\(\mathbb{Q}\) is not a Mackey group (Q471464)

The letters \(\mathbb{Z}\), \(\mathbb Q\) and \(\mathbb R\) stand for the integers, rationals and real numbers respectively with their usual topology and group structure. The symbol \(\mathbb T\) denotes the quotient group \(\mathbb R/\mathbb Z\) and \(\mathbb T_+=\{x+\mathbb Z\in \mathbb T: x\in [-\frac14,\frac14]\}\). Given a topological group \((G,\tau)\), its dual \((G,\tau)^\wedge\) or simply \(G^\wedge\) is the group of all continuous homomorphisms from \(G\) to \(\mathbb T\). Two group topologies \(\tau\) and \(\mu\) on a group \(G\) are called \textit{compatible} if \((G, \tau)^\wedge = (G, \mu)^\wedge\). A subset \(A\) of a topological group \(G\) is said to be quasi-convex if for any \(x\notin A\), there exists \(\chi\in G^\wedge\) such that \(\chi(A)\subset \mathbb T_+\) and \(\chi(x) \notin \mathbb T_+\). The group \(G\) is \textit{locally quasi-convex} if it has a local neighborhood base consisting of quasi-convex sets. A locally quasi-convex group \((G, \tau)\) is a \textit{Mackey group} if for any locally quasi-convex group topology \(\nu\) on \(G\), if \(\nu\) is compatible with \(\tau\), then \(\nu\subset \tau\). The main result of the paper is Theorem 1.1 which states that the group \(\mathbb Q\) of rational numbers is not a Mackey group.