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PreprintCC BY 4.0arXiv

Primeless proofs of the Menger and Rothberger games

  • Renan Maneli Mezabarba
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Abstract

We continue the study of the Menger and Rothberger games on lattices carried out in "Pointless proofs of the Menger and Rothberger games" (Topology Appl. 300 (2021), 107774). This time, we extend the earlier results by dropping some hypotheses that turned out to be unnecessary, and use Stone duality to recover known game characterizations for dense open families. We also give a formulation of Ufin for partially ordered sets and prove its game characterization without any lattice assumption. Finally, almost disjoint families give complete distributive lattices on which the selection principles and the corresponding games differ. Dias's results give the lower bounds cov(ℳ) and d for the least sizes of such counterexamples.

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Cite

BibTeXRIS
BibTeX
@misc{mezabarba2026primeless,
  title = {{Primeless proofs of the Menger and Rothberger games}},
  author = {Renan Maneli Mezabarba},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28780},
  eprint = {2609.28780},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28780},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Primeless proofs of the Menger and Rothberger games
AU  - Renan Maneli Mezabarba
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.28780
UR  - https://arxiv.org/abs/2609.28780
AB  - We continue the study of the Menger and Rothberger games on lattices carried out in "Pointless proofs of the Menger and Rothberger games" (Topology Appl. 300 (2021), 107774). This time, we extend the earlier results by dropping some hypotheses that turned out to be unnecessary, and use Stone duality to recover known game characterizations for dense open families. We also give a formulation of U_fin for partially ordered sets and prove its game characterization without any lattice assumption. Finally, almost disjoint families give complete distributive lattices on which the selection principles and the corresponding games differ. Dias's results give the lower bounds cov(ℳ) and d for the least sizes of such counterexamples.
ER  -