{
  "abstract": "We address three problems on separable quotients of topological groups posed by Leiderman, Morris, and Tkachenko in \\cite{LMT} published on Israel Journal of Mathematics. First, we construct in ZFC a connected Baire Pontryagin-reflexive dense subgroup of $\\T^{\\cc}$ whose countable subgroups are $h$-embedded and whose uncountable subgroups are dense. Its underlying abstract group is the circle group, and all its compact subsets are finite. Second, we construct a zero-dimensional Baire Pontryagin-reflexive example with the same subgroup properties whose underlying group is free abelian of rank $\\cc$. Both examples have no nontrivial separable Hausdorff quotient. Third, for the group constructed in their Theorem~3.5, we determine every closed subgroup of every finite power up to an integral change of coordinates and prove that every countable subgroup of every Hausdorff quotient of a finite power is $h$-embedded and closed. The same conclusions hold for our free Baire reflexive example. These results answer Problem~1.25 negatively, Problem~3.12 affirmatively and realize all three regularity properties in Problem~3.14 simultaneously in \\cite{LMT}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26049",
  "author_total": 2,
  "authors": [
    {
      "name": "Li-Hong Xie"
    },
    {
      "name": "Jiang Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bed634dsdfnx5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26049",
  "fulltext_url": null,
  "id": "bed634dsdfnx5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26049",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GN"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7214093036",
  "pdf_url": "https://arxiv.org/pdf/2609.26049",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bed634dsdfnx5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26049",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26049"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Three Problems on Separable Quotients of Precompact Abelian Groups",
  "title_text": "Three Problems on Separable Quotients of Precompact Abelian Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/bed634dsdfnx5",
  "venue": "arXiv",
  "versions": []
}