σ-Hereditarily Closure-Preserving I-sn-Networks, I-sn-Metrizability, and Local HCP-Finiteness

Let \(\mathcal{I}\) be an admissible ideal on \(\mathbb{N}\). If \(y_n\to_{\mathcal{I}} x\) in a space \(X\) and \(\{n:y_n\ne x\}\in\mathcal{I}^{+}\), 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 hereditarily closure-preserving (HCP) family. If no such sequence exists at \(x\), then \(\{x\}\) itself is an \(\mathcal{I}\)-sequential neighborhood. These two alternatives imply that every space with a \(\sigma\)-point-discrete \(\mathcal{I}\)-\(sn\)-network is \(\mathcal{I}\)-\(snf\)-countable. Together with existing characterizations, this proves that every regular space with a \(\sigma\)-HCP \(\mathcal{I}\)-\(sn\)-network is \(\mathcal{I}\)-\(sn\)-metrizable, answering Problem~5.2 of Zhou–Liu–Liu–Lin for every admissible ideal.We say that \(\mathcal{I}\) has local HCP-finiteness if, in every \(T_1\) space, the convergence and support assumptions above imply that each HCP family of \(\mathcal{I}\)-sequential neighborhoods of \(x\) is finite. A countable Rudin--Keisler antichain of free ultrafilters gives an admissible ideal without this property, so the countability bound cannot in general be strengthened to finiteness. Every \(\mathfrak{p}^{-}\) ideal has local HCP-finiteness. We prove preservation theorems for Fubini products, heterogeneous Fubini sums with cross-row Kat\v{e}tov absorption, and suitable increasing unions. These results establish local HCP-finiteness for the finite Fubini powers of \(\mathrm{Fin}\), Katˇetov’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 \(\mathrm{Fin}_\omega\). All of the listed examples except \(\mathrm{Fin}\) fail \(\mathfrak{p}^{-}\).