Remarks on the Bohr-torsion topology of a locally compact abelian group (Q2419425)

Let \((G,t)\) be a topological abelian group. The \textit{Bohr topology} \(t^+\) on \(G\) is the weakest topology on \(G\) that makes the homomorphisms (\textit{characters}) \(\phi:G\rightarrow \mathbb{T}\) continuous, where \(\mathbb{T}\) denotes the group of complex numbers of modulus \(1\) under multiplication. This topology might be trivial but when the group is locally compact, it is Hausdorff. In fact, to have \(t^+\) Hausdorff it is necessary and sufficient that the continuous characters of \(G\) separate the points of \(G\). If \(\mathcal{T}\) denotes the torsion subgroup of \(\mathbb{T}\), the weakest topology on \(G\) that makes the characters \(\phi:G\rightarrow \mathcal{T}\) continuous is called the \textit{Bohr-torsion topology}, and it is denoted by \(t^{\oplus}\). Clearly \(t^{\oplus}\subseteq t^+\subseteq t\). As \(t^+\), the topology \(t^{\oplus}\) is Hausdorff if and only if the continuous homomorphisms from \(G\) to \(\mathcal{T}\) separate the points of \(G\). Remark that if \(G\) is locally compact \(t^+\) is Hausdorff, however \(t^{\oplus}\) is not necessarily Hausdorff in general. For instance it is trivial for connected groups. This paper focuses on solving when \(t^{\oplus}\) is Hasudorff for locally compact abelian groups and on the behavior of closed subgroups and Hausdorff quotients. Recall that two subsets \(A\) and \(B\) of a topological space \(X\) are \textit{completely separated} if there is a continuous function \(f: X\rightarrow [0,1]\) such that \(f(A)=\{0\}\) and \(f(B)=\{1\}\). The space \(X\) is called \textit{strongly zero-dimensional} if for any pair \(A\) and \(B\) of completely separated subsets of \(X\), there exists an open-and-closed set \(U\) of \(X\) such that \(A\subseteq U\subseteq X\setminus B\). The author obtains the following equivalences for a locally compact abelian group \((G,t)\): (1) \(G\) is strongly zero-dimensional. (2) \(G\) is zero-dimensional. (3) \(G\) has an open compact zero-dimensional subgroup. (4) \((G,t^{\oplus})\) is Hausdorff. (5) \((G,t^{\oplus})\) is zero-dimensional. (6) \((G,t^{\oplus})\) is strongly zero-dimensional. Another property that holds for locally compact groups is that both \(t\) and \(t^+\) have the same closed subgroups. When \(t^{\oplus}\) is Hausdorff, \(t\) and \(t^{\oplus}\) have the same closed subgroups as well. A similar result for \(t^{\oplus}\), which was proved for \(t^+\) by the author in a previous paper, is the following one: Let \((G,t)\) be a locally compact abelian group and let \(H\) be a \(t\)-closed subgroup of \(G\). If \(t^{\oplus}\) is Hausdorff, then (i) \(\, H\) is \(t^{\oplus}\)-closed. (ii) The identity map from \((H,{(t|_H)}^{\oplus})\) to \((H,t^{\oplus}|_H)\) is a topological isomorphism. (iii) The identity map from \((G/H,(t_{G/H})^{\oplus})\) to \((G,t^{\oplus})/(H,t^{\oplus}|_H)\) is a topological isomorphism.