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

1. Introduction

Throughout, spaces are $T_1$ unless explicitly stated otherwise, and $\mathbb{N}=\{0,1,2,\dots \}$. On any countably infinite carrier, $\mathrm{Fin}$ denotes the ideal of finite subsets of that carrier. An admissible ideal$\mathcal{I}$ on $\mathbb{N}$ satisfies

$$\mathrm{Fin}\subseteq \mathcal{I}⊊\mathcal{P}\left(\mathbb{N}\right),$$

and is closed under subsets and finite unions. Put

$${\mathcal{I}}^{★}=\{A\subseteq \mathbb{N}:\mathbb{N}\setminus A\in \mathcal{I}\},{\mathcal{I}}^{+}=\mathcal{P}\left(\mathbb{N}\right)\setminus \mathcal{I}.$$

A sequence $\left(x_n\right)$ in a space X is $\mathcal{I}$-convergent to $x\in X$, written $x_n{\to }_{\mathcal{I}}x$, if

$$\{n\in \mathbb{N}:x_n\notin U\}\in \mathcal{I}$$

for every neighborhood U of x. Ideal convergence for sequences in metric spaces was introduced by Kostyrko, Šalát and Wilczyński [17]. Lahiri and Das [21] extended it to topological spaces. Zhou, Liu and Lin studied topological spaces determined by $\mathcal{I}$-convergence [43]. Lin investigated ${\mathcal{I}}_{sn}$-open sets, $\mathcal{I}$-neighborhood spaces and $\mathcal{I}$-quotient spaces [24]. Zhou–Lin related $\mathcal{I}$-$cs$-networks to images of metric spaces under $\mathcal{I}$-covering maps [39]. Liu, Lin and Zhou obtained metric-image characterizations for spaces defined by ideal convergence [31].

A set $P\subseteq X$ is an $\mathcal{I}$-sequential neighborhood of x if $x_n{\to }_{\mathcal{I}}x$ implies $\{n:x_n\notin P\}\in \mathcal{I}$ [42, Definition 2.5]. An $\mathcal{I}$-$sn$-network is a family $\mathcal{P}={\bigcup }_{x\in X}{\mathcal{P}}_x$ such that, for each $x\in X$, every member of ${\mathcal{P}}_x$ contains x, ${\mathcal{P}}_x$ is directed downward under finite intersections, ${\mathcal{P}}_x$ is a local network at x, and every member of ${\mathcal{P}}_x$ is an $\mathcal{I}$-sequential neighborhood of x [40, Definition 5.1]. Lin introduced ordinary $sn$-networks [23], and Lin–Yan used them in the study of sequence-covering maps of metric spaces [26, Definition 4.2]. Ge studied $sn$-metrizability and countable $sn$-networks [7,8,9]. Luo [32] proved a mapping theorem for $sn$-metrizable spaces and compared several countability conditions on $sn$-networks. Ge–Lin characterized g-metrizable spaces by images of semi-metric spaces [10], and Lin–Ge gave image characterizations of $sn$-metrizable spaces [25]. Lin–Yun [27] give general background on generalized metric spaces. Lin–Zhang [28] survey point-countable covers, sequence-covering mappings and HCP families.

Following Zhou–Lin–Zhang [40, Definition 4.1], a space is $\mathcal{I}$-$snf$-countable if each point has a countable local network consisting of $\mathcal{I}$-sequential neighborhoods. Zhou–Liu–Liu–Lin call a space $\mathcal{I}$-$sn$-metrizable if it is regular and has a $\sigma$-locally finite $\mathcal{I}$-$sn$-network [41, Definition 2.4]. They posed the following problem [41, Problem 5.2].

Problem 5.2. Let X be a regular space with a $\sigma$-hereditarily closure-preserving $\mathcal{I}$-$sn$-network. Is X$\mathcal{I}$-$sn$-metrizable?

A family $\mathcal{A}$ of subsets of X is closure-preserving if $cl(\bigcup \mathcal{B})={\bigcup }_{B\in \mathcal{B}}clB$ for every subfamily $\mathcal{B}\subseteq \mathcal{A}$. It is hereditarily closure-preserving (HCP) if, for every choice of subpieces $H\left(A\right)\subseteq A$$(A\in \mathcal{A})$, the family $\left\{H\right(A):A\in \mathcal{A}\}$ is closure-preserving [11, Definition 2.1]. A family $\mathcal{A}$ is point-discrete, also called weakly hereditarily closure-preserving, if $\{a_A:A\in \mathcal{A}\}$ is closed discrete for every choice $a_A\in A$$(A\in \mathcal{A})$ [30, p. 199]. A family is σ-point-discrete if it is a countable union of point-discrete families. A family is $\sigma$-HCP if it is a countable union of HCP families. Every subfamily of an HCP family is HCP. In a $T_1$ space every HCP family is point-discrete. For every singleton selection, the selected singleton family is closure-preserving. Since singletons are closed in a $T_1$ space, every subset of the selected set is closed in X, so the selected set is closed discrete. Classical work on HCP families in metrization and generalized metrization includes Burke–Engelking–Lutzer [2] and Gruenhage [12]. For $\sigma$-HCP k-networks, Foged showed that a regular space is a closed image of a metric space exactly when it is Fréchet and has such a network [6]. Tanaka showed that a regular space is g-metrizable exactly when it is weakly first countable and has such a network [36]. Junnila–Yun gave a necessary and sufficient condition for a regular space with a $\sigma$-HCP k-network to be an ℵ-space [15]. Liu [29] surveys spaces with $\sigma$-HCP k-networks.

The first main result answers Problem 5.2 for every admissible ideal. Its pointwise countability lemma requires only point-discreteness, not HCP. Suppose $y_n{\to }_{\mathcal{I}}x$ and the support away from x, $\{n:y_n\ne x\}$, is $\mathcal{I}$-positive. For a point-discrete family $\mathcal{H}$ of $\mathcal{I}$-sequential neighborhoods of x, consider the indices n for which $y_n$ belongs to infinitely many members of $\mathcal{H}$. Point-discreteness forces this exceptional index set to belong to $\mathcal{I}$. Outside it, each $y_n$ belongs to only finitely many members of $\mathcal{H}$, while for every $P\in \mathcal{H}$ the set $\{n:y_n\in P,y_n\ne x\}$ is $\mathcal{I}$-positive. Hence $\mathcal{H}$ is countable. If no $\mathcal{I}$-convergent sequence at x has $\mathcal{I}$-positive support away from x, then $\left\{x\right\}$ itself is an $\mathcal{I}$-sequential neighborhood. Applying these two alternatives at each point gives $\mathcal{I}$-$snf$-countability for every space with a $\sigma$-point-discrete $\mathcal{I}$-$sn$-network.

Once $\mathcal{I}$-$snf$-countability is established, existing characterizations complete the metrization argument. Zhou–Lin–Zhang [40, Lemma 5.2]] show that every $\mathcal{I}$-$sn$-network is an ordinary $sn$-network. Hence a $\sigma$-HCP $\mathcal{I}$-$sn$-network is a $\sigma$-HCP ordinary $sn$-network. Ge’s characterization [8, Lemma 2.2] gives ordinary $sn$-metrizability. Zhou–Liu–Liu–Lin [41, Theorem 3.6] proved that $\mathcal{I}$-$sn$-metrizability is equivalent to the conjunction of $\mathcal{I}$-$snf$-countability and ordinary $sn$-metrizability. Theorem 3.2(2) therefore gives an affirmative answer to Problem 5.2.

At the ordinary level, Ge–Shen–Ge [11, Theorem 3.1] work throughout with regular $T_1$ spaces and prove $sn$-first countability for spaces with a $\sigma$-weakly HCP $sn$-network. The weak-HCP condition in their theorem is the point-discrete condition used here. Lin–Shen [22] later prove the stronger conclusion that every space with a $\sigma$-point-discrete $sn$-network has a $\sigma$-compact-finite $sn$-network. Since every $\mathcal{I}$-$sn$-network is an ordinary $sn$-network [40, Lemma 5.2], these results already give ordinary $sn$-first countability once the ideal structure is forgotten. They do not by themselves give $\mathcal{I}$-$snf$-countability, because the resulting ordinary $sn$-network is not asserted to consist of $\mathcal{I}$-sequential neighborhoods. A particularly close local result is Liu–Lin–Li’s Lemma 17(1) [30]. Their paper assumes Hausdorff spaces, and the lemma shows that a point-discrete family is finite whenever one nontrivial convergent sequence is eventually contained in every member. For $\mathcal{I}$-sequential neighborhoods, the exceptional index set may instead be infinite and may depend on the member. The local lemma below gives the corresponding countability bound under this weaker control. Ge’s earlier HCP subsequence lemma [8, Lemma 2.4] is another point-selection precursor.

The countability conclusion cannot in general be strengthened to finiteness. Theorem 3.4 constructs an admissible ideal $\mathcal{I}$ for which $\mathsf{HCF}\left(\mathcal{I}\right)$ fails. The witness consists of a zero-dimensional Hausdorff space X, a point $p\in X$, an $\mathcal{I}$-convergent sequence to p with $\mathcal{I}$-positive support away from p, and a countably infinite HCP family of $\mathcal{I}$-sequential neighborhoods of p. The construction uses a one-point topology defined by an ideal and a countable Rudin–Keisler antichain of free ultrafilters [35].

With Problem 5.2 settled at the countability level, the rest of the paper turns to the stronger local HCP-finiteness property. Every $P^{-}$ ideal has local HCP-finiteness by a finite diagonal selection argument. Local HCP-finiteness can nevertheless persist through constructions that need not preserve $P^{-}$. A Fubini product $\mathcal{K}\otimes \mathcal{J}$ has local HCP-finiteness whenever $\mathcal{K}$ has $P^{-}$ and $\mathcal{J}$ has local HCP-finiteness. A heterogeneous version allows the row ideals to vary under a cross-row Katětov absorption condition. These results give $\mathsf{HCF}\left({\mathrm{Fin}}^{\omega }\right)$ at Katětov’s first limit stage [16, p. 240] and $\mathsf{HCF}\left({\mathrm{Fin}}^{\omega +r}\right)$ at its finite successors.

