EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6572 ISSN 1307-5543 – ejpam.com Published by New York Business Global Almost Quasi τ ⋆(σ1, σ2)-Continuous and Weakly Quasi τ ⋆(σ1, σ2)-Continuous Functions Butsakorn Kong-ied1, 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 two classes of continuous functions defined between an ideal topological space and a bitopological space, called almost quasi τ⋆(σ1, σ2)-continuous functions and weakly quasi τ⋆(σ1, σ2)-continuous functions. Furthermore, several characterizations and some properties concerning almost quasi τ⋆(σ1, σ2)-continuous functions and weakly quasi τ⋆(σ1, σ2)- continuous functions are investigated. Moreover, the relationships between almost quasi τ⋆(σ1, σ2)- continuity and weak quasi τ⋆(σ1, σ2)-continuity are considered. 2020 Mathematics Subject Classifications: 54C05, 54C08 Key Words and Phrases: Almost quasi τ⋆(σ1, σ2)-continuous function, weakly quasi τ⋆(σ1, σ2)- continuous function 1. Introduction In 1961, Marcus [1] introduced the concept of quasi continuous functions. Popa [2] introduced and investigated the notion of almost quasi continuous functions. Neubrun- novaá [3] showed that quasi continuity is equivalent to semi-continuity due to Levine [4]. Popa and Stan [5] introduced and studied the notion of weakly quasi continuous functions. Weak quasi continuity is implied by quasi continuity and weak continuity [6] which are independent of each other. It is shown in [7] that weak quasi continuity is equivalent to weak semi-continuity due to Arya and Bhamini [8] and Kar and Bhattacharyya [9]. In 1990, Janković and Hamlett [10] introduced the concept of I -open sets in ideal topolog- ical spaces. Abd El-Monsef et al. [11] introduced and studied the concepts of I -closed sets and I -continuous functions. Semi-I -open sets, pre-I -open sets, α-I -open sets, β- I -open sets and δ-I -open sets play an important role in the research of generalizations of continuity in ideal topological spaces. Using these notions many authors introduced ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6572 Email addresses: butrakorn.k@msu.ac.th (B. Kong-ied), 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) B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6572 2 of 12 and studied various types of generalizations of continuity for functions. Hatir and Noiri [12] introduced and investigated the notions of weakly pre-I -open sets and weakly pre- I -continuous functions. Moreover, Hatir and Noiri [13] investigated further properties of semi-I -open sets and semi-I -continuous functions. On the other hand, the present authors introduced and investigated the concepts of pı-continuous functions [14], weakly pı-continuous functions [14], (τ1, τ2)-continuous functions [15], almost (τ1, τ2)-continuous functions [16] and weakly (τ1, τ2)-continuous functions [17]. Kong-ied et al. [18] intro- duced and studied the notion of almost quasi (τ1, τ2)-continuous functions. Chiangpradit et al. [19] introduced and investigated the concept of weakly quasi (τ1, τ2)-continuous func- tions. In this paper, we introduce new classes of functions between an ideal topological space and a bitopological space, namely almost quasi τ⋆(σ1, σ2)-continuous functions and weakly quasi τ⋆(σ1, σ2)-continuous functions. We also investigate several characterizations of almost quasi τ⋆(σ1, σ2)-continuous