Linear σ-additivity and some applications

We show that countable increasing unions preserve a large family of well-studied covering properties, which are not necessarily s-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 (1996), Gruenhage and Szeptycki (2005), Tsaban andZdomskyy (2008), and Tsaban (2006), (2007), and construct topological groups with very strong combinatorial properties. © 2011 American Mathematical Society.