Combinatorics of open covers (VIII)

Selection principles and hyperspace topologies

“…In [3] a covering property of a space (U fin (Γ, Ω) in the notation from [12]) which is intermediate between the Hurewicz property and the Menger property was characterized as an S fin property, namely as S fin (Ω, O wgp ). In [5] the property S fin (K, O k-wgp ) was considered.…”

Selection principles in relator spaces

“…A cover U of a space X is a weakly groupable cover [4] if it is a union of countably many nite, pairwise disjoint subfamilies U n such that for each nite subset F of X there is an n such that F ⊂ ∪ U n . In other words, if we put U = {U n : n ∈ N}, then there is a sequence n 1 < n 2 < · · · < n k < · · · of natural numbers such that each nite subset F of X is contained in…”

Relative versions of some star-selection principles

“…An open cover U of X is groupable [12] if it can be expressed as a countable union of nite, pairwise disjoint subfamilies U n , n ∈ N, such that each x ∈ X belongs to ∪ U n for all but nitely many n. An open cover U of X is weakly groupable [2] if it can be expressed as a countable union of nite, pairwise disjoint subfamilies U n , n ∈ N, such that each nite subset F of X, F ⊂ ∪ U n for some n.…”

“…For other notions and terminology we follow [8]. For general theory of selection principles see [2], [3], [1], [12], [13] and for some relative selection principles see [6], [10].…”