A second permanence route comes from increasing unions. If ${\mathcal{J}}_0\subseteq {\mathcal{J}}_1\subseteq \dots$ are admissible ideals on one countable carrier and their union is a proper ideal, then local HCP-finiteness passes to the union when cofinally many stages are both K-uniform and locally HCP-finite. Applied to the tree-derived hierarchy of Pelayo Gómez [34, Theorems 3.4 and 3.5], this gives $\mathsf{HCF}\left({\mathcal{H}}_{<\omega }\right)$. It also gives $\mathsf{HCF}\left({\mathrm{Fin}}_{\omega }\right)$ for the inductive limit in the Debs–Saint Raymond hierarchy [4, Section 6.2]. The finite Fubini powers ${\mathrm{Fin}}^{\otimes r}$ for $r\ge 2$, the ideals ${\mathrm{Fin}}^{\omega +r}$ for $r<\omega$, ${\mathcal{H}}_{<\omega }$, and ${\mathrm{Fin}}_{\omega }$ fail $P^{-}$. Together, the Fubini and increasing-union routes establish stability of local HCP-finiteness beyond the class of $P^{-}$ ideals.

Section 2 collects the ideal-theoretic facts used throughout. Section 3 isolates universal point-discrete countability, resolves Problem 5.2, and separates that countability phenomenon from the stronger local HCP-finiteness property. Section 4 develops the Fubini permanence route, while Section 5 develops the increasing-union route and its two rank-$\omega$ applications.

2. Preliminaries

2.1. Ideals, Convergence, and Separation Rank

The general-topology conventions follow Engelking [5]. For $A,B\subseteq \mathbb{N}$, write $A{\subseteq }^{*}B$ if $A\setminus B$ is finite. When an ideal is defined on another countable set such as $\mathbb{N}\times \mathbb{N}$, it is transported to $\mathbb{N}$ by a fixed bijection. The ideal properties considered here are invariant under such reindexing.

For an ideal $\mathcal{A}$ on a countably infinite carrier D, write $dom\left(\mathcal{A}\right)=D$ and ${\mathcal{A}}^{+}=\mathcal{P}\left(D\right)\setminus \mathcal{A}$, and let ${\mathcal{A}}^{★}=\{D\setminus A:A\in \mathcal{A}\}$ denote its dual filter. The ideal $\mathcal{A}$ is admissible if it contains every finite subset of D and is proper. It is tall if every infinite subset of D contains an infinite member of $\mathcal{A}$. When working on the carrier D itself, a D-indexed family ${\left(x_d\right)}_{d\in D}$ in a space X is $\mathcal{A}$-convergent to $x\in X$ if $\{d\in D:x_d\notin U\}\in \mathcal{A}$ for every neighborhood U of x. A set $P\subseteq X$ is an $\mathcal{A}$-sequential neighborhood of x if $\{d\in D:x_d\notin P\}\in \mathcal{A}$ for every such $\mathcal{A}$-convergent family. This agrees with the preceding $\mathbb{N}$-indexed definitions after reindexing. For such an indexed family, its support away from x is $\{d\in D:x_d\ne x\}$. Regard $\mathcal{P}\left(D\right)$ as $2^D$ via characteristic functions and equip it with the product topology. For an analytic ideal $\mathcal{A}$, its Borel separation rank$rk\left(\mathcal{A}\right)$ is the least $\alpha <{\omega }_1$ for which there is $S\in {\Sigma }_{1+\alpha }^0$ with $\mathcal{A}\subseteq S$ and $S\cap {\mathcal{A}}^{★}=⌀$. This is the dual-ideal form of Debs–Saint Raymond’s separation rank [4, Definition 3.1], under the convention recalled by Kwela [18, p. 1].

The metrization argument in Section 3.1 uses Zhou–Lin–Zhang’s implication from $\mathcal{I}$-$sn$-networks to ordinary $sn$-networks.

Lemma 2.1

([40, Lemma 5.2]). For every admissible ideal $\mathcal{I}$, every $\mathcal{I}$-$sn$-network is an ordinary $sn$-network.

2.2. The $P^{-}$ Property

Hrušák, Meza-Alcántara, Thümmel and Uzcátegui [14, Theorem 3.8] give a Katětov-order characterization of the standard $P^{-}$ property. The equivalent partition form in Definition 2.2 appears in Camargo–Uzcátegui [3, Theorem 3.1(v)] and Uzcátegui Aylwin [37, Theorem 8.2(v)].

Definition 2.2.

An admissible ideal $\mathcal{I}$ has $P^{-}$ if, whenever $S\in {\mathcal{I}}^{+}$ is partitioned as

$$S=\underset{n\in \mathbb{N}}{⨆}{E}_n,E_n\in \mathcal{I},$$

there is $A\in {\mathcal{I}}^{+}$, $A\subseteq S$, such that $A\cap E_n$ is finite for every n.

Lemma 2.3.

For an admissible ideal $\mathcal{I}$, the following are equivalent.

(i)

$\mathcal{I}$ has $P^{-}$.

(ii)

For every $S\in {\mathcal{I}}^{+}$ and every sequence $\left(G_n\right)$ in ${\mathcal{I}}^{★}$, there are finite sets

$$F_n\subseteq S\cap G_n(n\in \mathbb{N})$$

such that ${\bigcup }_n{F}_n\in {\mathcal{I}}^{+}$.

Proof. 

Assume first that $\mathcal{I}$ has $P^{-}$. Given $S\in {\mathcal{I}}^{+}$ and $G_n\in {\mathcal{I}}^{★}$, replace $G_n$ by $G_0\cap \dots \cap G_n$ and put $T_n=S\cap G_n$, $T={\bigcap }_n{T}_n$. If $T\in {\mathcal{I}}^{+}$, partition T into finite sets $F_n$. Then $F_n\subseteq S\cap G_n$ and ${\bigcup }_n{F}_n=T$.

Suppose $T\in \mathcal{I}$. The sets

$$E_0=(S\setminus T_0)\cup T,E_{n+1}=T_n\setminus T_{n+1}$$

form a partition of S into members of $\mathcal{I}$. By $P^{-}$, choose $A\in {\mathcal{I}}^{+}$, $A\subseteq S$, with $A\cap E_n$ finite for every n. Set $F_n=A\cap E_{n+1}$. Then $F_n\subseteq T_n\subseteq S\cap G_n$ and

$$\bigcup _n{F}_n=A\setminus (A\cap E_0)\in {\mathcal{I}}^{+}.$$

Conversely, suppose the finite-dual form holds and write $S={⨆}_n{E}_n$ with $S\in {\mathcal{I}}^{+}$ and $E_n\in \mathcal{I}$. Put

$$G_n=(dom\left(\mathcal{I}\right)\setminus S)\cup \bigcup _{i\ge n}{E}_i.$$

Then $G_n\in {\mathcal{I}}^{★}$. Choose finite $F_n\subseteq S\cap G_n$ with $A={\bigcup }_n{F}_n\in {\mathcal{I}}^{+}$. For each m,

$$A\cap E_m\subseteq \bigcup _{n\le m}{F}_n,$$

so $A\cap E_m$ is finite. This proves the partition condition in Definition 2.2. □

Lemma 2.4.

Let $\mathcal{A}$ be an admissible ideal on a countably infinite set D, and let

$$D=\underset{m\in \mathbb{N}}{⨆}{C}_m.$$

Suppose $C_m\in \mathcal{A}$ for every m, and every $B\subseteq D$ such that $B\cap C_m$ is finite for every m belongs to $\mathcal{A}$. Then $\mathcal{A}$ does not have $P^{-}$.

Proof. 

Put $G_n={\bigcup }_{m\ge n}{C}_m$. The complement of $G_n$ is a finite union of members of $\mathcal{A}$, so $G_n\in {\mathcal{A}}^{★}$. If finite $F_n\subseteq G_n$ are chosen, then for fixed m only $F_0,\dots ,F_m$ can meet $C_m$. Hence ${\bigcup }_n{F}_n$ meets every $C_m$ in a finite set and therefore belongs to $\mathcal{A}$. Applying Lemma 2.3 with the positive set D shows that $\mathcal{A}$ does not have $P^{-}$. □

In the dual-filter formulation, Debs–Saint Raymond [4, Lemma 7.4] obtain a stronger embedding conclusion under a more general finite-intersection hypothesis. Only the finite-dual consequence in Lemma 2.4 is used in Section 4 and Section 5.

Two standard conditions stronger than $P^{-}$ are the additive property $\left(AP\right)$ and $P^{+}$. The property $\left(AP\right)$ means that for every sequence $\left(E_k\right)$ in $\mathcal{I}$ there is $E\in \mathcal{I}$ with $E_k{\subseteq }^{*}E$ for every k. Equivalently, $\mathrm{AP}(\mathcal{I},\mathrm{Fin})$ holds [33, Lemma 3.9 and Definition 3.10]. The property $P^{+}$ means that every decreasing sequence $\left(S_n\right)$ in ${\mathcal{I}}^{+}$ has $S\in {\mathcal{I}}^{+}$ with $S{\subseteq }^{*}{S}_n$ for all n [14, p. 2023].

Both conditions imply $P^{-}$. Let $S={⨆}_n{E}_n\in {\mathcal{I}}^{+}$ with $E_n\in \mathcal{I}$. Under $\left(AP\right)$, choose $E\in \mathcal{I}$ with $E_k\setminus E$ finite for every k. Then $A=S\setminus E\in {\mathcal{I}}^{+}$ and $A\cap E_k$ is finite. Under $P^{+}$, put $B_k={\bigcup }_{n\ge k}{E}_n$. Each $B_k$ belongs to ${\mathcal{I}}^{+}$, since $S\setminus B_k$ is a finite union of members of $\mathcal{I}$. Choose $B\in {\mathcal{I}}^{+}$ with $B{\subseteq }^{*}{B}_k$ for every k and set $A=B\cap S$. Since $B{\subseteq }^{*}{B}_0=S$, the set $B\setminus S$ is finite, and hence $A\in {\mathcal{I}}^{+}$. Also,

$$A\cap E_k\subseteq B\setminus B_{k+1},$$

so $A\cap E_k$ is finite.

2.3. Fubini Products and K-Uniformity

For ideals $\mathcal{K},\mathcal{J}$ on countable sets D and E, respectively, their Fubini product is the ideal on $D\times E$ defined by

$$A\in \mathcal{K}\otimes \mathcal{J}⟺\{d\in D:A_d\notin \mathcal{J}\}\in \mathcal{K},$$

where $A_d=\{e\in E:(d,e)\in A\}$ [20]. If $\mathcal{K}$ and $\mathcal{J}$ are admissible, then so is $\mathcal{K}\otimes \mathcal{J}$. For $r\ge 1$ and ideals ${\mathcal{I}}_1,\dots ,{\mathcal{I}}_r$, the product is right-associated. For $r=1$ the iterate is ${\mathcal{I}}_1$. For $r\ge 2$, it is defined by

$${\mathcal{I}}_1\otimes \cdots \otimes {\mathcal{I}}_r={\mathcal{I}}_1\otimes ({\mathcal{I}}_2\otimes \cdots \otimes {\mathcal{I}}_r).$$

When all factors are $\mathrm{Fin}$, write ${\mathrm{Fin}}^{\otimes r}$ for the r-fold Fubini product.

