New results on totally bounded groups
A countable infinite disjoint family of finite subsets of a topological space is called weak sequence convergent to a point if each neighborhood of this point intersects almost all members of the family. The paper contains results concerning the topology of arbitrary totally bounded topological groups: Theorem 2. In an infinite group, for any point, there is a weak sequence that is convergent to this point in an arbitrary totally bounded group topology. Theorem 3. In a countably infinite totally bounded group, there exists a countable infinite disjoint family \(\mathfrak{F}\) of finite subsets such that, for any totally bounded group topology, each nonempty open subset with respect to this topology intersects almost all members of \(\mathfrak{F.}\) Hence \(\mathfrak{F}\) is a weak sequence that simultaneously converges in this topology to all points of this group. It is proved that in any countable infinite totally bounded group there exists an almost disjoint family, of cardinality 2\(^{\aleph _{0}}\) , of subsets that are dense in the arbitrary totally bounded group topology.
scientific article; zbMATH DE number 1470984