Linear $\sigma$-additivity and some applications

TL;DR

This paper demonstrates that countable increasing unions preserve various covering properties, introduces applications of combinatorial and forcing methods, and constructs topological groups with strong combinatorial features.

Contribution

It establishes the preservation of covering properties under countable unions and applies combinatorial and forcing techniques to solve open problems and construct groups with enhanced properties.

Findings

Countable increasing unions preserve many covering properties.

Solved several open problems in topology and combinatorics.

Constructed topological groups with strong combinatorial characteristics.

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.