Abstract
ℐ‐open sets were introduced and studied by Janković and Hamlett (1990) to generalize the well‐known Banach category theorem. Quasi‐ℐ‐openness was introduced and studied by Abd El‐Monsef et al. (2000). These are ∗‐dense‐in‐itself sets of the ideal spaces. In this note, properties of these sets are further investigated and characterizations of these sets are given. Also, their relation with ℐ‐dense sets and ℐ‐locally closed sets is discussed. Characterizations of completely codense ideals are given in terms of semi‐preopen sets.
Highlights
Introduction and preliminariesThe subject of ideals in topological spaces has been studied by Kuratowski [12] and Vaidyanathaswamy [20]
Given a topological space (X, τ) with an ideal Ᏽ on X and if ℘(X) is the set of all subsets of X, a set operator (·)∗ : ℘(X) → ℘(X), called a local function [12] of A with respect to Ᏽ and τ, is defined as follows: for A ⊂ X, A∗(Ᏽ, τ) = {x ∈ X | U ∩ A ∈ Ᏽ for every U ∈ τ(x)}, where τ(x) = {U ∈ τ | x ∈ U }
The largest semiopen set contained in A is called the semi-interior of A and is denoted by sint(A)
Summary
Introduction and preliminariesThe subject of ideals in topological spaces has been studied by Kuratowski [12] and Vaidyanathaswamy [20]. A subset A of a space (X, τ) is semiopen [13] if there exists an open set G such that G ⊂ A ⊂ cl(G) or, equivalently, A ⊂ cl(int(A)). A subset A of an ideal space (X, τ, Ᏽ) is τ∗-closed [10] (resp., ∗-dense in itself [9], ∗-perfect [9]) if A∗ ⊂ A (resp., A ⊂ A∗, A = A∗). A subset A of an ideal space (X, τ, Ᏽ) is Ᏽ-locally closed, [5] if A = G ∩ V , where G is open and V is ∗-perfect.
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