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More on dominated and microscopic sets

  • Ondřej Zindulka
  • Piotr Nowakowski
  • Cristina Villanueva-Segovia
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Abstract

Let S be a family of sequences decreasing to zero. A set E in a metric space is S-dominated if, for every s∈ S, there exists a countable cover {En} of E such that diam En<sn for every n. We continue the study of families of dominated sets initiated in the previous paper. We look, e.g., into Cartesian products of dominated sets with strong measure and interaction of microscopic and porous sets. Based upon a tight relationship of Hausdorff measures and dominated sets, cardinal invariants of the ideals of dominated sets are determined and a related problem of Kwela is resolved.

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Cite

BibTeXRIS
BibTeX
@misc{zindulka2026more,
  title = {{More on dominated and microscopic sets}},
  author = {Ondřej Zindulka and Piotr Nowakowski and Cristina Villanueva-Segovia},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39942},
  eprint = {2609.39942},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39942},
}
RIS
TY  - UNPB
TI  - More on dominated and microscopic sets
AU  - Ondřej Zindulka
AU  - Piotr Nowakowski
AU  - Cristina Villanueva-Segovia
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39942
UR  - https://arxiv.org/abs/2609.39942
AB  - Let S be a family of sequences decreasing to zero. A set E in a metric space is S-dominated if, for every s∈ S, there exists a countable cover {E_n} of E such that diam E_n<s_n for every n. We continue the study of families of dominated sets initiated in the previous paper. We look, e.g., into Cartesian products of dominated sets with strong measure and interaction of microscopic and porous sets. Based upon a tight relationship of Hausdorff measures and dominated sets, cardinal invariants of the ideals of dominated sets are determined and a related problem of Kwela is resolved.
ER  -