Completeness properties of group topologies for \(\mathbb{R}\)
The authors study several complete Hausdorff group topologies on the real line. They recall that the character group of the discrete group \(\mathbb Q\) is a torsion-free divisible group of cardinality \(\mathfrak c\) and hence algebraically isomorphic to \(\mathbb R\) and show that this topology is not comparable with the usual topology \({\mathcal U}\) on \(\mathbb R\). Afterwards, they consider group topologies of the form \({\mathcal U}\times {\mathcal T}\) on \(\mathbb R\times \mathbb Z\) where \({\mathcal T}\) is some Hausdorff group topology on \(\mathbb Z\). They compute the completion of the quotient topology on \(\mathbb R\) induced by the mapping \((\mathbb R,{\mathcal U})\times (\mathbb Z,{\mathcal T})\to \mathbb R,\;(x,y)\mapsto x+y\) and apply the result to topologies \({\mathcal T}\) induced by \(T\)-sequences on \(\mathbb Z\) to obtain complete group topologies on \(\mathbb R\) which are strictly weaker than the usual topology.
scientific article; zbMATH DE number 6473618