functions and weakly quasi τ⋆(σ1, σ2)-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 [20] 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 [20] 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 [20] of A and is denoted by τ1τ2-Int(A). Lemma 1. [20] 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 [21] (resp. (τ1, τ2)s- open [22], (τ1, τ2)p-open [22], (τ1, τ2)β-open [22]) 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 [23] if A ⊆ B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6572 3 of 12 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 [24] (resp. (τ1, τ2)s-closure [22], α(τ1, τ2)-closure [25]) 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 [24] (resp. (τ1, τ2)s-interior [22], α(τ1, τ2)-interior [25]) of A and is denoted by (τ1, τ2)-pInt(A) (resp. (τ1, τ2)-sInt(A), α(τ1, τ2)-Int(A)). Lemma 2. [18] For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: (1) (τ1, τ2)-sCl(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∪A [17]; (2) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A. For a subset A of a bitopological space (X, τ1, τ2), a point x ∈ X is called a (τ1, τ2)θ- cluster point 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 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 if (τ1, τ2)θ-Cl(A) = 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 of X contained in A is called the (τ1, τ2)θ-interior of A and is denoted by (τ1, τ2)θ-Int(A) [21]. Lemma 3. [21] For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: (1) If A is τ1τ2-open in X, then τ1τ2-Cl(A) = (τ1, τ2)θ-Cl(A). (2) (τ1, τ2)θ-Cl(A) is τ1τ2-closed in X. An ideal I on a topological space (X, τ) is a nonempty collection of subsets of X satisfying the following properties: (1) A ∈ I and B ⊆ A imply B ∈ I ; (2) A ∈ I and B ∈ I imply A ∪ B ∈ I . A topological space (X, τ) with an ideal I on X is called an ideal topological space and is denoted by (X, τ,I ). For an ideal topological space (X, τ,I ) and a subset A of X, A⋆(I ) is defined as follows: A⋆(I ) = {x ∈ X : U ∩A ̸∈ I for every open neighbourhood U of x}. In case there is no chance for confusion, A⋆(I ) is simply written as A⋆. In [26], A⋆ is called the local function of A with respect to I and τ and Cl⋆(A) = A⋆∪A defines a Kuratowski closure operator for a topology τ⋆(I ) finer than τ . A subset A is said to be ⋆-closed [10] if A⋆ ⊆ A. The interior of a subset A in (X, τ⋆(I )) is denoted by Int⋆(A). A subset A of an ideal topological space (X, τ,I ) is said to be semi⋆-I -open [27] (resp. semi-I -open [13]) if A ⊆ Cl(Int⋆(A)) (resp. A ⊆ Cl⋆(Int(A))). The complement of a semi⋆-I -open (resp. semi-I -open) set is said to be semi⋆-I -closed [27] (resp. semi-I -closed [13]). A subset A B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6572 4 of 12 of an ideal topological space (X, τ,I ) is said to be semi-I ⋆-open [28] if A ⊆ Cl⋆(Int⋆(A)). The complement of a semi-I ⋆-open set is called semi-I ⋆-closed [28]. For a subset A of an ideal topological space (X, τ,I ), the intersection of all semi-I -closed sets containing A is called the semi-I ⋆-closure [28] of A and is denoted by sCl⋆(A) (sClI ⋆(A) [28]). The union of all semi-I -open sets contained in A is called the semi-I ⋆-interior [28] of A and is denoted by sInt⋆(A) (sIntI ⋆(A) [28]). Lemma 4. [28] For a subset A of a an ideal topological space (X, τ,I ), the following properties hold: (1) sCl⋆(A) = A ∪ Int⋆(Cl⋆(A)); (2) sInt⋆(A) = A ∩ Cl⋆(Int⋆(A)). 