For ideals $\mathcal{A}$ on X and $\mathcal{B}$ on Y, write $\mathcal{A}{\le }_K\mathcal{B}$ if there is a map $f:Y\to X$ such that $f^{-1}\left[C\right]\in \mathcal{B}$ for every $C\in \mathcal{A}$. For $A\subseteq X$, let $\mathcal{A}\upharpoonright A=\{C\cap A:C\in \mathcal{A}\}$, viewed as an ideal on A. Following Hrušák [13, p. 37], an ideal $\mathcal{A}$ is K-uniform if

$$\mathcal{A}\upharpoonright A{\le }_K\mathcal{A}\mathrm{for}\mathrm{every}A\in {\mathcal{A}}^{+}.$$

Zhou–Lin–Zhang [40, Definition 5.3] use the same condition for ideal convergence. An ideal $\mathcal{A}$ is homogeneous if $\mathcal{A}\upharpoonright A\cong \mathcal{A}$ for every $A\in {\mathcal{A}}^{+}$ [20, Definition 1.3]. Every homogeneous ideal is K-uniform.

3. Local Point-Discrete Countability and Local HCP-Finiteness

3.1. Local Point-Discrete Countability and Problem 5.2

The metrization problem requires local countability, not local finiteness. The next lemma isolates a stronger countability phenomenon already forced by point-discreteness along one ideal-convergent sequence.

Lemma 3.1.

Let $\mathcal{I}$ be an admissible ideal, let X be a space, not necessarily $T_1$, and let $x\in X$. Suppose

$$y_n{\to }_{\mathcal{I}}x,S=\{n:y_n\ne x\}\in {\mathcal{I}}^{+}.$$

If $\mathcal{H}$ is a point-discrete family of $\mathcal{I}$-sequential neighborhoods of x, then $\mathcal{H}$ is countable.

Proof. 

For $n\in S$, put

$$\mathcal{H}\left(n\right)=\{P\in \mathcal{H}:y_n\in P\},$$

and let

$$D=\{n\in S:\mathcal{H}(n\left)\mathrm{is}\mathrm{infinite}\right\}.$$

First, $D\in \mathcal{I}$. Suppose otherwise that $D\in {\mathcal{I}}^{+}$. Since $\mathcal{I}$ is admissible, D is infinite. Enumerate it without repetition as $D=\{n_k:k\in \mathbb{N}\}$. Recursively choose pairwise distinct

$$P_k\in \mathcal{H}\left(n_k\right).$$

This is possible because each $\mathcal{H}\left(n_k\right)$ is infinite and only finitely many members have been excluded at stage k.

Every member of $\mathcal{H}$ contains x, since the constant sequence at x is $\mathcal{I}$-convergent. Select x from $P_0$, select $y_{n_k}$ from $P_k$ for $k\ge 1$, and select x from every remaining member of $\mathcal{H}$. Point-discreteness then makes

$$A=\left\{x\right\}\cup \{y_{n_k}:k\ge 1\}$$

a closed discrete set. Since $D\subseteq S$, none of the points $y_{n_k}$ equals x. Moreover, $D\setminus \left\{n_0\right\}\in {\mathcal{I}}^{+}$ because $\left\{n_0\right\}\in \mathcal{I}$. Hence

$$x\in cl\{y_{n_k}:k\ge 1\}.$$

A neighborhood U of x disjoint from $\{y_{n_k}:k\ge 1\}$ would imply

$$D\setminus \left\{n_0\right\}\subseteq \{n:y_n\notin U\}\in \mathcal{I},$$

contrary to $D\setminus \left\{n_0\right\}\in {\mathcal{I}}^{+}$. This contradicts the discreteness of A at x. Thus $D\in \mathcal{I}$.

Now fix $P\in \mathcal{H}$. Since P is an $\mathcal{I}$-sequential neighborhood of x,

$$E_P=\{n:y_n\notin P\}\in \mathcal{I}.$$

Put

$$G_P=\{n\in S:y_n\in P\}=S\setminus E_P.$$

The set $G_P$ belongs to ${\mathcal{I}}^{+}$. Otherwise

$$S=(S\cap E_P)\cup G_P\in \mathcal{I},$$

a contradiction. Since $D\in \mathcal{I}$, also $G_P\setminus D\in {\mathcal{I}}^{+}$, so there is some $n\in S\setminus D$ with $y_n\in P$. Therefore

$$P\in \mathcal{H}\left(n\right)$$

for some $n\in S\setminus D$, and hence

$$\mathcal{H}\subseteq \bigcup _{n\in S\setminus D}\mathcal{H}\left(n\right).$$

Each family $\mathcal{H}\left(n\right)$ on the right is finite by the definition of D, and $S\setminus D$ is countable. Thus $\mathcal{H}$ is countable. □

Theorem 3.2.

Let $\mathcal{I}$ be an admissible ideal.

(1)

Every space, not necessarily $T_1$, with a σ-point-discrete $\mathcal{I}$-$sn$-network is $\mathcal{I}$-$snf$-countable.

(2)

Every regular space with a σ-hereditarily closure-preserving $\mathcal{I}$-$sn$-network is $\mathcal{I}$-$sn$-metrizable.

Proof. 

For (1), let

$$\mathcal{P}=\bigcup _{m\in \mathbb{N}}{\mathcal{P}}_m$$

be a $\sigma$-point-discrete $\mathcal{I}$-$sn$-network on X, where each ${\mathcal{P}}_m$ is point-discrete. Fix $x\in X$.

Suppose first that there is an $\mathcal{I}$-convergent sequence $y_n{\to }_{\mathcal{I}}x$ such that

$$\{n:y_n\ne x\}\in {\mathcal{I}}^{+}.$$

For each m, the trace

$${\mathcal{P}}_m\cap {\mathcal{P}}_x$$

is a point-discrete family of $\mathcal{I}$-sequential neighborhoods of x. Lemma 3.1 shows that every such trace is countable. Hence

$${\mathcal{P}}_x=\bigcup _{m\in \mathbb{N}}({\mathcal{P}}_m\cap {\mathcal{P}}_x)$$

is countable. Since ${\mathcal{P}}_x$ is already a downward directed local network consisting of $\mathcal{I}$-sequential neighborhoods, it is a countable local $\mathcal{I}$-$sn$-network at x.

Suppose instead that every sequence $y_n{\to }_{\mathcal{I}}x$ satisfies

$$\{n:y_n\ne x\}\in \mathcal{I}.$$

Then $\left\{x\right\}$ is an $\mathcal{I}$-sequential neighborhood of x. The singleton family $\left\{\right\{x\left\}\right\}$ is downward directed and is a local network at x, because $\left\{x\right\}$ is contained in every neighborhood of x. Thus x again has a countable local $\mathcal{I}$-$sn$-network. Therefore X is $\mathcal{I}$-$snf$-countable.

For (2), let X be regular and let $\mathcal{P}$ be a $\sigma$-HCP $\mathcal{I}$-$sn$-network. By Lemma 2.1, the same family is a $\sigma$-HCP ordinary $sn$-network. Ge’s characterization [8, Lemma 2.2] gives ordinary $sn$-metrizability. Since X is $T_1$ and every HCP family in a $T_1$ space is point-discrete, part (1) gives $\mathcal{I}$-$snf$-countability. Zhou–Liu–Liu–Lin’s characterization [41, Theorem 3.6] therefore implies that X is $\mathcal{I}$-$sn$-metrizable. □

Theorem 3.2(2) answers Problem 5.2 of Zhou–Liu–Liu–Lin [41, Problem 5.2]. Part (1) gives the stronger point-discrete countability statement. The remainder of the paper studies the stronger local HCP-finiteness property.

3.2. Local HCP-Finiteness Can Fail

Definition 3.3.

An admissible ideal $\mathcal{I}$ has the local HCP-finiteness property, denoted $\mathsf{HCF}\left(\mathcal{I}\right)$, if the following holds. Whenever X is a $T_1$ space, $x\in X$, $\mathcal{H}$ is an HCP family of $\mathcal{I}$-sequential neighborhoods of x, and $y_n{\to }_{\mathcal{I}}x$ with

$$S=\{n:y_n\ne x\}\in {\mathcal{I}}^{+},$$

then $\mathcal{H}$ is finite.

Definition 3.3 asks for finiteness on HCP families, whereas Lemma 3.1 gives universal countability already for the broader class of point-discrete families. Ideal isomorphisms preserve local HCP-finiteness by transporting indexed sequences along the carrier bijection, and preserve K-uniformity by conjugating Katětov maps with that bijection.

Theorem 3.4.

There is an admissible ideal $\mathcal{I}$ on $\mathbb{N}$ for which $\mathsf{HCF}\left(\mathcal{I}\right)$ fails. More precisely, the failure is witnessed by a zero-dimensional Hausdorff space X, a point $p\in X$, an $\mathcal{I}$-convergent sequence to p with $\mathcal{I}$-positive support away from p, and a countably infinite HCP family of $\mathcal{I}$-sequential neighborhoods of p.

Proof. 

Use the convention that $\mathcal{V}{\le }_{\mathrm{RK}}\mathcal{U}$ when $\mathcal{V}$ is the image of $\mathcal{U}$ under a map. Shelah’s theorem, presented by Shelah and Rudin [35, pp. 199–200], gives $2^{2^{{\aleph }_0}}$ pairwise Rudin–Keisler incomparable ultrafilter types on $\mathbb{N}$. The principal type is Rudin–Keisler below every ultrafilter type, so none of these types is principal. Choose countably many representatives, all free, transport them to pairwise disjoint countably infinite sets $D_0,D_1,\dots$, and denote the resulting ultrafilters by

$${\mathcal{U}}_0,{\mathcal{U}}_1,\dots .$$

Put

$${\mathcal{K}}_m=\mathcal{P}\left(D_m\right)\setminus {\mathcal{U}}_m$$

for $m\in \mathbb{N}$, let $D={⨆}_{m\in \mathbb{N}}{D}_m$, and define ideals on D by

$$\mathcal{K}=\left\{A\subseteq D:A\cap D_m\in {\mathcal{K}}_m\mathrm{for}\mathrm{every}m\in \mathbb{N}\right\}$$

and

$$\mathcal{I}=\left\{A\subseteq D:\{m:A\cap D_m\notin {\mathcal{K}}_m\}\in \mathrm{Fin}\right\}.$$

Both families are ideals, and every finite $A\subseteq D$ belongs to both. Since $D_m\notin {\mathcal{K}}_m$ for every m, one has $D\notin \mathcal{K}$, while $\{m:D_m\notin {\mathcal{K}}_m\}=\mathbb{N}\notin \mathrm{Fin}$, so $D\notin \mathcal{I}$. Hence both ideals are admissible, and $\mathcal{K}\subseteq \mathcal{I}$. The carrier D may be transported to $\mathbb{N}$ by a bijection, as in the conventions of Section 2.

