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Selection principles and the Scheepers diagram in an uncountable setting — arXiv:2609.33856

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.33856
License (as archived): CC0-1.0
Archived: 2026-10-07T23:47:21.409Z
SHA-256 of the archived text: 225a8fb54f1932fcc79ca8ebcaabf39b28c4f956ef22653b7925398b9b7b87e9
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How to cite

Selection principles and the Scheepers diagram in an uncountable setting — arXiv:2609.33856 — arXiv (en), https://arxiv.org/abs/2609.33856 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_7e51ca2f7fd2 (Metraipolis archive dc_7e51ca2f7fd2, sha256 225a8fb54f1932fcc79ca8ebcaabf39b28c4f956ef22653b7925398b9b7b87e9)

BibTeX

@misc{dc7e51ca2f7fd2,
  title = {Selection principles and the Scheepers diagram in an uncountable setting — arXiv:2609.33856},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.33856},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_7e51ca2f7fd2; sha256 225a8fb54f1932fcc79ca8ebcaabf39b28c4f956ef22653b7925398b9b7b87e9},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.33856. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We continue the study of selection principles in the uncountable setting initiated in [18] and further developed in [1]. We investigate several problems posed in [1] concerning the Scheepers diagram, the $\kappa$-Hurewicz Conjecture, and related questions on $\kappa$-Menger and $K_\kappa$-spaces. We determine the Scheepers diagram in the uncountable setting up to three unresolved implications and compute the critical cardinalities of several properties occurring in the diagram. For weakly compact $\kappa$, we show that there is a model of ZFC in which the $\kappa$-Hurewicz Conjecture fails; in particular, it cannot hold under GCH at $\kappa$. Moreover, any model in which $\kappa$ is weakly compact and the $\kappa$-Hurewicz Conjecture holds must satisfy $2^{\mathfrak{b}_\kappa} = 2^\kappa$. We also study the existence of $\mathfrak{b}_\kappa$-scale sets which are $K_\kappa$-spaces and derive a necessary tree-theoretic condition for such an example. Under the additional assumption of the Perfect Subtree Property, no $\mathfrak{b}_\kappa$-scale set is a $K_\kappa$-space. Consequently, under this assumption, every $\mathfrak{b}_\kappa$-scale set yields an alternative example of a $\kappa$-Menger space which is not a $K_\kappa$-space.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-29 in math.LO).
Categorie arXiv: General Topology (math.GN) ; Logic (math.LO).
Pagina del lavoro: https://arxiv.org/abs/2609.33856 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T23:26:18.475Z dalla pagina https://arxiv.org/catchup/math.LO/2026-09-29?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).