We continue to investigate various diagonalization properties for sequences of open covers of separable metrizable spaces introduced in Part I. These properties generalize classical ones of Rothberger, Menger, Hurewicz, and Gerlits-Nagy. In particular, we show that most of the properties introduced in Part I are indeed distinct. We characterize two of the new properties by showing that they are...

Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (τ -covers). We deal with two types of combinatorial questions which arise from this study. (1) Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum. (2) We study the additiv...

Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (τ -covers). We deal with two types of combinatorial questions which arise from this study. 1. Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum. 2. We study the additivit...

In this work, we introduce the notions of Star-?K and absolutely spaces which allow us to unify results among several properties in theory star selection principles on small spaces. particular, selective versions Menger, Hurewicz Rothberger property (a) regarding size space. Connections other well-known are mentioned. Furthermore, absolute version neighbourhood principle introduced. As an appli...

The additivity number of a topological property (relative to a given space) is the minimal number of subspaces with this property whose union does not have the property. The most well-known case is where this number is greater than א0, i.e. the property is σ-additive. We give a rather complete survey of the known results about the additivity numbers of a variety of topological covering properti...

We explore the connections between selection games on Hausdorff spaces and their corresponding Vietoris space of compact subsets. These considerations offer a similar relationship as well-known ?-covers X ordinary open covers finite powers X. The primary utility this method is to establish relationships with k-covers Particularly, we show that some commonly studied principles are equivalent rel...

We study when the product of separable metric spaces has the selective screenability property, the Menger property, or the Rothberger property. Our results imply The product of a Lusin set and (1) a Sierpinski set always has the Menger property (Corollary 12); (2) a γ-set always has the Rothberger property (Corollary 13). All topological spaces considered in this paper are assumed to be separab...