Let

$$X=D\cup \left\{p\right\}.$$

Declare every point of D isolated, and take

$$V_A=\left\{p\right\}\cup (D\setminus A),A\in \mathcal{K},$$

as a neighborhood base at p. This is a neighborhood base because $V_A\cap V_B=V_{A\cup B}$ and $\mathcal{K}$ is an ideal. Since $X\setminus V_A=A\subseteq D$ is open, every $V_A$ is clopen. For $d\in D$, the singleton $\left\{d\right\}$ is open by isolation and closed because $X\setminus \left\{d\right\}=V_{\left\{d\right\}}$ is open. Hence X has a clopen base and is zero-dimensional. The disjoint open neighborhoods $V_{\left\{d\right\}}$ and $\left\{d\right\}$ separate p from d, while distinct points of D are separated by singleton neighborhoods. Thus X is Hausdorff. The D-indexed family $y_d=d$ is $\mathcal{I}$-convergent to p, because

$$\{d:y_d\notin V_A\}=A\in \mathcal{K}\subseteq \mathcal{I}$$

for every $A\in \mathcal{K}$. After reindexing D by a bijection with $\mathbb{N}$, this is a sequence in the original definition of $\mathcal{I}$-convergence. The D-indexed notation is retained to make the row structure explicit. Its support away from p is all of D, and

$$D\notin \mathcal{I}$$

because every row $D_m$ is ${\mathcal{K}}_m$-positive.

For $n\in \mathbb{N}$, put

$$P_n=\left\{p\right\}\cup \bigcup _{m\ge n}{D}_m.$$

Since $P_n\setminus P_{n+1}=D_n\ne ⌀$ for every n, the family $\{P_n:n\in \mathbb{N}\}$ is countably infinite. This family is HCP. Fix an arbitrary subfamily. For each selected $P_n$, choose $H_n\subseteq P_n$, and put $H_n=⌀$ for the remaining indices. Since every point of D is isolated, closure preservation can fail only at p. If $p\in cl{H}_n$ for some n, then p belongs to both $cl\left({\bigcup }_n{H}_n\right)$ and ${\bigcup }_{n}cl{H}_n$. Otherwise $p\notin cl{H}_n$ for every n. Then $H_n\subseteq D$. For each n, choose a basic neighborhood $V_{A_n}$ of p disjoint from $H_n$. Then $H_n\subseteq A_n\in \mathcal{K}$, and hence $H_n\in \mathcal{K}$. For a fixed m, only the finitely many indices $n\le m$ can contribute points from $D_m$. Hence

$$\left(\bigcup _n{H}_n\right)\cap D_m$$

is a finite union of members of ${\mathcal{K}}_m$ and therefore belongs to ${\mathcal{K}}_m$. Thus ${\bigcup }_n{H}_n\in \mathcal{K}$, so $p\notin cl{\bigcup }_n{H}_n$. Closure preservation at the isolated points of D is automatic.

It remains to show that every $P_n$ is an $\mathcal{I}$-sequential neighborhood of p. Let $z_d{\to }_{\mathcal{I}}p$ be an arbitrary D-indexed family and write $f\left(d\right)=z_d$. Fix $r\in \mathbb{N}$ and put

$$E=f^{-1}\left[D_r\right].$$

The claim is that $E\in \mathcal{I}$. Suppose not. Then

$$M=\{m:E\cap D_m\in {\mathcal{U}}_m\}$$

is infinite. For $m\in M$, put $E_m=E\cap D_m$ and let

$$f_m:E_m⟶D_r$$

be the restriction of f. Since $E_m\in {\mathcal{U}}_m$, the trace

$${\mathcal{U}}_m\upharpoonright E_m=\{A\cap E_m:A\in {\mathcal{U}}_m\}$$

is an ultrafilter on $E_m$. Let

$${\mathcal{V}}_m=\{B\subseteq D_r:f_m^{-1}\left[B\right]\in {\mathcal{U}}_m\upharpoonright E_m\}$$

be its image under $f_m$.

Fix $B\in {\mathcal{K}}_r$, and let $A_B\subseteq D$ have rth section B and all other sections empty. Then $A_B\in \mathcal{K}$. Since $X\setminus V_{A_B}=A_B$ and $z_d{\to }_{\mathcal{I}}p$,

$$f^{-1}\left[A_B\right]\in \mathcal{I}.$$

For $m\in M$,

$$f_m^{-1}\left[B\right]=f^{-1}\left[A_B\right]\cap D_m.$$

Consequently, for all but finitely many $m\in M$,

$$f_m^{-1}\left[B\right]\in {\mathcal{K}}_m.$$

For each such m, since $E_m\in {\mathcal{U}}_m$, membership in the trace ${\mathcal{U}}_m\upharpoonright E_m$ agrees with membership in ${\mathcal{U}}_m$ for subsets of $E_m$. Hence $B\notin {\mathcal{V}}_m$ for all but finitely many $m\in M$. If $C\in {\mathcal{U}}_r$, then $D_r\setminus C\in {\mathcal{K}}_r$. Applying the preceding conclusion to $B=D_r\setminus C$ and using that each ${\mathcal{V}}_m$ is an ultrafilter gives $C\in {\mathcal{V}}_m$ for all but finitely many $m\in M$. Enumerate $M=\{m_k:k\in \mathbb{N}\}$ without repetition and regard $\beta D_r$ as the space of ultrafilters on the discrete set $D_r$. For $C\subseteq D_r$, write $\widehat{C}=\{\mathcal{W}\in \beta D_r:C\in \mathcal{W}\}$. The clopen sets $\widehat{C}$ with $C\in {\mathcal{U}}_r$ form a neighborhood base at ${\mathcal{U}}_r$ in $\beta D_r$, so

$${\mathcal{V}}_{m_k}⟶{\mathcal{U}}_r.$$

Every convergent sequence in $\beta D_r$ is eventually constant [5, Corollary 3.6.15]. Hence ${\mathcal{V}}_{m_k}={\mathcal{U}}_r$ for all sufficiently large k.

Choose a sufficiently large k with $m_k\ne r$. By construction, ${\mathcal{V}}_{m_k}$ is a Rudin–Keisler image of ${\mathcal{U}}_{m_k}\upharpoonright E_{m_k}$. Since $E_{m_k}\in {\mathcal{U}}_{m_k}$, the trace ${\mathcal{U}}_{m_k}\upharpoonright E_{m_k}$ is Rudin–Keisler equivalent to ${\mathcal{U}}_{m_k}$. The inclusion $E_{m_k}\hookrightarrow D_{m_k}$ maps the trace ultrafilter to ${\mathcal{U}}_{m_k}$, while any retraction $D_{m_k}\to E_{m_k}$ that fixes $E_{m_k}$ maps ${\mathcal{U}}_{m_k}$ to the trace. It follows that

$${\mathcal{U}}_r{\le }_{\mathrm{RK}}{\mathcal{U}}_{m_k},$$

contrary to the choice of the ultrafilters. Therefore $f^{-1}\left[D_r\right]\in \mathcal{I}$ for every $r\in \mathbb{N}$.

Finally,

$$X\setminus P_n=\bigcup _{r<n}{D}_r.$$

The inverse image of this finite union belongs to $\mathcal{I}$, so $P_n$ is an $\mathcal{I}$-sequential neighborhood of p. The family $\{P_n:n\in \mathbb{N}\}$, together with the $\mathcal{I}$-convergent family $y_d=d$ constructed above, witnesses the failure of $\mathsf{HCF}\left(\mathcal{I}\right)$. □

Because the witness family in Theorem 3.4 is HCP, hence point-discrete, that theorem shows that the countability conclusion of Lemma 3.1 cannot in general be strengthened to finiteness. Local HCP-finiteness requires additional structure on the ideal.

3.3. $P^{-}$ Implies Local HCP-Finiteness

Lemma 3.5.

If $\mathcal{I}$ has $P^{-}$, then $\mathsf{HCF}\left(\mathcal{I}\right)$ holds.

Proof. 

Suppose $\mathcal{H}$ is infinite and choose pairwise distinct $P_0,P_1,\dots \in \mathcal{H}$. Let $y_n{\to }_{\mathcal{I}}x$ and $S=\{n:y_n\ne x\}\in {\mathcal{I}}^{+}$. Since each $P_j$ is an $\mathcal{I}$-sequential neighborhood of x,

$$G_j=\{n:y_n\in P_j\}\in {\mathcal{I}}^{★}.$$

By Lemma 2.3, choose finite

$$F_j\subseteq S\cap G_j$$

such that $F={\bigcup }_j{F}_j\in {\mathcal{I}}^{+}$. Put

$$H\left(P_j\right)=\{y_n:n\in F_j\}\subseteq P_j$$

and set $H\left(P\right)=⌀$ for $P\in \mathcal{H}\setminus \{P_j:j\in \mathbb{N}\}$. Each $H\left(P_j\right)$ is finite and contained in $X\setminus \left\{x\right\}$, so $x\notin clH\left(P_j\right)$ because X is $T_1$.

On the other hand, $F\in {\mathcal{I}}^{+}$ and $y_n{\to }_{\mathcal{I}}x$ imply

$$x\in cl\{y_n:n\in F\}=cl\left(\bigcup _{j}H\left(P_j\right)\right).$$

Hereditary closure preservation gives

$$cl\left(\bigcup _{j}H\left(P_j\right)\right)=\bigcup _{j}clH\left(P_j\right),$$

a contradiction. □

Thus $P^{-}$ gives a basic sufficient condition for local HCP-finiteness. Section 4 and Section 5 show how the property persists beyond $P^{-}$ through Fubini constructions and increasing unions.

4. Fubini Constructions

4.1. Fubini Products

Local HCP-finiteness is preserved by Fubini products with a $P^{-}$ outer factor, although $P^{-}$ need not be preserved.

Lemma 4.1.

