Combinatorial covering properties in an uncountable setting: canonical examples

2026/06/15 by Ayelet Amsalem, Adi Jarden, Michał Pawlikowski +3 · 1 voice
Mathematics · #math.GN #math.LO

paper · pdf

Abstract

We provide examples of spaces satisfying generalized combinatorial covering properties such as the Hurewicz, Menger, and γ-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 σ-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 γ-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.

Citations

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Discussions

• Ayelet Amsalem, Adi Jarden, Micha{\l} Pawlikowski, Piotr Szewczak, Boaz Tsaban, Lyubomyr Zdomskyy: Combinatorial covering properties in an uncountable setting: canonical examples https://arxiv.org/abs [bsky, 0 points, 0 comments]

Related

• The combinatorics of open covers (II)
• Menger's and Hurewicz's Problems: Solutions from "The Book" and refinements
• Small Hurewicz and Menger sets which have large continuous images
• Mildly version of Hurewicz Basis covering property and Hurewicz measure zero spaces
• Unbounded towers and products