Minimally almost periodic group topology on infinite countable Abelian groups: A solution to Comfort's question

TL;DR

The paper proves that every unbounded countable Abelian group can be equipped with a complete Hausdorff minimally almost periodic topology, solving Comfort's longstanding question in topological group theory.

Contribution

It provides a construction of MinAP topologies for all unbounded countable Abelian groups, answering a key open problem in the field.

Findings

Any unbounded countable Abelian group admits a MinAP topology.

Bounded infinite Abelian groups have MinAP topologies iff all Ulm-Kaplansky invariants are infinite.

Countably infinite groups can have a complete MinAP topology.

Abstract

For any countable subgroup $H$ of an unbounded Abelian group $G$ there is a complete Hausdorff group topology $\tau$ such that $H$ is the von Neumann radical of $(G,\tau)$. In particular, we obtain the positive answer to Comfort's question: any unbounded countable Abelian group admits a complete Hausdorff minimally almost periodic (MinAP) group topology. A bounded infinite Abelian group admits a MinAP group topology if and only if all its leading Ulm-Kaplansky invariants are infinite. If, in addition, $G$ is countably infinite, a MinAP group topology can be chosen to be complete.