Let $\mathcal{K}$ and $\mathcal{J}$ be admissible ideals. Every $(\mathcal{K}\otimes \mathcal{J})$-sequential neighborhood of x is a $\mathcal{J}$-sequential neighborhood of x.

Proof. 

Let P be a $(\mathcal{K}\otimes \mathcal{J})$-sequential neighborhood of x, and let $z_j{\to }_{\mathcal{J}}x$. Define the array $w_{i,j}=z_j$. For each neighborhood U of x the set

$$\{(i,j):w_{i,j}\notin U\}=dom\left(\mathcal{K}\right)\times \{j:z_j\notin U\}$$

belongs to $\mathcal{K}\otimes \mathcal{J}$, so $w_{i,j}{\to }_{\mathcal{K}\otimes \mathcal{J}}x$. Hence

$$dom\left(\mathcal{K}\right)\times \{j:z_j\notin P\}\in \mathcal{K}\otimes \mathcal{J}.$$

Since $\mathcal{K}$ is proper, this is possible only if $\{j:z_j\notin P\}\in \mathcal{J}$. □

Theorem 4.2.

Let $\mathcal{K}$ and $\mathcal{J}$ be admissible ideals. If $\mathcal{K}$ has $P^{-}$ and $\mathsf{HCF}\left(\mathcal{J}\right)$ holds, then

$$\mathsf{HCF}(\mathcal{K}\otimes \mathcal{J})$$

holds.

Proof. 

Let X be $T_1$, let $x\in X$, let $\mathcal{H}$ be an HCP family of $(\mathcal{K}\otimes \mathcal{J})$-sequential neighborhoods of x, and let

$$y_{i,j}{\to }_{\mathcal{K}\otimes \mathcal{J}}x$$

with

$$S=\{(i,j):y_{i,j}\ne x\}\in {(\mathcal{K}\otimes \mathcal{J})}^{+}.$$

Write $S_i=\{j:y_{i,j}\ne x\}$ and

$$C=\{i:S_i\in {\mathcal{J}}^{+}\}.$$

Then $C\in {\mathcal{K}}^{+}$. Suppose that $\mathcal{H}$ is infinite and choose pairwise distinct $P_0,P_1,\dots \in \mathcal{H}$.

By Lemma 2.3, each $P_n$ is a $\mathcal{J}$-sequential neighborhood of x. If for some $i\in C$ the row ${\left(y_{i,j}\right)}_j$ is $\mathcal{J}$-convergent to x, then $S_i\in {\mathcal{J}}^{+}$ and $\mathsf{HCF}\left(\mathcal{J}\right)$ forces the HCP family $\{P_n:n\in \mathbb{N}\}$ to be finite, a contradiction.

No row indexed by C can therefore be $\mathcal{J}$-convergent to x. For each $i\in C$ choose a neighborhood $U_i$ of x such that

$$B_i=\{j:y_{i,j}\notin U_i\}\in {\mathcal{J}}^{+}.$$

For each n, since $P_n$ is a $(\mathcal{K}\otimes \mathcal{J})$-sequential neighborhood,

$$D_n=\{(i,j):y_{i,j}\notin P_n\}\in \mathcal{K}\otimes \mathcal{J}.$$

Thus

$$E_n=\{i:{\left(D_n\right)}_i\notin \mathcal{J}\}\in \mathcal{K},G_n=dom\left(\mathcal{K}\right)\setminus E_n\in {\mathcal{K}}^{★}.$$

By Lemma 2.3 applied to $C\in {\mathcal{K}}^{+}$ and $\left(G_n\right)$, there are finite

$$R_n\subseteq C\cap G_n$$

with $R={\bigcup }_n{R}_n\in {\mathcal{K}}^{+}$. For $i\in R_n$ put

$$A_{n,i}=B_i\setminus {\left(D_n\right)}_i.$$

Since ${\left(D_n\right)}_i\in \mathcal{J}$ and $B_i\in {\mathcal{J}}^{+}$, $A_{n,i}\in {\mathcal{J}}^{+}$. Define

$$H\left(P_n\right)=\{y_{i,j}:i\in R_n,j\in A_{n,i}\}\subseteq P_n$$

and set $H\left(P\right)=⌀$ for the remaining members of $\mathcal{H}$. Because $R_n$ is finite,

$$V_n=\bigcap _{i\in R_n}{U}_i$$

is a neighborhood of x, with the empty intersection interpreted as X, and $H\left(P_n\right)\cap V_n=⌀$. Hence $x\notin clH\left(P_n\right)$ for every n.

Let

$$F=\bigcup _n\bigcup _{i\in R_n}\left(\left\{i\right\}\times A_{n,i}\right).$$

For every $i\in R$ the section $F_i$ contains some $\mathcal{J}$-positive $A_{n,i}$. Therefore

$$\{i:F_i\notin \mathcal{J}\}\supseteq R\in {\mathcal{K}}^{+},$$

so $F\notin \mathcal{K}\otimes \mathcal{J}$. The convergence of $\left(y_{i,j}\right)$ gives

$$x\in cl\{y_{i,j}:(i,j)\in F\}=cl\left(\bigcup _{n}H\left(P_n\right)\right),$$

contradicting hereditary closure preservation. □

Every Fubini product of two admissible ideals fails $P^{-}$. After reindexing the outer domain as $\mathbb{N}$, let

$$C_i=\left\{i\right\}\times dom\left(\mathcal{J}\right).$$

Each $C_i$ belongs to $\mathcal{K}\otimes \mathcal{J}$, and every set meeting each $C_i$ finitely has all sections in $\mathcal{J}$. Lemma 2.4 applies. Equivalently, after reindexing the countable carriers, $\mathrm{Fin}\otimes \mathrm{Fin}{\le }_K\mathcal{K}\otimes \mathcal{J}$, and the Katětov boundary for $P^{-}$ [14, Theorem 3.8(4)] gives the same conclusion.

Corollary 4.3.

Let $r\ge 1$ and let ${\mathcal{I}}_1,\dots ,{\mathcal{I}}_r$ be admissible ideals. If ${\mathcal{I}}_k$ has $P^{-}$ for $1\le k<r$ and $\mathsf{HCF}\left({\mathcal{I}}_r\right)$ holds, then

$$\mathsf{HCF}({\mathcal{I}}_1\otimes \dots \otimes {\mathcal{I}}_r)$$

holds. If $r\ge 2$, the product ideal does not have $P^{-}$.

Proof. 

For $r=1$, the local HCP-finiteness assertion is the hypothesis. For $r>1$, apply Theorem 4.2 inductively from the innermost factor outward. The preceding Fubini-product observation proves the last assertion. □

Corollary 4.4.

For every $r\ge 1$, $\mathsf{HCF}\left({\mathrm{Fin}}^{\otimes r}\right)$ holds. For every $r\ge 2$, ${\mathrm{Fin}}^{\otimes r}$ does not have $P^{-}$.

Proof. 

For every sequence $\left(E_k\right)$ of finite sets, $E=⌀$ satisfies $E_k{\subseteq }^{*}E$ for all k. Thus $\mathrm{Fin}$ has $\left(AP\right)$ and hence $P^{-}$. Lemma 3.5 gives $\mathsf{HCF}\left(\mathrm{Fin}\right)$, so Corollary 4.3 applies. □

Thus local HCP-finiteness is not confined to $P^{-}$ ideals.

4.2. Heterogeneous Fubini Sums

The product theorem treats a fixed inner ideal. At limit stages such as ${\mathrm{Fin}}^{\omega }$, the inner ideal varies from row to row. The Fubini-sum notation follows Kwela [18, Subsection 2.2]. Let $\mathcal{K}$ be an admissible ideal on a countably infinite set I, and let ${\mathcal{J}}_i$ be an admissible ideal on a countably infinite set $D_i$ for each $i\in I$. Regard ${⨆}_{i\in I}{D}_i$ as the tagged union ${\bigcup }_{i\in I}(\left\{i\right\}\times D_i)$ and, for $A\subseteq {⨆}_{i\in I}{D}_i$, put $A_i=\{d\in D_i:(i,d)\in A\}$. Define

$$\mathcal{S}=\mathcal{K}-\sum _{i\in I}{\mathcal{J}}_i$$

by

$$A\in \mathcal{S}⟺\{i\in I:A_i\notin {\mathcal{J}}_i\}\in \mathcal{K}.$$

Under these assumptions, $\mathcal{S}$ is admissible. Heredity and finite-union closure follow from the ideal axioms. Every finite set has only finite sections, so all of its sections lie in the corresponding row ideals. The whole carrier is excluded because its bad-row set is $I\notin \mathcal{K}$. For $i\in I$ and $E\in {\mathcal{J}}_i^{+}$, set

$$R(i,E)=\{j\in I:{\mathcal{J}}_i\upharpoonright E{\le }_K{\mathcal{J}}_j\}.$$

The global cross-row absorption condition is

$$R(i,E)\in {\mathcal{K}}^{+}(i\in I,E\in {\mathcal{J}}_i^{+}).$$

(4.1)

For comparison with (4.1), fix ideals $\mathcal{I}$ and $\mathcal{J}$. Kwela–Lesner–Tryba [19, Theorem 6.1] prove that $\mathcal{I}\upharpoonright A{\le }_K\mathcal{J}$ for every $A\in {\mathcal{I}}^{+}$ if and only if every $\mathcal{J}$-continuous function preserves $\mathcal{I}$-convergence.

In the constant-row case $D_i=D$ and ${\mathcal{J}}_i=\mathcal{J}$ for all i, condition (4.1) reduces to K-uniformity of $\mathcal{J}$. Theorem 4.2 does not require this additional uniformity assumption. Theorem 4.6 requires this absorption condition only for rows in some $W\in {\mathcal{K}}^{★}$.

Lemma 4.5.

Let $W\subseteq I$ and suppose $R(i,E)\in {\mathcal{K}}^{+}$ for every $i\in W$ and every $E\in {\mathcal{J}}_i^{+}$. Then every $\mathcal{S}$-sequential neighborhood of x is a ${\mathcal{J}}_i$-sequential neighborhood of x for each $i\in W$.

Proof. 