3. Almost quasi τ ⋆(σ1, σ2)-continuous functions In this section, we introduce the concept of almost quasi τ⋆(σ1, σ2)-continuous func- tions. Moreover, some characterizations of almost quasi τ⋆(σ1, σ2)-continuous functions are considered. Definition 1. A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be almost quasi τ⋆(σ1, σ2)- continuous at a point x ∈ X if for every σ1σ2-open set V of Y containing f(x) and each ⋆-open set U of X containing x, there exists a nonempty ⋆-open set G such that G ⊆ U and f(G) ⊆ (σ1, σ2)-sCl(V ). A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be almost quasi τ⋆(σ1, σ2)-continuous if f is almost quasi τ⋆(σ1, σ2)-continuous at each point of X. Theorem 1. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost quasi τ⋆(σ1, σ2)-continuous at x ∈ X; (2) for every σ1σ2-open set V of Y containing f(x), there exists a semi-I ⋆-open set U of X containing x such that f(U) ⊆ (σ1, σ2)-sCl(V ); (3) x ∈ sInt⋆(f−1((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y containing f(x); (4) x ∈ Cl⋆(Int⋆(f−1((σ1, σ2)-sCl(V )))) for every σ1σ2-open set V of Y containing f(x). Proof. (1) ⇒ (2): Let U (x) be the family of all ⋆-open sets of X containing x. Let V be any σ1σ2-open set of Y containing f(x). For eachH ∈ U (x), there exists a nonempty ⋆- open set GH such that GH ⊆ H, f(GH) ⊆ (σ1, σ2)-sCl(V ). Let W = ∪{GH | H ∈ U (x)}. Then, W is ⋆-open in X and x ∈ Cl⋆(W ). Put U = W ∪ {x}. Then, U is a semi-I ⋆-open set of X containing x and f(U) ⊆ (σ1, σ2)-sCl(V ). (2) ⇒ (3): Let V be any σ1σ2-open set of Y containing f(x). Then, there exists a semi-I ⋆-open set U of X containing x such that f(U) ⊆ (σ1, σ2)-sCl(V ). Thus, x ∈ U ⊆ f−1((σ1, σ2)-sCl(V )) B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6572 5 of 12 and hence x ∈ U ⊆ sInt⋆(f−1((σ1, σ2)-sCl(V ))). (3) ⇒ (4): Let V be any σ1σ2-open set of Y containing f(x). Thus by (3), x ∈ sInt⋆(f−1((σ1, σ2)-sCl(V ))). Now, put U = sInt⋆(f−1((σ1, σ2)-sCl(V ))). Then, we have U is semi-I ⋆-open in X and by Lemma 4, x ∈ U ⊆ Cl⋆(Int⋆(U)) ⊆ Cl⋆(Int⋆(f−1((σ1, σ2)-sCl(V )))). (4) ⇒ (1): Let U be any ⋆-open set of X containing x and V be any σ1σ2-open set of Y containing f(x). Then, we have x ∈ Cl⋆(Int⋆(f−1((σ1, σ2)-sCl(V )))) and hence U ∩ Int⋆(f−1((σ1, σ2)-sCl(V ))) ̸= ∅. Put W = U ∩ Int⋆(f−1((σ1, σ2)-sCl(V ))). Then, W is a nonempty ⋆-open set of X such that W ⊆ U , f(W ) ⊆ (σ1, σ2)-sCl(V ). This shows that f is almost quasi τ⋆(σ1, σ2)-continuous at x. Theorem 2. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost quasi τ⋆(σ1, σ2)-continuous; (2) for each x ∈ X and every σ1σ2-open set V of Y containing f(x), there exists a semi-I ⋆-open set U of X containing x such that f(U) ⊆ (σ1, σ2)-sCl(V ); (3) f−1(V ) is semi-I ⋆-open in X for every (σ1, σ2)r-open set V of Y ; (4) f−1(V ) ⊆ sInt⋆(f−1((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y ; (5) sCl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ; (6) f−1(V ) ⊆ Cl⋆(Int⋆(f−1((σ1, σ2)-sCl(V )))) for every σ1σ2-open set V of Y . Proof. (1) ⇒ (2): The proof follows from Theorem 1. (2) ⇒ (3): Let V be any (σ1, σ2)r-open set of Y and x ∈ f−1(V ). Then, f(x) ∈ V and there exists a semi-I ⋆-open set U of X containing x such that f(U) ⊆ V . Thus, x ∈ U ⊆ f−1(V ) and hence x ∈ sInt⋆(f−1(V )). Therefore, f−1(V ) ⊆ sInt⋆(f−1(V )). This shows that f−1(V ) is semi-I ⋆-open in X. (3) ⇒ (4): Let V be any σ1σ2-open set of Y and x ∈ f−1(V ). Then, we have f(x) ∈ V ⊆ (σ1, σ2)-sCl(V ). Thus, x ∈ f−1((σ1, σ2)-sCl(V )). By Lemma 2, (σ1, σ2)-sCl(V ) is (σ1, σ2)r-open in Y . By (3), f−1((σ1, σ2)-sCl(V )) is semi-I ⋆-open in X and so x ∈ sInt⋆(f−1((σ1, σ2)-sCl(V ))). Thus, f−1(V ) ⊆ sInt⋆(f−1((σ1, σ2)-sCl(V ))). (4) ⇒ (5): Let B be any subset of Y . Then, we have Y − σ1σ2-Cl(B) is σ1σ2-open in Y . Thus by (4) and Lemma 2, X − f−1(σ1σ2-Cl(B)) = f−1(Y − σ1σ2-Cl(B)) B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6572 6 of 12 ⊆ sInt⋆(f−1((σ1, σ2)-sCl(Y − σ1σ2-Cl(B)))) = sInt⋆(X − f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) = X − sCl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) and hence sCl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ f−1(σ1σ2-Cl(B)). (5) ⇒ (6): Let V be any σ1σ2-open set of Y . Then, Y −V is σ1σ2-closed in Y . By (5) and Lemma 4, Int⋆(Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(Y − V ))))) ⊆ f−1(Y − V ) = X − f−1(V ). Moreover, we have Int⋆(Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(Y − V ))))) = Int⋆(Cl⋆(f−1(Y − σ1σ2-Int(σ1σ2-Cl(V ))))) = Int⋆(Cl⋆(X − f−1((σ1, σ2)-sCl(V )))) = X − Cl⋆(Int⋆(f−1((σ1, σ2)-sCl(V )))). Thus, f−1(V ) ⊆ Cl⋆(Int⋆(f−1((σ1, σ2)-sCl(V )))). (6) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). By (6), we have x ∈ f−1(V ) ⊆ Cl⋆(Int⋆(f−1((σ1, σ2)-sCl(V )))) and by Lemma 4, x ∈ f−1(V ) ⊆ sInt⋆(f−1((σ1, σ2)-sCl(V ))). Put U = sInt⋆(f−1((σ1, σ2)-sCl(V ))). Then, U is semi-I ⋆-open set of X containing x such that f(U) ⊆ (σ1, σ2)-sCl(V ). This shows that f is almost quasi τ⋆(σ1, σ2)-continuous. Theorem 3. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is almost quasi τ⋆(σ1, σ2)-continuous; (2) sCl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) sCl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ; (4) f−1(V ) ⊆ sInt⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) for every (σ1, σ2)p-open set V of Y . Proof. The proof follows from Theorem 2. B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6572 7 of 12 4. Weakly quasi τ ⋆(σ1, σ2)-continuous functions In this section, we introduce the notion of weakly quasi τ⋆(σ1, σ2)-continuous functions. Furthermore, several characterizations of weakly quasi τ⋆(σ1, σ2)-continuous functions are discussed. Definition 2. A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be weakly quasi τ⋆(σ1, σ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing f(x) and each ⋆-open set U of X containing x, there exists a nonempty ⋆-open set G such that G ⊆ U , f(G) ⊆ σ1σ2-Cl(V ). A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be weakly quasi τ⋆(σ1, σ2)-continuous if f is weakly quasi τ⋆(σ1, σ2)-continuous at each point of X. Remark 1. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following implication holds: almost quasi τ⋆(σ1, σ2)-continuity ⇒ weakly quasi τ⋆(σ1, σ2)-continuity. The converse of the implication is not true in general. We give an example for the implication as follows. Example 1. Let X = {1, 2, 3} with a topology τ = {∅, {1}, {2, 3}, X} and an ideal I = {∅, {1}}. Let Y = {a, b, c} with topologies σ1 = {∅, {a}, {b}, {a, b}, {a, c}, Y } and σ2 = {∅, {a}, {b}, {a, b}, Y }. A function f : (X, τ,I ) → (Y, σ1, σ2) is defined as follows: f(1) = a, f(2) = b and f(3) = c. Then, f is weakly quasi τ⋆(σ1, σ2)-continuous but f is not almost quasi τ⋆(σ1, σ2)-continuous. Theorem 4. