Open Access. © 2022 Bishwambhar Roy and Ritu Sen, published by De Gruyter. This work is licensed under the Creative Com- mons Attribution 4.0 License. Topol. Algebra Appl. 2022; 10:154–160 Research Article Open Access Bishwambhar Roy* and Ritu Sen On ω * -open sets and decomposition of continuity https://doi.org/10.1515/taa-2022-0121 Received 17 July, 2019; accepted 29 August, 2022 Abstract: In this paper our main interest is to introduce a new topology on an ideal topological space. We have studied some properties of this newly defined topology. It is shown that X is Lindel¨ of with respect to this new topology if and only if it is Lindel¨ of with respect to the old one. We have introduced and studied ω * -continuity and some related notions. Finally we have given decomposition theorem for continuity and ω * -continuity. Keywords: countable set, ideal, ω * -open set, ω * -continuity MSC: 54A10, 54C08 1 Introduction The concept of ideals on topological spaces was studied by Kuratowski [1] and Vaidyana-thasamay [2] which is one of the important area of research in the branch of mathematics. After then different mathematicians applied the concept of ideals in topological spaces (see [2–4]). Hdeib [5] in 1982 introduced the concept of ω- open sets to study ω-closed mappings. Meanwhile Al-Hawary et. al. [6] introduced the concept of ω 0 -sets on topological spaces. Recall that an ideal [1] I on a topological space (X , τ) is a non-empty collection of subsets of X with the following properties : (i) A  B and B ∈ I ⇒ A ∈ I (ii) A ∈ I, B ∈ I ⇒ A ∪ B ∈ I. A topological space (X , τ) with an ideal I on it is denoted by (X , τ , I) and known as an ideal topological space. Let I be an ideal on a topological space (X , τ). For any A  X, A * (I, τ)= {x ∈ X : A ∩ U ∉ I for each open set U containing x}. If there is no confusion, we will write A * for A * (I, τ). The operator cl * : P(X) → P(X) defined by cl * (A)= A ∪ A * is a Kuratowski closure operator [3] and hence gives a topology τ * (called *-topology) on X which is finer than the original one. Basic open sets of τ * are given by {U \ A : U ∈ τ , A ∈ I}, where I is the ideal on the topological space (X , τ) (see [3] for details). A subset A of a topological space (X , τ) is called ω-open [7] if for each x ∈ A, there exists some open set U of X containing x such that U \ A is countable. The collection of all ω-open sets of a topological space (X , τ) is denoted by τ ω which is a topology on X finer than τ. A subset A of a topological space (X , τ) is called ω 0 -open [6] if for each x ∈ A, there exists some open set U of X containing x such that U \ int(A) is countable. In this paper we have introduced another topology denoted by ω * O(X) other than the given topology τ and the ideal topology τ * on X. Recall that a function f :(X , τ) → (Y , σ) is called ω-continuous [7] if f −1 (V ) is ω-open for each open subset V of (Y , σ). In section 2, we have introduced the concept of ω * -open sets and studied the properties of such sets. We have shown that (X , τ) is Lindel ¨ of if and only if (X , ω * O(X)) is so. In section 3, we have introduced the concept of ω * -continuous functions and some related functions. We have shown that Lindel ¨ ofnes is preserved under *Corresponding Author: Bishwambhar Roy: Department of Mathematics Women’s Christian College 6, Greek Church Row, Kolkata-700 026,India, E-mail : bishwambhar_roy@yahoo.co.in Ritu Sen: Department of Mathematics, Presidency University, 86/1 College Street, Kolkata-700073, India, E-mail : ritu_sen29@yahoo.co.in