On Star-K-I-Hurewicz Property

A space X is said to have the star-K-I-Hurewicz property (SKIH) [Tyagi, B. K.—Singh, S.—Bhardwaj, M. Ideal analogues of some variants of Hurewicz property, Filomat 33 (2019), no. 9, 2725–2734] if for each sequence (Un : n ∈ ℕ) of open covers of X there is a sequence (Kn : n ∈ ℕ) of compact subsets of X such that for each x ∈ X, {n ∈ ℕ : x ∉ St(Kn, Un )} ∈ I, where I is the proper admissible ideal of ℕ. In this paper, we continue to investigate the relationship between the SKIH property and other related properties and study the topological properties of the SKIH property.

The Absolutely Strongly Star-Hurewicz Property with Respect to an Ideal

Aspace X is said to have the absolutely strongly star -𝒤-Hurewicz (ASS𝒤H) property if for each sequence (𝒰n : n ∈ 𝕅)of opencovers of X and each dense subset Y of X, there is a sequence (Fn : n ∈ 𝕅) of finite subsets of Y such that for each x ∈ X, {n ∈ 𝕅 : x ∉ St(Fn, 𝒰n)}∈ 𝒤, where 𝒤 is the proper admissible ideal of 𝕅. In this paper, we investigate the relationship between the ASS𝒤H property and other related properties and study the topological properties of the ASS𝒤H property. This paper generalizes several results of Song [25] to the larger class of spaces having the ASS𝒤H properties.

Counterexample of theorems on star versions of Hurewicz property

In this paper, an example contradicting Theorem 4.5 and Theorem 5.3 is provided and these theorems are proved under some extra hypothesis.

Ideal analogues of some variants of the Hurewicz property

In this paper, we continue the study on the ideal analogues of several variations of the Hurewicz property introduced by Das et al. [4, 6, 7] for example, the I-Hurewicz (IH), the star I-Hurewicz (SIH), the weakly I-Hurewicz (WIH) and the weakly star-I-Hurewicz (WSIH). It is shown that several implications in the relationship diagram of their concepts are reversible under certain conditions, for instance; (1) If a paracompact Hausdorff space has the WSIHproperty, then it has theWIHproperty. (2) If the complement of dense set has the IHproperty, then theWIHproperty implies the IHproperty and (3) If the complement of dense set has the SIH property, then the WSIH property implies the SIH property. In addition, we introduce the ideal analogues of some new variations of the Hurewicz property called the mildly I-Hurewicz and the star K-I-Hurewicz properties and explore their relationships with other variants of the I-Hurewicz property. We also study the preservation properties under certain mappings.

PRODUCTS OF HUREWICZ SPACES IN THE LAVER MODEL

This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We illustrate this interplay by proving that in the Laver model for the consistency of the Borel’s conjecture, the product of any two metrizable spaces with the Hurewicz property has the Menger property.