Fubini and Increasing-Union Stability of Local HCP-Finiteness for I-sn-Networks
Abstract
Let $\mathcal{I}$ be an admissible ideal on $\mathbb{N}$, and let $\operatorname{Fin}$ denote the ideal of finite subsets of $\mathbb{N}$. 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 central condition is local HCP-finiteness, denoted $\operatorname{HCF}(\mathcal{I})$. The standard $\mathfrak{p}^{-}$ property implies this condition. If $\mathcal{K}$ has $\mathfrak{p}^{-}$ and $\mathcal{J}$ has local HCP-finiteness, then $\mathcal{K}\otimes\mathcal{J}$ has local HCP-finiteness. For varying inner ideals, the heterogeneous sum has local HCP-finiteness if the outer ideal has $\mathfrak{p}^{-}$ and, outside a $\mathcal{K}$-small set of rows, local HCP-finiteness and cross-row Kat\v{e}tov absorption hold. At Kat\v{e}tov's first limit stage, the heterogeneous theorem proves $\operatorname{HCF}(\operatorname{Fin}^{\omega})$, while constant-inner lifting gives $\operatorname{HCF}(\operatorname{Fin}^{\omega+r})$ for $0<r<\omega$. The question therefore has an affirmative answer for each of these ideals, although none has $\mathfrak{p}^{-}$. For an increasing sequence of admissible ideals on a fixed countable set whose union is a proper ideal, local HCP-finiteness passes to the union if cofinally many stages are $K$-uniform and have local HCP-finiteness. Two rank-$\omega$ limit ideals satisfy the hypotheses of the increasing-union theorem. For the…
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