The size of the poset of compatible locally quasi-convex topologies on locally compact abelian groups

Abstract

For a Hausdorff locally quasi-convex abelian group G, let C(G) be the poset of all Hausdorff locally quasi-convex group topologies on its underlying group having the same continuous characters as G. We prove that, whenever G is non-precompact, C(G) contains an order-isomorphic copy of (Pow(cont),⊆), where cont=2^ℵ₀. The embedding takes values between the Bohr topology and the original topology. Consequently, every infinite discrete abelian group D satisfies |C(D)|=widthC(D)=2^{2^|D|}. The discrete reduction for locally compact abelian groups then yields exact cardinality and width formulas for all such groups; in particular, both invariants equal 2^cont for every non-compact σ-compact locally compact abelian group. These results answer Questions 6.1--6.3 and 6.5--6.7, and the locally compact case of Question 6.4, posed by L.~Außenhofer and D.~Dikranjan in \cite{AD20}. The embedding also applies to non-compact complete metrizable locally quasi-convex groups. A non-compact precompact nuclear group with a unique compatible topology shows that non-compactness alone does not suffice in the nuclear setting. Finally, the compatible poset of R^N is not order-isomorphic to that of any discrete abelian group.

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BibTeX RIS

BibTeX

@misc{huang2026size,
  title = {{The size of the poset of compatible locally quasi-convex topologies on locally compact abelian groups}},
  author = {ZhouXiang Huang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12116},
  eprint = {2610.12116},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12116},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The size of the poset of compatible locally quasi-convex topologies on locally compact abelian groups
AU  - ZhouXiang Huang
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12116
UR  - https://arxiv.org/abs/2610.12116
AB  - For a Hausdorff locally quasi-convex abelian group G, let C(G) be the poset of all Hausdorff locally quasi-convex group topologies on its underlying group having the same continuous characters as G. We prove that, whenever G is non-precompact, C(G) contains an order-isomorphic copy of (Pow(cont),⊆), where cont=2^ℵ_0. The embedding takes values between the Bohr topology and the original topology. Consequently, every infinite discrete abelian group D satisfies |C(D)|=widthC(D)=2^2^|D|. The discrete reduction for locally compact abelian groups then yields exact cardinality and width formulas for all such groups; in particular, both invariants equal 2^cont for every non-compact σ-compact locally compact abelian group. These results answer Questions 6.1--6.3 and 6.5--6.7, and the locally compact case of Question 6.4, posed by L.~Außenhofer and D.~Dikranjan in \cite{AD20}. The embedding also applies to non-compact complete metrizable locally quasi-convex groups. A non-compact precompact nuclear group with a unique compatible topology shows that non-compactness alone does not suffice in the nuclear setting. Finally, the compatible poset of R^N is not order-isomorphic to that of any discrete abelian group.
ER  -

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