σ-hereditarily closure-preserving ℐ-sn-networks, ℐ-sn-metrizability, and local HCP-finiteness
Abstract
Let ℐ be an admissible ideal on ℕ. Zhou, Liu, Liu, and Lin asked whether every regular space with a σ-hereditarily closure-preserving ℐ-sn-network is ℐ-sn-metrizable. The answer is affirmative for every admissible ideal. The proof rests on a local countability theorem. If yn→ℐx and {n| yn≠ x}∈ℐ+, then every point-discrete family of ℐ-sequential neighborhoods of x is countable. In a T1 space, this applies in particular to every HCP family. If no ℐ-convergent sequence to x has ℐ-positive support away from x, then {x} is an ℐ-sequential neighborhood. It follows that every space with a σ-point-discrete ℐ-sn-network is 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 Fin, Katětov's first limit-stage ideal Finω and its finite successors, the tree-derived rank-ω ideal ℋ<ω, and the inductive-limit ideal Finω. Every listed example other than Fin fails P−.
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BibTeX
@misc{hu2026σhereditarily,
title = {{$σ$-hereditarily closure-preserving $\mathcal I$-$sn$-networks, $\mathcal I$-$sn$-metrizability, and local HCP-finiteness}},
author = {Xing-Yu Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30317},
eprint = {2609.30317},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30317},
}
RIS
TY - UNPB
TI - σ-hereditarily closure-preserving ℐ-sn-networks, ℐ-sn-metrizability, and local HCP-finiteness
AU - Xing-Yu Hu
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.30317
UR - https://arxiv.org/abs/2609.30317
AB - Let ℐ be an admissible ideal on ℕ. Zhou, Liu, Liu, and Lin asked whether every regular space with a σ-hereditarily closure-preserving ℐ-sn-network is ℐ-sn-metrizable. The answer is affirmative for every admissible ideal. The proof rests on a local countability theorem. If y_n→_ℐx and {n| y_n≠ x}∈ℐ^+, then every point-discrete family of ℐ-sequential neighborhoods of x is countable. In a T_1 space, this applies in particular to every HCP family. If no ℐ-convergent sequence to x has ℐ-positive support away from x, then {x} is an ℐ-sequential neighborhood. It follows that every space with a σ-point-discrete ℐ-sn-network is 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 Fin, Katětov's first limit-stage ideal Fin^ω and its finite successors, the tree-derived rank-ω ideal ℋ_<ω, and the inductive-limit ideal Fin_ω. Every listed example other than Fin fails P^−.
ER -