Star-Hurewicz and related properties

Abstract

Abstract. We continue the investigation of star selection principles first considered in [9]. We are concentrated onto star versions of the Hurewicz covering property and star selection principles related to the classes of open covers which have been recently introduced.2000 AMS Classification: 54D20.

Relative versions of some star-selection principles

“…, y p ) ∈ St (B n , W n ) ⊂ St (S p n , W n ) and consequently H ⊂ St (S n , U n ). Corollary 2.4 [4]. If for every n ∈ N, X n is SSM, then SS * f in (O, Ω) holds.…”

“…Since each countable γ-cover is groupable, the proof follows. Corollary 2.3 [4]. For a space X, the following conditions are equivalent:…”

“…Let A be an almost disjoint family of cardinality < b. In [4] it is proved that A is relatively SSH in Ψ(A); since A is closed in Ψ(A), we have that A is relatively closed SSH in Ψ(A). Since the subspace A of Ψ(A) is the discrete space of cardinality b, it can not be SSH.…”

“…In [4] some relative versions of SSH property were introduced. In particular it was introduced the following denition: Definition 2.11 [4].…”

“…In [4] some relative versions of SSH property were introduced. In particular it was introduced the following denition: Definition 2.11 [4]. A subset Y of a space X is relatively SSH in X if for each sequence (U n : n ∈ N) of open covers of X there is a sequence (A n : n ∈ N) of nite subsets of X such that each point y ∈ Y belongs to all but nitely many sets St (A n , U n ) (i.e., if SS * f in (O X , Γ Y X ) holds).…”