Linear $\sigma$-additivity and some applications

Abstract

We show that countable increasing unions preserve a large family of well-studied covering properties, which are not necessarily sigma-additive. Using this, together with infinite-combinatorial methods and simple forcing theoretic methods, we explain several phenomena, settle problems of Just, Miller, Scheepers and Szeptycki [COC2], Gruenhage and Szeptycki [FUfin], Tsaban and Zdomskyy [SFT], and Tsaban [o-bdd, OPiT], and construct topological groups with very strong combinatorial properties.

Cited by in corpus (6)

• Mathias forcing and combinatorial covering properties of filters
• Selective covering properties of product spaces
• Pointwise convergence of partial functions: The Gerlits-Nagy Problem
• Null sets and combinatorial covering properties
• On the interplay between productively Menger and productively Hurewicz spaces in models of $\mathfrak{b}=\mathfrak{d}$
• Omission of Intervals: Deducing covering properties of subsets of the real line from their combinatorial structure