Combinatorial covering properties in an uncountable setting: canonical examples

Abstract

We provide examples of spaces satisfying generalized combinatorial covering properties such as the Hurewicz, Menger, and ${\hat{I}}^3$-properties in an uncountable setting. Our approach is motivated by canonical constructions from the classical countable case, including the examples of Bartoszyński and Shelah separating the Hurewicz property from $\ddot{I}ƒ$-compactness, the examples of Tsaban and Zdomskyy separating the Hurewicz and Menger properties, and Tsaban's construction of a nontrivial set of reals with the ${\hat{I}}^3$-property. We focus on the genuinely nontrivial aspects of these higher-cardinal generalizations, uncovering several open problems whose nature appears substantially different from that of their countable counterparts.