ s(σ1, σ2)θ-Cl(Y − σ1σ2-Cl(V )) and hence x ̸∈ f−1(s(σ1, σ2)θ-Cl(Y − σ1σ2-Cl(V ))). By (3), x ̸∈ (τ1, τ2)-pCl(f −1(Y − σ1σ2-Cl(V ))). There exists a (τ1, τ2)p-open set U of X containing x such that U ∩f−1(Y −σ1σ2-Cl(V )) = ∅; hence f(U)∩ (Y −σ1σ2-Cl(V )) = ∅. This shows that f(U) ⊆ σ1σ2-Cl(V ). Thus by Theorem 8, f is almost contra-(τ1, τ2)p- continuous. P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6038 8 of 11 Theorem 10. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost contra-(τ1, τ2)p-continuous; (2) f−1(σ1σ2-Cl(V )) is (τ1, τ2)p-open in X for every (σ1, σ2)β-open set V of Y ; (3) f−1(σ1σ2-Cl(V )) is (τ1, τ2)p-open in X for every (σ1, σ2)s-open set V of Y ; (4) f−1(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X for every (σ1, σ2)p-open set V of Y . Proof. (1) ⇒ (2): Let V be any (σ1, σ2)β-open set of Y . Then, σ1σ2-Cl(V ) is (σ1, σ2)r- closed in Y , by Theorem 1 we have f−1(σ1σ2-Cl(V )) is (τ1, τ2)p-open in X. (2) ⇒ (3): This is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open. (3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . Then, Y − σ1σ2-Cl(V ) is (σ1, σ2)r- closed and hence Y − σ1σ2-Cl(V ) is (σ1, σ2)s-open. Thus by (3), f−1(σ1σ2-Cl(Y − σ1σ2-Int(σ1σ2-Cl(V )))) is (τ1, τ2)p-open in X. Since X − f−1(σ1σ2-Int(σ1σ2-Cl(V ))) = f−1(Y − σ1σ2-Int(σ1σ2-Cl(V ))) = f−1(σ1σ2-Cl(Y − σ1σ2-Int(σ1σ2-Cl(V )))), we have f−1(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X. (4) ⇒ (1): Let V be any (σ1, σ2)r-open set of Y . Then, V is (σ1, σ2)p-open in Y . By (4), we have f−1(V ) = f−1(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X. It follows from Theorem 1 that f is almost contra-(τ1, τ2)p-continuous. Lemma 3. For a bitopological space (X, τ1, τ2), the following properties hold: (1) α(τ1, τ2)-Cl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)β-open set V of X; (2) (τ1, τ2)-pCl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)s-open set V of X; (3) (τ1, τ2)-sCl(V ) = τ1τ2-Int(τ1τ2-Cl(V )) for every (τ1, τ2)p-open set V of X. Corollary 1. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost contra-(τ1, τ2)p-continuous; (2) f−1(α(σ1, σ2)-Cl(V )) is (τ1, τ2)p-open in X for every (σ1, σ2)β-open set V of Y ; (3) f−1((σ1, σ2)-pCl(V )) is (τ1, τ2)p-open in X for every (σ1, σ2)s-open set V of Y ; (4) f−1((σ1, σ2)-sCl(V )) is (τ1, τ2)p-closed in X for every (σ1, σ2)p-open set V of Y . P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6038 9 of 11 Proof. This is an immediate consequence of Theorem 10 and Lemma 3. Definition 12. [20] 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. [20] 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 11. If a function f : (X, τ1, τ2) → (Y, σ1, σ2) is almost contra-(τ1, τ2)p-continuous, then f is almost 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 (σ1, σ2)r-closed set of Y containing f(x). 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