An ideal version of the star-C-Hurewicz covering property

Abstract

A space X is said to have the star-C-I-Hurewicz (SCIH) property if for each sequence (U n : n ∈ N) of open covers of X there is a sequence (K n : n ∈ N) of countably compact subsets of X such that for each x ∈ X, {n ∈ N : x St(K n , U n)} ∈ I, where I is a proper admissible ideal of N. We investigate the relationships among the SCIH and related properties. We study the topological properties of the SCIH property. This paper generalizes several results of [21, 24] to the larger class of spaces having the SCIH property. The star-C-I-Hurewicz game is introduced. It is shown that, in a paracompact Hausdorff space X, if TWO has a winning strategy in the SCIH game on X, then TWO has a winning strategy in the I-Hurewicz game on X.

The almost I-Hurewicz covering property

“…Maio and Kocinac [17] introduced certain types of open covers and selection principles using the ideal of asymptotic density zero of N. Das, Chandra and Kocinac (see [1][2][3][4][5]) studied the open covers and selection principles using arbitrary ideals of N (also see [25]). Further Das et al (see [1,5]) defined the ideal analogues of some variants of the Hurewicz property such as the I-Hurewicz, the star-I-Hurewicz and the weakly I-Hurewicz, where I is the proper admissible ideal of N. Recently authors continued (see [21,25]) the study of Hurewicz covering properties using ideal.…”

On Star-K-I-Hurewicz Property

“…Di Maio and Kocinac [14] introduced the statistical analogues of certain types of open covers and selection principles using actually the ideal of asymptotic density zero sets of N. Das, Kocinac and Chandra [2,3] extended this study to the arbitrary ideal of N. Using the notions of ideals, they started a more general approach to study certain results of open covers and selection principles. Further, Das et al [4] studied the ideal analogues of the Hurewicz, the star-Hurewicz, and the strongly star-Hurewicz properties called them the I-Hurewicz (see [1-3, 17, 19, 24]), the star-I-Hurewicz and the strongly star-I--Hurewicz properties, respectively, where I is the proper admissible ideal of N. In [18,25], Singh et al introduced the ideal versions of the star-K-Hurewicz and the star-C-Hurewicz properties called the star-K-I-Hurewicz (SKIH) and the star-C-I-Hurewicz properties, respectively.…”

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

“…Using the notions of ideals, they started a more general approach to study certain results of open covers and selection principles. Further, Das et al [8] studied the ideal version of the SSH property called the strongly star-I-Hurewicz (SSIH) property, where I is the proper admissible ideal of N. Singh, Tyagi and Bhardwaj [22,27] studied the ideal version of star-K-Hurewicz and star-C-Hurewicz properties.…”