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$σ$-hereditarily closure-preserving $\mathcal I$-$sn$-networks, $\mathcal I$-$sn$-metrizability, and local HCP-finiteness — arXiv:2609.30317

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.30317
License (as archived): CC0-1.0
Archived: 2026-10-08T03:11:48.712Z
SHA-256 of the archived text: 91526124bc3c0f1f066a67227832a864ae893d942c5c686c4f70333b1f2dde9f
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$σ$-hereditarily closure-preserving $\mathcal I$-$sn$-networks, $\mathcal I$-$sn$-metrizability, and local HCP-finiteness — arXiv:2609.30317 — arXiv (en), https://arxiv.org/abs/2609.30317 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_f7d367752cd8 (Metraipolis archive dc_f7d367752cd8, sha256 91526124bc3c0f1f066a67227832a864ae893d942c5c686c4f70333b1f2dde9f)

BibTeX

@misc{dcf7d367752cd8,
  title = {$σ$-hereditarily closure-preserving $mathcal I$-$sn$-networks, $mathcal I$-$sn$-metrizability, and local HCP-finiteness — arXiv:2609.30317},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.30317},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_f7d367752cd8; sha256 91526124bc3c0f1f066a67227832a864ae893d942c5c686c4f70333b1f2dde9f},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.30317. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Let $\mathcal I$ be an admissible ideal on $\mathbb N$. Zhou, Liu, Liu, and Lin asked whether every regular space with a $\sigma$-hereditarily closure-preserving $\mathcal I$-$sn$-network is $\mathcal I$-$sn$-metrizable. The answer is affirmative for every admissible ideal. The proof rests on a local countability theorem. If $y_n\to_{\mathcal I}x$ and $\{n\mid y_n\ne x\}\in\mathcal I^+$, then every point-discrete family of $\mathcal I$-sequential neighborhoods of $x$ is countable. In a $T_1$ space, this applies in particular to every HCP family. If no $\mathcal I$-convergent sequence to $x$ has $\mathcal I$-positive support away from $x$, then $\{x\}$ is an $\mathcal I$-sequential neighborhood. It follows that every space with a $\sigma$-point-discrete $\mathcal I$-$sn$-network is $\mathcal I$-$snf$-countable. This conclusion, together with the characterization theorems of Ge and of Zhou, Liu, Liu, and Lin, settles Problem 5.2. The countability bound cannot in general be strengthened to finiteness. A countable Rudin--Keisler antichain of free ultrafilters gives an admissible ideal for which local HCP-finiteness fails. Every $P^{-}$ ideal has local HCP-finiteness. Preservation theorems for Fubini products, heterogeneous Fubini sums under cross-row Katětov absorption, and suitable increasing unions extend local HCP-finiteness beyond the $P^{-}$ class. They apply to the finite Fubini powers of $\mathrm{Fin}$, Katětov's first limit-stage ideal $\mathrm{Fin}^{\omega}$ and its finite successors, the tree-derived rank-$\omega$ ideal $\mathcal H_{<\omega}$, and the inductive-limit ideal $\mathrm{Fin}_{\omega}$. Every listed example other than $\mathrm{Fin}$ fails $P^{-}$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-28 in math.LO).
Categorie arXiv: General Topology (math.GN) ; Logic (math.LO).
Pagina del lavoro: https://arxiv.org/abs/2609.30317 — 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 2026-10-08T02:47:39.409Z dalla pagina https://arxiv.org/catchup/math.LO/2026-09-28?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).