Some remarks on open covers and selection principles using ideals

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

“…A space X is said to have the I-Hurewicz property [4,6,20] (in short, IH) if for each sequence (U n : n ∈ N) of open covers of X there is a sequence (V n : n ∈ N) such that for each n ∈ N, V n is a finite subset of U n and for each x ∈ X, {n ∈ N : x V n } ∈ I. A space X is said to have the star-I-Hurewicz property [6] (in short, SIH) if for each sequence (U n : n ∈ N) of open covers of X there is a sequence (V n : n ∈ N) such that for each n ∈ N, V n is a finite subset of U n and for each x ∈ X, {n ∈ N : x St(∪V n , U n )} ∈ I.…”

“…An open cover U of X is said to be large [13,19] if for each point there are infinitely many sets in U containing that point. A countable open cover U = {U n : n ∈ N} of X is said to be an I-large cover [7] if for each x ∈ X the set {n ∈ N : x ∈ U n } ∈ I + . The collection of all I-large cover will be denoted by I − Λ.…”

“…The collection of all I-large cover will be denoted by I − Λ. An open cover U of X is said to be I-groupable [4] if it can be expressed in the form U = n∈N U n , where U n 's are finite, pairwise disjoint and for each x ∈ X, {n ∈ N :…”

“…Our next result gives certain conditions under which I-groupabable open cover coincides with large open covers. For this we need the following theorem from Das, Kočinac and Chandra [7]. Theorem 5.4.…”

“…Di Maio and Kočinac [9] introduced certain types of open covers and selection principles using the ideal of asymptotic density zero sets in N. Now a natural question arises whether this investigation can be extended to arbitrary ideals instead of special type of ideal. Das, Chandra and Kočinac [2][3][4][5]7] studied the open covers and selection principle using arbitrary ideals of N. Further Das et al [4,6] defined the ideal analogues of some variants of the Hurewicz property such as the I-Hurewicz (IH), the star-I-Hurewicz (SIH) and the strongly star-I-Hurewicz (SSIH), where I is a proper admissible ideal of N. Following this study, the authors [28] investigated the ideal analogue of the star-K-Hurewicz property called the star-K-I-Hurewicz (SKIH) (also see [29]).…”