Fix $i\in W$, and suppose that an $\mathcal{S}$-sequential neighborhood P is not a ${\mathcal{J}}_i$-sequential neighborhood of x. Choose $z_d{\to }_{{\mathcal{J}}_i}x$ such that

$$E=\{d\in D_i:z_d\notin P\}\in {\mathcal{J}}_i^{+}.$$

Then $R=R(i,E)\in {\mathcal{K}}^{+}$. For each $j\in R$, choose

$$f_j:D_j⟶E$$

witnessing ${\mathcal{J}}_i\upharpoonright E{\le }_K{\mathcal{J}}_j$, and define

$$w_{j,e}=\left\{\begin{array}{cc}{z}_{f_j\left(e\right)},\hfill & j\in R,\hfill \\ x,\hfill & j\notin R.\hfill \end{array}\right.$$

If U is a neighborhood of x, then $B_U=\{d:z_d\notin U\}\in {\mathcal{J}}_i$, and for $j\in R$,

$$\{e:w_{j,e}\notin U\}=f_j^{-1}[B_U\cap E]\in {\mathcal{J}}_j.$$

For $j\notin R$ and $e\in D_j$, $w_{j,e}=x\in U$, so the corresponding bad section is empty. Hence $w{\to }_{\mathcal{S}}x$. On the other hand, for every $j\in R$ the whole jth row lies outside P. Thus the set of rows on which w is outside P is $\mathcal{K}$-positive, contradicting the $\mathcal{S}$-sequential-neighborhood property of P. □

Theorem 4.6.

Let

$$\mathcal{S}=\mathcal{K}-\sum _{i\in I}{\mathcal{J}}_i.$$

If $\mathcal{K}$ has $P^{-}$ and, for some $W\in {\mathcal{K}}^{★}$, $\mathsf{HCF}\left({\mathcal{J}}_i\right)$ holds for every $i\in W$ and $R(i,E)\in {\mathcal{K}}^{+}$ for every $i\in W$ and every $E\in {\mathcal{J}}_i^{+}$, then $\mathsf{HCF}\left(\mathcal{S}\right)$ holds.

Proof. 

Let X be $T_1$, let $x\in X$, let $\mathcal{H}$ be an HCP family of $\mathcal{S}$-sequential neighborhoods of x, and let

$$y_{i,d}{\to }_{\mathcal{S}}x,S=\{(i,d):y_{i,d}\ne x\}\in {\mathcal{S}}^{+}.$$

Put

$$S_i=\{d:y_{i,d}\ne x\},C=\{i:S_i\in {\mathcal{J}}_i^{+}\}\in {\mathcal{K}}^{+}.$$

Since $W\in {\mathcal{K}}^{★}$, also $C_W=C\cap W\in {\mathcal{K}}^{+}$. Suppose $\mathcal{H}$ is infinite and choose pairwise distinct $P_0,P_1,\dots \in \mathcal{H}$. By Lemma 4.5, every $P_n$ is a ${\mathcal{J}}_i$-sequential neighborhood of x for every $i\in W$.

If some row ${\left(y_{i,d}\right)}_{d\in D_i}$ with $i\in C_W$ is ${\mathcal{J}}_i$-convergent to x, then $S_i\in {\mathcal{J}}_i^{+}$ and $\mathsf{HCF}\left({\mathcal{J}}_i\right)$ forces $\mathcal{H}$ to be finite. Hence no row indexed by $C_W$ is ${\mathcal{J}}_i$-convergent. For each $i\in C_W$, choose a neighborhood $U_i$ of x such that

$$B_i=\{d:y_{i,d}\notin U_i\}\in {\mathcal{J}}_i^{+}.$$

For each n, since $P_n$ is an $\mathcal{S}$-sequential neighborhood,

$${\Delta }_n=\{(i,d):y_{i,d}\notin P_n\}\in \mathcal{S}.$$

Thus

$$E_n=\{i:{\left({\Delta }_n\right)}_i\notin {\mathcal{J}}_i\}\in \mathcal{K},G_n=I\setminus E_n\in {\mathcal{K}}^{★}.$$

The finite-dual form of $P^{-}$ gives finite

$$R_n\subseteq C_W\cap G_n$$

such that $R={\bigcup }_n{R}_n\in {\mathcal{K}}^{+}$. For $i\in R_n$, put

$$A_{n,i}=B_i\setminus {\left({\Delta }_n\right)}_i\in {\mathcal{J}}_i^{+}$$

and define

$$H\left(P_n\right)=\{y_{i,d}:i\in R_n,d\in A_{n,i}\}\subseteq P_n,$$

with $H\left(P\right)=⌀$ for the remaining members of $\mathcal{H}$. Since $R_n$ is finite,

$$V_n=\bigcap _{i\in R_n}{U}_i$$

is a neighborhood of x, with the empty intersection interpreted as X, and $H\left(P_n\right)\cap V_n=⌀$. Hence $x\notin clH\left(P_n\right)$.

Let

$$F=\bigcup _n\bigcup _{i\in R_n}\left(\left\{i\right\}\times A_{n,i}\right).$$

Every $i\in R$ has $F_i\notin {\mathcal{J}}_i$, so $F\notin \mathcal{S}$. The $\mathcal{S}$-convergence of y gives

$$x\in cl\{y_{i,d}:(i,d)\in F\}=cl\left(\bigcup _{n}H\left(P_n\right)\right),$$

contradicting hereditary closure preservation. Therefore $\mathcal{H}$ is finite. □

If, for some $W\in {\mathcal{K}}^{★}$, every ${\mathcal{J}}_i$ with $i\in W$ is K-uniform and

$$\{j\in I:{\mathcal{J}}_i{\le }_K{\mathcal{J}}_j\}\in {\mathcal{K}}^{+}(i\in W),$$

(4.2)

then the absorption hypothesis of Theorem 4.6 holds by transitivity of the Katětov order.

4.3. The First Limit Stage and Its Finite Successors

At Katětov’s first limit stage, the heterogeneous criterion applies to the row decomposition of ${\mathrm{Fin}}^{\omega }$. Katětov [16, p. 240] gave the original filter recursion. Kwela [18, Subsection 2.2] gives the corresponding ideal formulation and notation. The superscript notation distinguishes this hierarchy from the inductive-limit ideal ${\mathrm{Fin}}_{\omega }$ of Section 5. The hierarchy satisfies

$${\mathrm{Fin}}^1=\mathrm{Fin},{\mathrm{Fin}}^{\alpha +1}=\mathrm{Fin}\otimes {\mathrm{Fin}}^{\alpha }(0<\alpha <{\omega }_1),$$

and at a limit ordinal $0<\lambda <{\omega }_1$,

$${\mathrm{Fin}}^{\lambda }={\mathcal{I}}_{\lambda }-\sum _{0<\alpha <\lambda }{\mathrm{Fin}}^{\alpha },$$

where ${\mathcal{I}}_{\lambda }$ is the ideal of bounded subsets of $\lambda \setminus \left\{0\right\}$. At the first limit stage,

$${\mathrm{Fin}}^{\omega }=\mathrm{Fin}-\sum _{1\le n<\omega }{\mathrm{Fin}}^n.$$

(4.3)

For $1\le n<\omega$, ${\mathrm{Fin}}^n\cong {\mathrm{Fin}}^{\otimes n}$. Corollary 4.4 gives local HCP-finiteness. Kwela–Tryba [20, Example 2.3 and Proposition 2.9] proved that $\mathrm{Fin}$ is homogeneous and that Fubini products preserve homogeneity. Hence each finite power ${\mathrm{Fin}}^n$ is homogeneous and therefore K-uniform. If $1\le n\le m<\omega$, the projection onto the last n coordinates witnesses ${\mathrm{Fin}}^n{\le }_K{\mathrm{Fin}}^m$ [18, Subsection 2.4]. For each $1\le n<\omega$, the set $\{m:1\le m<\omega ,{\mathrm{Fin}}^n{\le }_K{\mathrm{Fin}}^m\}$ contains the tail $\{m:n\le m<\omega \}$ and is $\mathrm{Fin}$-positive. Taking $W=\{n:1\le n<\omega \}\in {\mathrm{Fin}}^{★}$, condition (4.2) holds for the outer ideal $\mathrm{Fin}$. Since $\mathrm{Fin}$ has $P^{-}$, Theorem 4.6 applies at the limit stage.

Corollary 4.7.

For every $r<\omega$,

$$\mathsf{HCF}\left({\mathrm{Fin}}^{\omega +r}\right)$$

holds. Moreover, ${\mathrm{Fin}}^{\omega +r}$ does not have $P^{-}$.

Proof. 

Equation (4.3) and Theorem 4.6 give $\mathsf{HCF}\left({\mathrm{Fin}}^{\omega }\right)$. The heterogeneous sum in (3) is admissible, and Fubini products preserve admissibility. For $s<\omega$, the successor recursion is

$${\mathrm{Fin}}^{\omega +(s+1)}=\mathrm{Fin}\otimes {\mathrm{Fin}}^{\omega +s}.$$

Thus Theorem 4.2 applies at each finite successor.

For the failure of $P^{-}$ at $r=0$, reindex the outer rows of (4.3) by $\mathbb{N}$ and let $C_m$ denote the mth tagged row. Each $C_m$ belongs to ${\mathrm{Fin}}^{\omega }$, and any set meeting each $C_m$ finitely has a finite section in every row and hence belongs to ${\mathrm{Fin}}^{\omega }$. Lemma 2.4 applies. For each $s<\omega$, the Fubini-product observation above applies to the displayed product ${\mathrm{Fin}}^{\omega +(s+1)}$. □

5. Increasing Unions and Rank-$\omega$ Limit Ideals

5.1. The Increasing-Union Theorem

The second permanence route replaces Fubini rows by an increasing chain. Its key step is a cross-ideal version of K-uniform reindexing. Zhang–Zhang [38, Theorem 2.5] used the same-ideal form, and Zhou–Lin–Zhang [40, Definition 5.3 and Lemma 5.5] later applied the reindexing to $\mathcal{I}$-$sn$ structures. In Lemma 5.1, the convergence ideal and the K-uniform stage ideal need not be the same.

Lemma 5.1.

Let $\mathcal{J}\subseteq \mathcal{I}$ be admissible ideals on the same countable set. If $\mathcal{J}$ is K-uniform, then every $\mathcal{I}$-sequential neighborhood of x is a $\mathcal{J}$-sequential neighborhood of x.

Proof. 

