Compact topologically torsion elements of topological abelian groups (Q2504941)

Let \((G,+)\) be a Hausdorff topological abelian group. An element \(x\in G\) is \textit{compact} if the closed subgroup of \(G\) generated by \(x\) is compact. The compact elements of \(G\) form a subgroup denoted by \(bG\). For a prime \(p\), an element \(x\in G\) is called a \textit{topologically \(p\)-torsion element} if \(\lim_{n\to\infty} p^n x=0\). The subgroup \(G_p\) of \(G\) consisting of all topologically \(p\)-torsion elements is the \textit{\(p\)-component of \(G\)}. An element \(x\in G\) is said to be a \textit{topologically torsion element} if \(\lim_{n\to\infty} n!x=0\). The subset of \(G\) containing all torsion elements is denoted by \(G!\). The set \(G!\) actually is a subgroup of \(G\) which contains the \(p\)-component \(G_p\) of \(G\) for every prime \(p\). Also, \(G!\) is a subset of \(bG\) provided \(G\) is locally compact. The author first proves a result concerning compact monothetic groups which allows him to state the main theorem of the paper: The subgroup of \(G\) generated by all compact elements which are topologically \(p\)-torsion for some prime \(p\) is a dense subgroup of \(\overline{bG}\). This result yields a new proof for the fact that in a locally compact abelian group \(G\) the subgroup \(G!\) is dense in \(bG\) (the latter is known to be closed in case of a locally compact group). Also, the main result, combined with Pontryagin duality, leads to new descriptions of the identity component of a locally compact abelian group.