$\sigma$-hereditarily closure-preserving $\mathcal I$-$sn$-networks, $\mathcal I$-$sn$-metrizability, and local HCP-finiteness

Abstract

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 Kat\v{e}tov absorption, and suitable increasing unions extend local HCP-finiteness beyond the $P^{-}$ class. They apply to the finite Fubini powers of $\mathrm{Fin}$, Kat\v{e}tov'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 $P^{-}$.