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PreprintCC BY-NC-ND 4.0arXiv

A nonmicroscopic Gδ set with only microscopic compact subsets

  • Martin Doležal
  • Václav Vlasák
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Abstract

We construct a nonmicroscopic Gδ subset of the real line all of whose compact subsets are microscopic, answering a question of Zindulka and Nowakowski. We first obtain such a set in the standard Cantor space by coding branches of a finitely branching tree together with suitable choices of marked nodes. The coding of the branches ensures that the resulting set is not microscopic, while the marked nodes allow us to construct covers showing that every compact subset is microscopic. The example is then transferred from the Cantor space to the real line by the binary-expansion map.

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Cite

BibTeXRIS
BibTeX
@misc{dolezal2026nonmicroscopic,
  title = {{A nonmicroscopic $G_δ$ set with only microscopic compact subsets}},
  author = {Martin Doležal and Václav Vlasák},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06301},
  eprint = {2610.06301},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06301},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - A nonmicroscopic G_δ set with only microscopic compact subsets
AU  - Martin Doležal
AU  - Václav Vlasák
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06301
UR  - https://arxiv.org/abs/2610.06301
AB  - We construct a nonmicroscopic G_δ subset of the real line all of whose compact subsets are microscopic, answering a question of Zindulka and Nowakowski. We first obtain such a set in the standard Cantor space by coding branches of a finitely branching tree together with suitable choices of marked nodes. The coding of the branches ensures that the resulting set is not microscopic, while the marked nodes allow us to construct covers showing that every compact subset is microscopic. The example is then transferred from the Cantor space to the real line by the binary-expansion map.
ER  -