On Hewitt groups and finer locally compact group topologies
The authors study the class \(\mathcal{H}\) of all Hewitt abelian topological groups. A topological group \((G,\tau)\) is called a Hewitt group if the only strictly finer than \(\tau\) locally compact group topology on \(G\) is the discrete topology. Hewitt showed that the reals \(\mathbb R\) and the circle group \(\mathbb T\) belong to \(\mathcal{H}\). \textit{N. W. Rickert} [Trans. Am. Math. Soc. 126, 225--235 (1967; Zbl 0189.32404)] and \textit{M. Rajagopalan} [Math. Ann. 176, 169--180 (1968; Zbl 0174.05601)] characterized the non-discrete abelian locally compact groups in the class \(\mathcal{H}\) as those that contain either \(\mathbb R\), \(\mathbb T\), or the compact group \(\mathbb J_p\) of \(p\)-adic integers as an open subgroup. The authors prove that a non-discrete topological abelian group \(H\) is a Hewitt group precisely when it satisfies one of the following conditions: (1) if \(H\) is non-locally compact, then it does not have infinite compact subgroups and pseudolines (i.e., subgroups isomorphic to a continuous injective image of \(\mathbb R\)); (2) if \(H\) is locally compact, then it is either topologically isomorphic to \(\mathbb R\times D\) for a discrete group \(D\) or contains an open subgroup isomorphic to either \(\mathbb J_p\) or \(\mathbb T\).
scientific article; zbMATH DE number 6473607