Let P be an $\mathcal{I}$-sequential neighborhood of x, and suppose $z_n{\to }_{\mathcal{J}}x$. If

$$D=\{n:z_n\notin P\}\in {\mathcal{J}}^{+},$$

K-uniformity gives a map $f:dom\left(\mathcal{J}\right)\to D$ such that

$$B\in \mathcal{J}\upharpoonright D⟹f^{-1}\left[B\right]\in \mathcal{J}.$$

Define $w_n=z_{f\left(n\right)}$. For every neighborhood U of x, the set

$$\{k:z_k\notin U\}\cap D$$

belongs to $\mathcal{J}\upharpoonright D$, and hence

$$\{n:w_n\notin U\}=f^{-1}[\{k:z_k\notin U\}\cap D]\in \mathcal{J}\subseteq \mathcal{I}.$$

Thus $w_n{\to }_{\mathcal{I}}x$. But $w_n\notin P$ for every n, contradicting that P is an $\mathcal{I}$-sequential neighborhood. Hence $D\in \mathcal{J}$. □

By Kwela–Tryba [20, Example 2.3 and Proposition 2.9], every ${\mathrm{Fin}}^{\otimes r}$ is homogeneous and hence K-uniform for $r\ge 1$. An increasing union of admissible ideals on the same carrier is admissible whenever the union is proper.

Theorem 5.2.

Let ${\mathcal{J}}_0\subseteq {\mathcal{J}}_1\subseteq \dots$ be admissible ideals on the same countable set, and suppose

$$\mathcal{I}=\bigcup _{n\in \mathbb{N}}{\mathcal{J}}_n$$

is proper. If for every N there is $n\ge N$ such that $\mathsf{HCF}\left({\mathcal{J}}_n\right)$ holds and ${\mathcal{J}}_n$ is K-uniform, then $\mathsf{HCF}\left(\mathcal{I}\right)$ holds.

Proof. 

Passing to a cofinal subsequence of stages with both properties and reindexing it by $\mathbb{N}$ does not change the union. Thus every ${\mathcal{J}}_n$ may be assumed to have local HCP-finiteness and be K-uniform.

Let X be $T_1$, let $x\in X$, let $\mathcal{H}$ be an HCP family of $\mathcal{I}$-sequential neighborhoods of x, and let

$$y_k{\to }_{\mathcal{I}}x,S=\{k:y_k\ne x\}\in {\mathcal{I}}^{+}.$$

Suppose $\mathcal{H}$ is infinite and choose pairwise distinct $P_0,P_1,\dots \in \mathcal{H}$. Put

$$D_n=\{k:y_k\notin P_n\}\in \mathcal{I}$$

and choose $d_n$ with $D_n\in {\mathcal{J}}_{d_n}$.

The sequence must converge with respect to ${\mathcal{J}}_N$ for some N. Otherwise, for every r there is a neighborhood U of x with $\{k:y_k\notin U\}\notin {\mathcal{J}}_r$. For each n put $r_n=max\{n,d_n\}$ and choose a neighborhood $U_n$ such that

$$B_n=\{k:y_k\notin U_n\}\notin {\mathcal{J}}_{r_n}.$$

Set $A_n=B_n\setminus D_n$. Since $D_n\in {\mathcal{J}}_{r_n}$, $A_n\notin {\mathcal{J}}_{r_n}$. Also

$$\{y_k:k\in A_n\}\subseteq P_n\setminus U_n,$$

so $x\notin cl\{y_k:k\in A_n\}$.

Let $A={\bigcup }_n{A}_n$. If $A\in \mathcal{I}$, choose M with $A\in {\mathcal{J}}_M$ and then choose $n\ge M$. Since $r_n\ge n\ge M$,

$$A_n\subseteq A\in {\mathcal{J}}_M\subseteq {\mathcal{J}}_{r_n},$$

a contradiction. Hence $A\in {\mathcal{I}}^{+}$. The $\mathcal{I}$-convergence of $\left(y_k\right)$ gives

$$x\in cl\{y_k:k\in A\}=cl\left(\bigcup _n\{y_k:k\in A_n\}\right),$$

contradicting hereditary closure preservation for the subpieces $\{y_k:k\in A_n\}$ of $P_n$. Therefore there is N such that

$$y_k{\to }_{{\mathcal{J}}_N}x.$$

By Lemma 5.1, every member of $\mathcal{H}$ is a ${\mathcal{J}}_N$-sequential neighborhood of x. Because $S\notin \mathcal{I}$, also $S\notin {\mathcal{J}}_N$. Thus $\mathsf{HCF}\left({\mathcal{J}}_N\right)$ implies that $\mathcal{H}$ is finite, a contradiction. □

5.2. The Tree-Derived Ideal ${\mathcal{H}}_{<\omega }$

The first rank-$\omega$ application of the increasing-union theorem is the tree-derived ideal ${\mathcal{H}}_{<\omega }$ of Pelayo Gómez. Fix a tree partition $\langle A_s:s\in {\mathbb{N}}^{<\omega }\rangle$ of $\mathbb{N}$ as in [34, Definition 2.1]. For $X\subseteq \mathbb{N}$, put

$$T_X^{\infty }=\{s\in {\mathbb{N}}^{<\omega }:|X\cap A_s|=\infty \}.$$

For $T\subseteq {\mathbb{N}}^{<\omega }$ define

$$D_{\mathrm{Fin}}\left(T\right)=\{s\in T:\{n:s⌢n\in T\}\mathrm{is}\mathrm{infinite}\},$$

and iterate this operator by $D_{\mathrm{Fin}}^0\left(T\right)=T$ and $D_{\mathrm{Fin}}^{n+1}\left(T\right)=D_{\mathrm{Fin}}\left(D_{\mathrm{Fin}}^n\left(T\right)\right)$. Following [34, Definition 3.1], set

$${\mathcal{H}}_n=\{X\subseteq \mathbb{N}:⌀\notin D_{\mathrm{Fin}}^n\left(T_X^{\infty }\right)\}(n<\omega ).$$

The resulting ideals are independent, up to ideal isomorphism, of the chosen tree partition [34, Proposition 2.4]. Pelayo Gómez [34, Theorems 3.4 and 3.5] proved

$${\mathcal{H}}_0⊊{\mathcal{H}}_1⊊\dots ,{\mathcal{H}}_n\cong {\mathrm{Fin}}^{\otimes (n+1)}(n<\omega ),$$

and

$${\mathcal{H}}_{<\omega }:=\bigcup _{n<\omega }{\mathcal{H}}_n.$$

Pelayo Gómez [34, Proposition 3.8] proved that ${\mathcal{H}}_{<\omega }$ is a proper tall ${\Sigma }_{\omega }^0$ ideal of separation rank exactly $\omega$.

Theorem 5.3.

The ideal ${\mathcal{H}}_{<\omega }$ has local HCP-finiteness and does not have $P^{-}$.

Proof. 

The isomorphisms ${\mathcal{H}}_n\cong {\mathrm{Fin}}^{\otimes (n+1)}$ and the admissibility of finite Fubini products show that each ${\mathcal{H}}_n$ is admissible. Corollary 4.4 and ideal-isomorphism invariance give local HCP-finiteness of each ${\mathcal{H}}_n$. By Kwela–Tryba [20, Example 2.3 and Proposition 2.9], ${\mathrm{Fin}}^{\otimes (n+1)}$ is homogeneous and hence K-uniform. Since K-uniformity is invariant under ideal isomorphism, ${\mathcal{H}}_n$ is K-uniform as well. Theorem 5.2 gives $\mathsf{HCF}\left({\mathcal{H}}_{<\omega }\right)$.

For the failure of $P^{-}$, let $\phi :\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ be an isomorphism from $\mathrm{Fin}\otimes \mathrm{Fin}$ onto ${\mathcal{H}}_1$ and put

$$C_i=\phi [\left\{i\right\}\times \mathbb{N}].$$

Each $C_i$ belongs to ${\mathcal{H}}_1$. If $B\subseteq \mathbb{N}$ meets each $C_i$ finitely, then ${\phi }^{-1}\left[B\right]$ has finite vertical sections, so $B\in {\mathcal{H}}_1\subseteq {\mathcal{H}}_{<\omega }$. Lemma 2.4 shows that ${\mathcal{H}}_{<\omega }$ does not have $P^{-}$. □

5.3. Support Lifts and the Inductive-Limit Ideal ${\mathrm{Fin}}_{\omega }$

The second rank-$\omega$ application is ${\mathrm{Fin}}_{\omega }$. Its finite stages are isomorphic to support lifts of finite Fubini powers. For an admissible ideal $\mathcal{I}$ on a countably infinite set D, define its support lift to be $\mathcal{I}\times 0$ in the standard Fubini-sum notation [1, Section 2.1]. Here $0=\{⌀\}$, which is not admissible. On $D\times \mathbb{N}$, the support lift is

$$\widehat{\mathcal{I}}=\{A\subseteq D\times \mathbb{N}:supp\left(A\right)\in \mathcal{I}\},supp\left(A\right)=\{d:A_d\ne ⌀\}.$$

Proposition 5.4.

Let $\mathcal{I}$ be an admissible ideal on a countably infinite set D. Then $\widehat{\mathcal{I}}$ is an admissible ideal. Moreover,

(a)

If $\mathsf{HCF}\left(\mathcal{I}\right)$ holds, then $\mathsf{HCF}\left(\widehat{\mathcal{I}}\right)$ holds.

(b)

If $\mathcal{I}$ is K-uniform, then $\widehat{\mathcal{I}}$ is K-uniform.

Proof. 

The support map is monotone under inclusion and satisfies $supp(A\cup B)=supp\left(A\right)\cup supp\left(B\right)$. Hence $\widehat{\mathcal{I}}$ is hereditary and closed under finite unions. Finite sets have finite support, while the whole carrier has support $D\notin \mathcal{I}$. Thus $\widehat{\mathcal{I}}$ is admissible.

For (a), let X be a $T_1$ space, let $x\in X$, let $\mathcal{H}$ be an HCP family of $\widehat{\mathcal{I}}$-sequential neighborhoods of x, and let $y_{d,r}{\to }_{\widehat{\mathcal{I}}}x$ satisfy

$$S=\{(d,r):y_{d,r}\ne x\}\in {\widehat{\mathcal{I}}}^{+}.$$

Then $T=supp\left(S\right)\in {\mathcal{I}}^{+}$. For each $d\in T$ choose $r_d$ with $y_{d,r_d}\ne x$ and put $z_d=y_{d,r_d}$, while $z_d=x$ for $d\notin T$. For every neighborhood U of x,

$$\{d:z_d\notin U\}\subseteq supp\{(d,r):y_{d,r}\notin U\}\in \mathcal{I},$$

so $z_d{\to }_{\mathcal{I}}x$ and $\{d:z_d\ne x\}=T\in {\mathcal{I}}^{+}$.

