Topological groups close to being \(\sigma\)-compact
A topological group \(G\) is called o-bounded if for every sequence \((U_i)_{i\geq 0}\) of neighborhoods in \(G\) there is a sequence \((F_i)_{i\geq 0}\) of finite subsets of \(G\) such that \(G=\bigcup_{i\geq 0}F_iU_i\). Strengthening this notion, one considers a game where the first player chooses a neighborhood \(U_i\), the second player chooses a finite set \(F_i\), then the first player chooses another neighborhood \(U_{i+1}\), and so on. The second player wins if \(G=\bigcup_{i\geq 0}F_iU_i\). The group \(G\) is called strictly o-bounded if there is a winning strategy for the second player. The class of (strictly) o-bounded groups is closed with respect to the passage to subgroups or to images under continuous homomorphisms. Every (subgroup of a) \(\sigma\)-compact group is strictly o-bounded. The paper under review discusses inclusion relations between the classes of (strictly) \text{o-bounded} groups and other classes of topological groups. In particular, examples are given of a group which is strictly o-bounded but not a subgroup of any \(\sigma\)-compact group, and of an o-bounded group which is not strictly o-bounded.
scientific article; zbMATH DE number 1422599