Subsets of ideal topological spaces

Abstract

Properties of α-I-open sets, t-I-sets, strong β-I-open sets, S βI -sets and S-I-sets in ideal topological spaces are discussed. Also, we define a new class of sets called semi-I-locally closed sets which contains the class of all Ilocally closed sets and is contained in the class of all semilocally closed sets.

A stronger form of locally closed set and its homeomorphic image

“…Definition 4. An L ⊆ Z is addressed as I-locally closed [6] (resp., semi-I-locally closed [12]) if we can present L as L = H ∩ K, where H ∈ T and K is ⋆-perfect (resp., L = H ∩ L ⋆ , where H is semi-open). An equivalent definition of L to be I-locally closed is L = H ∩ L ⋆ , where H ∈ T (see [12]).…”

“…Further, {ℓ 3 , ℓ 4 } is I-locally closed but not ⋆-Locally closed; {ℓ 2 , ℓ 4 } is semi-I-locally closed but not ⋆-Locally closed. Here, ⋆-Locally closed sets are precisely ∅, {ℓ 2 }, {ℓ 4 } and {ℓ 2 , ℓ 3 , ℓ 4 }, and these are also I-locally closed and hence, they are semi-⋆-locally closed (as we know from [12] that I-locally closed implies semi-I-locally closed). Because {ℓ 4 } is ⋆-Locally closed is locally closed and hence, λ-locally closed (since locally closed implies λ-locally closed [20]), whereas {d} in Example 2.3 of [20] λ-locally closed but not ⋆-Locally closed.…”

“…But the study of a locally closed set relative to an ideal (see [13]) is a new idea, and this has been introduced through this paper. The authors Jeyanthi et al [12] and the author Dontchev [6] have studied locally closed sets in terms of ideal, but these locally closed sets differ somewhat from the current one.…”

“…Locally closed set and its study is not a new idea in topology. This notion was disclosed by Bourbaki [3], and after that it has been extensively studied by a good number of mathematicians (see [7,12,20,21]). This study has been interesting because it generalizes both open and closed sets.…”