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• W2930287272 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." @default.
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• W2930287272 date "2019-06-01" @default.
• W2930287272 modified "2023-09-23" @default.
• W2930287272 title "Gaps in the lattices of topological group topologies" @default.
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• W2930287272 doi "https://doi.org/10.1016/j.topol.2019.03.023" @default.
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