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly quasi τ⋆(σ1, σ2)-continuous; (2) for each x ∈ X and every σ1σ2-open set V of Y containing f(x), there exists a semi-I ⋆-open set of X containing x such that f(U) ⊆ σ1σ2-Cl(V ); (3) Int⋆(Cl⋆(f−1(σ1σ2-Int(K)))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ; (4) f−1(V ) ⊆ sInt⋆(f−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (5) sCl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open sets V of Y . Proof. (1) ⇒ (2): Let U (x) be the family of all ⋆-open sets of X containing x. Let V be any σ1σ2-open set of Y containing f(x). For eachH ∈ U (x), there exists a nonempty ⋆- open set GH such that GH ⊆ H and f(GH) ⊆ σ1σ2-Cl(V ). Put W = ∪{GH | H ∈ U (x)}. Then, W is ⋆-open in X and x ∈ Cl⋆(W ). Let U = W ∪ {x}. Then, U is a semi-I ⋆-open set of X containing x and f(U) ⊆ σ1σ2-Cl(V ). (2) ⇒ (4): Let V be any σ1σ2-open set of Y and x ∈ f−1(V ). Then, f(x) ∈ V and there exists a semi-I ⋆-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). Thus, x ∈ U ⊆ sInt⋆(f−1(σ1σ2-Cl(V ))) B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6572 8 of 12 and hence f−1(V ) ⊆ sInt⋆(f−1(σ1σ2-Cl(V ))). (4) ⇒ (5): Let V be any σ1σ2-open set of Y . Then by (4), we have X − f−1(σ1σ2-Cl(V )) = f−1(Y − σ1σ2-Cl(V )) ⊆ sInt⋆(f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))) = sInt⋆(f−1(Y − σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ sInt⋆(f−1(Y − V )) = sInt⋆(X − f−1(V )) = X − sCl⋆(f−1(V )) and so sCl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )). (5) ⇒ (3): Let K be any σ1σ2-closed set of Y . Thus by (5) and Lemma 4, Int⋆(Cl⋆(f−1(σ1σ2-Int(K)))) ⊆ sCl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ f−1(σ1σ2-Cl(K)) = f−1(K). (3) ⇒ (4): Let V be any σ1σ2-open set of Y . By (3) and Lemma 4, X − sInt⋆(f−1(σ1σ2-Cl(V ))) = sCl⋆(f−1(Y − σ1σ2-Cl(V ))) ⊆ f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V ))) = f−1(Y − σ1σ2-Int(σ1σ2-Cl(V ))) ⊆ f−1(Y − V ) = X − f−1(V ) and hence f−1(V ) ⊆ sInt⋆(f−1(σ1σ2-Cl(V ))). (4) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). By (4), we have f−1(V ) ⊆ sInt⋆(f−1(σ1σ2-Cl(V ))). Put U = sInt⋆(f−1(σ1σ2-Cl(V ))). Then, U is a semi-I ⋆-open set of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). This shows that f is weakly quasi τ⋆(σ1, σ2)-continuous. Theorem 5. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly quasi τ⋆(σ1, σ2)-continuous; (2) sCl⋆(f−1(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (3) sCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every subset B of Y ; (4) sCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ; B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6572 9 of 12 (5) sCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (6) sCl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(K) for every (σ1, σ2)r-closed set K of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . Since (σ1, σ2)θ-Cl(B) is σ1σ2-closed in Y , by Theorem 4 we have Int⋆(Cl⋆(f−1(σ1σ2-Int((σ1, σ2)θ-Cl(B))))) ⊆ f−1((σ1, σ2)θ-Cl(B)) and by Lemma 4, sCl⋆(f−1(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)). (2) ⇒ (3): This is obvious since σ1σ2-Cl(B) ⊆ (σ1, σ2)θ-Cl(B) for every subset B of Y . (3) ⇒ (4): This is obvious since σ1σ2-Cl(V ) = (σ1, σ2)θ-Cl(V ) for every σ1σ2-open set V of Y . (4) ⇒ (5): Let V be any (σ1, σ2)p-open set of Y . Then, V ⊆ σ1σ2-Int(σ1σ2-Cl(V )) and σ1σ2-Cl(V ) = σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))). Now, put G = σ1σ2-Int(σ1σ2-Cl(V )), then G is σ1σ2-open in Y and σ1σ2-Cl(G) = σ1σ2-Cl(V ). Thus by (4), sCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )). (5) ⇒ (6): Let K be any (σ1, σ2)r-closed set of Y . Since σ1σ2-Int(K) is (σ1, σ2)p-open in Y and by (5), we have sCl⋆(f−1(σ1σ2-Int(K))) = sCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))))) ⊆ f−1(σ1σ2-Cl(σ1σ2-Int(K))) = f−1(K). (6) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, σ1σ2-Cl(V ) is (σ1, σ2)r-closed in Y and by (6), sCl⋆(f−1(V )) ⊆ sCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )). It follows from Theorem 4 that f is weakly quasi τ⋆(σ1, σ2)-continuous. Theorem 6. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly quasi τ⋆(σ1, σ2)-continuous; (2) sCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ; (3) sCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ; (4) sCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) 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, we have V ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))) B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6572 10 of 12 and hence σ1σ2-Cl(V ) = σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))). Since σ1σ2-Cl(V ) is (σ1, σ2)r- closed in Y and by Theorem 5, sCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )). (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 . Since σ1σ2-Cl(V ) is (σ1, σ2)r-closed, we have σ1σ2-Cl(V ) is (σ1, σ2)s-open in Y . Thus by (3), sCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ 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), sCl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )). It follows from Theorem 5 that f is weakly quasi τ⋆(σ1, σ2)-continuous. Theorem 7. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are equivalent: (1) f is weakly quasi τ⋆(σ1, σ2)-continuous; (2) Int⋆(Cl⋆(f−1(V ))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (3) sCl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ; (4) f−1(V ) ⊆ sInt⋆(f−1(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y . Proof. (1) ⇒ (2): Let V be any (σ1, σ2)p-open set of Y . Since f is weakly quasi τ⋆(σ1, σ2)-continuous, by Theorem 5 and Lemma 4 Int⋆(Cl⋆(f−1(V ))) ⊆ Int⋆(Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V ))))) ⊆ f−1(σ1σ2-Cl(V )). (2) ⇒ (3): Let V be any (σ1, σ2)p-open set of Y . By (2) and Lemma 4, we have sCl⋆(f−1(V )) = f−1(V ) ∪ Int⋆(Cl⋆(f−1(V ))) ⊆ f−1(σ1σ2-Cl(V )). (3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . Thus by (3), X − sInt⋆(f−1(σ1σ2-Cl(V ))) = sCl⋆(X − (f−1(σ1σ2-Cl(V )))) = sCl⋆(X − f−1(σ1σ2-Cl(V ))) = sCl⋆(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 ) ⊆ sInt⋆(f−1(σ1σ2-Cl(V ))). (4) ⇒ (1): Since every σ1σ2-open set is (σ1, σ2)p-open, this follows from Theorem 4. B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6572 11 of 12 Acknowledgements This research project was financially supported by Mahasarakham University. References [1] S. Marcus. Sur les fonctions quasicontinues au sense de S. Kempisty. Colloquium Mathematicum, 8:47–53, 1961. [2] V. Popa. On the decomposition of the quasi-continuity in topological spaces (Roma- nian). Studii şi Cercetǎri de Matematicǎ, 30:31–35, 1978. [3] A. Neubrunnová. On certain generalizations of the notion of continuity. Matematický