C-compactness modulo an ideal

In the present paper, we consider a topological space equipped with an ideal, a theme that has been treated by Vaidyanathaswamy [15] and Kuratowski [6] in their classical texts. An ideal on a set X is a nonempty subset of P(X), the power set of X , which is closed for subsets and finite unions. An ideal is also called a dual filter. {φ} and P(X) are trivial examples of ideals. Some useful ideals are (i) f , the ideal of all finite subsets of X , (ii) c, the ideal of all countable subsets of X , (iii) n , the ideal of all nowhere dense subsets in a topological space (X ,τ), and (iv) s, the set of all scattered sets in (X ,τ). For an ideal on X and A⊂ X , we denote the ideal {I ∩A : I ∈ } by A. A topological space (X ,τ) with an ideal on X is denoted by (X ,τ, ). For a subset A⊆ X , A∗( ,τ) (called the adherence of A modulo an ideal ) or A∗( ) or just A∗ is the set {x ∈ X : A∩U / ∈ for every open neighborhood U of x}. A∗( ,τ) has been called the local function of A with respect to in [6]. It is easy to see that (i) for the ideal {φ}, A∗ is the closure of A, (ii) for the ideal P(X), A∗ is φ, and (iii) for ideal f , A∗ is the set of all ω-accumulation points of A. For general properties of the operator ∗, we refer the readers to [5, 14]. Observe that the operator cl∗ : P(X)→P(X) defined by cl∗(A)=A∪A∗ is a Kuratowski closure operator on X and hence generates a topology τ∗( ) or just τ∗ on X finer than τ. As has already been observed, τ∗({φ}) = τ and τ∗(P(X)) = the discrete topology. A description of open sets in τ∗( ) as given in Vaidyanathaswamy [15] is given in the following.