Abstract
An ideal topological space is a triplet (X, τ, ℑ), where X is a nonempty set, τ is a topology on X, and ℑ is an ideal of subsets of X. In this paper, we introduce L∗-perfect, R∗-perfect, and C∗-perfect sets in ideal spaces and study their properties. We obtained a characterization for compatible ideals via R∗-perfect sets. Also, we obtain a generalized topology via ideals which is finer than τ using R∗-perfect sets on a finite set.
Highlights
Introduction and PreliminariesThe contributions of Hamlett and Jankovic [1,2,3,4] in ideal topological spaces initiated the generalization of some important properties in general topology via topological ideals
A nonempty collection of subsets of a set X is said to be an ideal on X, if it satisfies the following two conditions: (i) If A ∈ I and B ⊆ A, B ∈ I; (ii) If A ∈ I and B ∈ I, A∪B ∈ I
We obtain a new topology for the finite topological spaces which is finer than τ∗-topology
Summary
Introduction and PreliminariesThe contributions of Hamlett and Jankovic [1,2,3,4] in ideal topological spaces initiated the generalization of some important properties in general topology via topological ideals. Let (X, τ) be a topological space with an ideal I defined on X. A subset A of an ideal space (X, τ, I) is said to be (i) τ∗-closed [3] if A∗ ⊆ A, (ii) ∗-dense-in-itself [10] if A ⊆ A∗, (iii) I-open [11] if A ⊆ int(A∗), (iv) almost I-open [12] if A ⊆ cl(int(A∗)), (v) I-dense [7] if A∗ = X, (vi) almost strong β-I-open [13] if A ⊆ cl∗(int(A∗)), (vii) ∗-perfect [10] if A = A∗, (viii) regular I-closed [14] if A = (int(A))∗, (ix) an fI-set [15] if A ⊆ (int(A))∗.
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