Gaps in the lattices of topological group topologies

Abstract

We study gaps in the lattice of topological group topologies on a given abelian group and compare the properties of the topologies that form a gap. The emphasis is placed on the study of predecessors of the topology of a noncompact LCA group. It is shown that every Hausdorff predecessor, σ, of the topology τ of an LCA group G is finer than the Bohr topology of the group G and that the two topologies, τ and σ, have the same closed subgroups. This implies that the Bohr topology, τ+, of a noncompact LCA group (G,τ) is not a predecessor of τ. Complementing these results we prove that all predecessors of the topology of a Hausdorff topological abelian group G are Hausdorff provided that G is torsion free.We also show that if a topological abelian group (G,τ) contains a complete subgroup N such that the quotient group G/N is compact, then the predecessors of τ are in a one-to-one correspondence with the predecessors of τ↾N on N. In particular, if N is discrete and G/N is compact, then the predecessors of τ are in a one-to-one correspondence with the maximal topological group topologies on N.