C̆asopis, 23:374–380, 1973. [4] N. Levine. Semi-open sets and semi-continuity in topological spaces. The American Mathematical Monthly, 70:36–41, 1963. [5] V. Popa and C. Stan. On a decomposition of quasicontinuity in topological spaces. Studii şi Cercetǎri de Matematicǎ, 25:41–43, 1973. [6] N. Levine. A decomposition of continuity in topological spaces. The American Math- ematical Monthly, 68:44–46, 1961. [7] T. Noiri. Properties of some weak forms of continuity. International Journal of Mathematics and Mathematical Sciences, 10:97–111, 1987. [8] S. P. Arya and M. P. Bhamini. Some weaker forms of semi-continuous functions. Ganita, 33:124–134, 1982. [9] A. Kar and P. Bhattacharyya. Weakly semi-continuous functions. The Journal of the Indian Academy of Mathematics, 8:83–93, 1986. [10] D. Janković and T. R. Hamlett. New topologies from old via ideals. The American Mathematical Monthly, 97:295–310, 1990. [11] M. E. Abd El-Monsef, E. F. Lashien, and A. A. Nasef. On I -open sets and I - continuous functions. Kyungpook Mathematical Journal, 32:21–30, 1992. [12] E. Hatir and T. Noiri. Weakly pre-I-open sets and decomposition of continuity. Acta Mathematica Hungarica, 106(3):227–238, 2005. [13] E. Hatir and T. Noiri. On decompositions of continuity via idealization. Acta Math- ematica Hungarica, 96:341–349, 2002. [14] C. Boonpok. pı-continuity and weak pı-continuity. Carpathian Mathematical Publi- cations, 17(1):171–186, 2025. [15] C. Boonpok and N. Srisarakham. (τ1, τ2)-continuity for functions. Asia Pacific Jour- nal of Mathematics, 11:21, 2024. [16] C. Boonpok and P. Pue-on. Characterizations of almost (τ1, τ2)-continuous multi- functions. International Journal of Analysis and Applications, 22:33, 2024. [17] C. Boonpok and C. Khanarong. On weakly (τ1, τ2)-continuous functions. European Journal of Pure and Applied Mathematics, 17(1):416–425, 2024. [18] B. Kong-ied, S. Sompong, and C. Boonpok. Almost quasi (τ1, τ2)-continuous func- tions. Asia Pacific Journal of Mathematics, 11:64, 2024. B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6572 12 of 12 [19] M. Chiangpradit, S. Sompong, and C. Boonpok. Weakly quasi (τ1, τ2)-continuous functions. International Journal of Analysis and Applications, 22:125, 2024. [20] 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. [21] C. Viriyapong and C. Boonpok. (τ1, τ2)α-continuity for multifunctions. Journal of Mathematics, 2020:6285763, 2020. [22] C. Boonpok. (τ1, τ2)δ-semicontinuous multifunctions. Heliyon, 6:e05367, 2020. [23] N. Viriyapong, S. Sompong, and C. Boonpok. (τ1, τ2)-extremal disconnectedness in bitopological spaces. International Journal of Mathematics and Computer Science, 19(3):855–860, 2024. [24] N. Viriyapong, S. Sompong, and C. Boonpok. Upper and lower s-(τ1, τ2)p-continuous multifunctions. European Journal of Pure and Applied Mathematics, 17(3):2210–2220, 2024. [25] C. Viriyapong, S. Sompong, and C. Boonpok. Upper and lower slight α(τ1, τ2)- continuity. European Journal of Pure and Applied Mathematics, 17(3):2142–2154, 2024. [26] K. Kuratowski. Topology, Vol. I. Academic Press, New York, 1966. [27] E. Ekici and T. Noiri. ⋆-extremally disconnected ideal topological spaces. Acta Mathematica Hungarica, 122:81–90, 2009. [28] C. Boonpok. Weak quasi continuity for multifunctions in ideal topological spaces. Advances in Mathematics: Scientific Journal, 9(1):339–355, 2020.