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

Let \(I\) be an admissible ideal on \(N\). If \(y_n→I x\) in a space \(X\) and \(\{n:y_n≠ x\}⁺\), then every point-discrete family of \(I\)-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 \(I\)-sequential neighborhood. These two alternatives imply that every space with a \(σ\)-point-discrete \(I\)-\(sn\)-network is \(I\)-\(snf\)-countable. Together with existing characterizations, this proves that every regular space with a \(σ\)-HCP \(I\)-\(sn\)-network is \(I\)-\(sn\)-metrizable, answering Problem~5.2 of Zhou–Liu–Liu–Lin for every admissible ideal.We say that \(I\) has local HCP-finiteness if, in every \(T_1\) space, the convergence and support assumptions above imply that each HCP family of \(I\)-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 \(p⁻\) ideal has local HCP-finiteness. We prove preservation theorems for Fubini products, heterogeneous Fubini sums with cross-row Kat{e}tov absorption, and suitable increasing unions. These results establish local HCP-finiteness for the finite Fubini powers of \(Fin\), Katˇetov’s first limit-stage ideal \(Fin^ω\) and its finite successors, the tree-derived rank-\(ω\) ideal \(H<ω\), and the inductive-limit ideal \(Fin_ω\). All of the listed examples except \(Fin\) fail \(p⁻\).