A locally quasi-convex abelian group without Mackey topology

TL;DR

This paper constructs the first example of a countable reflexive, $k_ ext{omega}$, locally quasi-convex abelian group that lacks a strongest compatible topology, challenging existing assumptions.

Contribution

It introduces a novel example of a locally quasi-convex abelian group without a Mackey topology, specifically using the Graev free abelian group over a convergent sequence.

Findings

Provides the first such example of a group without Mackey topology

Shows the group is countable, reflexive, and $k_ ext{omega}$

Demonstrates limitations of existing topological group structures

Abstract

We give the first example of a locally quasi-convex (even countable reflexive and $k_\omega$) abelian group $G$ which does not admit the strongest compatible locally quasi-convex group topology. Our group $G$ is the Graev free abelian group $A_G(\mathbf{s})$ over a convergent sequence $\mathbf{s}$.