σ-Hereditarily Closure-Preserving I-sn-Networks, I-sn-Metrizability, and Local HCP-Finiteness
Abstract
Let \(\mathcal{I}\) be an admissible ideal on \(\mathbb{N}\). Zhou--Liu--Liu--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 stronger local countability lemma. 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 HCP family. If no \(\mathcal{I}\)-convergent sequence to \(x\) has \(\mathcal{I}\)-positive support away from \(x\), then the singleton \(\{x\}\) is itself an \(\mathcal{I}\)-sequential neighborhood. Consequently every space with a \(\sigma\)-point-discrete \(\mathcal{I}\)-\(sn\)-network is \(\mathcal{I}\)-\(snf\)-countable. For a regular space with a \(\sigma\)-HCP \(\mathcal{I}\)-\(sn\)-network, Ge's characterization of \(sn\)-metrizability and Zhou--Liu--Liu--Lin's characterization of \(\mathcal{I}\)-\(sn\)-metrizability then settle Problem~5.2 affirmatively. The countability bound cannot in general be strengthened to finiteness. A countable Rudin–Keisler antichain of free ultrafilters gives an admissible ideal \(\mathcal{I}\) for which \(\mathrm{HCF}(\mathcal{I})\) fails. Thus local HCP-finiteness is not automatic for admissible ideals. Every \(\mathfrak{p}^{-}\) ideal has local…
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