On certain subsets of a group. II (Q2710544)

In this paper the authors continue their investigations started in Part I [Quest. Answers Gen. Topology 17, 183-197 (1999; Zbl 0966.54001)]. In Chapter 1 totally bounded groups are characterized: A topological group \(G\) is totally bounded if and only if no non-empty open subset of \(G\) is small. If \(G\) is countable infinite, then it is totally bounded if and only if every discrete resp. nowhere dense subset is small. In Chapter 2 the authors strengthen I. V. Protasov's answer to a question of van Douwen [\textit{I. V. Protasov}, Math. Notes 55, 101-102 (1994; Zbl 0836.22003)]: In a totally bounded group each non-small subset contains a non-closed discrete subset. In their earlier paper (loc. cit.) the authors conjectured that any totally bounded group has a small dense subset. They prove this conjecture for countable Abelian groups, where further partial answers to questions from their mentioned paper are given. The main result of Chapter 4 is the following: In an infinite Abelian group there exist two disjoint large subsets which are dense with respect to any totally bounded group topology. The paper ends with results on the set of all points where a given subset of a topological group is not small (the authors call a subset \(A\) of a topological group \(G\) not small at a point \(x\in G\) if for each neighborhood \(V\) of \(x\) the set \(A\cap V\) is not small).