If P is a $\widehat{\mathcal{I}}$-sequential neighborhood of x and $w_d{\to }_{\mathcal{I}}x$, the constant-row array $v_{d,r}=w_d$ is $\widehat{\mathcal{I}}$-convergent to x. Hence

$$\{d:w_d\notin P\}=supp\{(d,r):v_{d,r}\notin P\}\in \mathcal{I}.$$

Thus every member of $\mathcal{H}$ is an $\mathcal{I}$-sequential neighborhood. Applying $\mathsf{HCF}\left(\mathcal{I}\right)$ to $\left(z_d\right)$ shows that $\mathcal{H}$ is finite, proving (a).

For (b), let $A\in {\widehat{\mathcal{I}}}^{+}$ and put $T=supp\left(A\right)\in {\mathcal{I}}^{+}$. Choose $g:dom\left(\mathcal{I}\right)\to T$ witnessing $\mathcal{I}\upharpoonright T{\le }_K\mathcal{I}$, and, for each d, fix $r_d$ with $(g\left(d\right),r_d)\in A$. Define $F:D\times \mathbb{N}\to A$ by

$$F(d,n)=(g\left(d\right),r_d).$$

If $E\in \widehat{\mathcal{I}}\upharpoonright A$, then $supp\left(E\right)\in \mathcal{I}$ and $supp\left(E\right)\subseteq T$. Also,

$$supp\left(F^{-1}\left[E\right]\right)\subseteq g^{-1}[supp\left(E\right)]\in \mathcal{I}.$$

Hence $F^{-1}\left[E\right]\in \widehat{\mathcal{I}}$, proving (b). □

For finite $n\ge 1$, Kwela’s subscript convention agrees with Katětov’s superscript convention, ${\mathrm{Fin}}_n={\mathrm{Fin}}^n$. At the limit stage $\omega$, ${\mathrm{Fin}}_{\omega }$ and ${\mathrm{Fin}}^{\omega }$ denote different constructions [18]. In Kwela’s notation, put

$$D_{\omega }=\underset{m\ge 1}{⨆}{\mathbb{N}}^m,$$

and, for $1\le k\le m$, let ${\pi }_{k,m}:{\mathbb{N}}^m\to {\mathbb{N}}^k$ be the projection onto the last k coordinates. Using Kwela’s full-domain convention [18, Definition 2.1 and Subsection 2.4], define

$${\mathcal{L}}_k=\left\{M\subseteq D_{\omega }:\exists B\in {\mathrm{Fin}}^{\otimes k}\forall m\ge k{\pi }_{k,m}[M\cap {\mathbb{N}}^m]\subseteq B\right\}.$$

Then ${\mathrm{Fin}}_{\omega }={\bigcup }_{k\ge 1}{\mathcal{L}}_k$. Kwela [18, Subsection 2.1 and Corollary 3.5] showed that this limit is an admissible ideal.

Lemma 5.5.

For every $k\ge 1$,

$${\mathcal{L}}_k\subseteq {\mathcal{L}}_{k+1}\mathrm{and}{\mathcal{L}}_k\cong \widehat{{\mathrm{Fin}}^{\otimes k}}.$$

Proof. 

If $M\in {\mathcal{L}}_k$ is witnessed by $B\in {\mathrm{Fin}}^{\otimes k}$, then $\mathbb{N}\times B\in {\mathrm{Fin}}^{\otimes (k+1)}$ witnesses M at stage $k+1$. Hence ${\mathcal{L}}_k\subseteq {\mathcal{L}}_{k+1}$.

For the isomorphism, for $s\in {\mathbb{N}}^k$ put

$$F_s=\underset{m\ge k}{⨆}\{t\in {\mathbb{N}}^m:{\pi }_{k,m}\left(t\right)=s\}.$$

The $F_s$ form a partition of ${⨆}_{m\ge k}{\mathbb{N}}^m$ into countably infinite sets, and

$$M\in {\mathcal{L}}_k⟺\{s:M\cap F_s\ne ⌀\}\in {\mathrm{Fin}}^{\otimes k}.$$

The possibly empty lower block ${⨆}_{1\le m<k}{\mathbb{N}}^m$ is unrestricted. Fix $s_0\in {\mathbb{N}}^k$, map each $F_s$ for $s\ne s_0$ bijectively onto $\left\{s\right\}\times \mathbb{N}$, and map $F_{s_0}$ together with the lower block bijectively onto $\left\{s_0\right\}\times \mathbb{N}$. Under the resulting bijection, the support of the image of M differs from $\{s:M\cap F_s\ne ⌀\}$ by at most the singleton $\left\{s_0\right\}$. Since ${\mathrm{Fin}}^{\otimes k}$ contains all finite sets, this is an ideal isomorphism from ${\mathcal{L}}_k$ onto $\widehat{{\mathrm{Fin}}^{\otimes k}}$. □

Theorem 5.6.

The ideal ${\mathrm{Fin}}_{\omega }$ has local HCP-finiteness, does not have $P^{-}$, and has separation rank ω.

Proof. 

By Corollary 4.4, every ${\mathrm{Fin}}^{\otimes k}$ has local HCP-finiteness. Kwela–Tryba [20, Example 2.3 and Proposition 2.9] show that every finite Fubini power of $\mathrm{Fin}$ is homogeneous and hence K-uniform. Proposition 5.4 therefore shows that $\widehat{{\mathrm{Fin}}^{\otimes k}}$ is admissible, has local HCP-finiteness, and is K-uniform. Lemma 5.5 and ideal-isomorphism invariance transfer these properties to every ${\mathcal{L}}_k$. Theorem 5.2 gives

$$\mathsf{HCF}\left({\mathrm{Fin}}_{\omega }\right).$$

For the failure of $P^{-}$, let $\psi :{\mathbb{N}}^2\times \mathbb{N}\to D_{\omega }$ witness $\widehat{\mathrm{Fin}\otimes \mathrm{Fin}}\cong {\mathcal{L}}_2$ and put

$$C_i=\psi \left[\{((i,j),r):j,r\in \mathbb{N}\}\right].$$

For each i, the support of ${\psi }^{-1}\left[C_i\right]$ is contained in the row with first coordinate i, so $C_i\in {\mathcal{L}}_2$. If $B\subseteq D_{\omega }$ meets every $C_i$ finitely, then each vertical section of $supp\left({\psi }^{-1}\left[B\right]\right)$ is finite. Hence ${\psi }^{-1}\left[B\right]\in \widehat{\mathrm{Fin}\otimes \mathrm{Fin}}$ and $B\in {\mathcal{L}}_2\subseteq {\mathrm{Fin}}_{\omega }$. Lemma 2.4 shows that ${\mathrm{Fin}}_{\omega }$ does not have $P^{-}$.

Debs–Saint Raymond give separation rank $\omega$ at their first limit stage [4, Theorem 6.5]. Kwela [18, Subsection 2.4] discusses a domain ambiguity in the original limit-stage presentation, so the full-domain convention used here is checked separately. For the upper bound, the support map $A\mapsto supp\left(A\right)$ is Baire class one as the pointwise limit of its continuous second-coordinate truncations. Every finite Fubini power has finite Borel complexity [4, Proposition 6.4], as also noted by Kwela [18, p. 1]. Since preimages of finite-level Borel sets under Baire class one maps are again of finite Borel class, each support lift $\widehat{{\mathrm{Fin}}^{\otimes k}}$ is of finite Borel class. The carrier bijection in Lemma 5.5 induces a homeomorphism of the corresponding power-set spaces, so every ${\mathcal{L}}_k$ is of finite Borel class. Hence

$${\mathrm{Fin}}_{\omega }=\bigcup _{k\ge 1}{\mathcal{L}}_k\in {\Sigma }_{\omega }^0.$$

Thus ${\mathrm{Fin}}_{\omega }$ is analytic. As a proper ideal, it is disjoint from its dual filter, so it is itself a ${\Sigma }_{\omega }^0$ separator and $rk\left({\mathrm{Fin}}_{\omega }\right)\le \omega$.

For the lower bound, fix $k\ge 1$ and define ${\Phi }_k:2^{{\mathbb{N}}^k}\to 2^{D_{\omega }}$ by

$${\Phi }_k\left(A\right)=\underset{m\ge k}{⨆}{\pi }_{k,m}^{-1}\left[A\right].$$

Each coordinate of ${\Phi }_k\left(A\right)$ depends on a single coordinate of A, so ${\Phi }_k$ is continuous. If $A\in {\mathrm{Fin}}^{\otimes k}$, then ${\Phi }_k\left(A\right)\in {\mathcal{L}}_k\subseteq {\mathrm{Fin}}_{\omega }$, with witness A. If $A\in {\left({\mathrm{Fin}}^{\otimes k}\right)}^{★}$, then ${\mathbb{N}}^k\setminus A\in {\mathrm{Fin}}^{\otimes k}$ and

$$D_{\omega }\setminus {\Phi }_k\left(A\right)=\underset{1\le m<k}{⨆}{\mathbb{N}}^m\bigsqcup {\Phi }_k({\mathbb{N}}^k\setminus A)\in {\mathcal{L}}_k,$$

with witness ${\mathbb{N}}^k\setminus A$. Hence ${\Phi }_k\left(A\right)\in {\mathrm{Fin}}_{\omega }^{★}$. Therefore, if $\alpha <{\omega }_1$ and $S\in {\Sigma }_{1+\alpha }^0$ separates ${\mathrm{Fin}}_{\omega }$ from ${\mathrm{Fin}}_{\omega }^{★}$, then ${\Phi }_k^{-1}\left[S\right]\in {\Sigma }_{1+\alpha }^0$ separates ${\mathrm{Fin}}^{\otimes k}$ from ${\left({\mathrm{Fin}}^{\otimes k}\right)}^{★}$. Hence

$$rk\left({\mathrm{Fin}}^{\otimes k}\right)\le rk\left({\mathrm{Fin}}_{\omega }\right).$$

Debs–Saint Raymond’s finite-stage computation [4, Theorem 6.5] gives $rk\left({\mathrm{Fin}}^{\otimes k}\right)=k$. Hence $rk\left({\mathrm{Fin}}_{\omega }\right)\ge k$ for every $k\ge 1$, so $rk\left({\mathrm{Fin}}_{\omega }\right)\ge \omega$, and equality follows. □

6. Conclusion

The local point-discrete countability lemma separates the metrization problem from the stronger finiteness theory. For every admissible ideal $\mathcal{I}$, every point-discrete family of $\mathcal{I}$-sequential neighborhoods of a point x is countable whenever there is an $\mathcal{I}$-convergent sequence to x with $\mathcal{I}$-positive support away from x. If no such sequence exists at x, the singleton $\left\{x\right\}$ is an $\mathcal{I}$-sequential neighborhood. These two alternatives give $\mathcal{I}$-$snf$-countability for every space carrying a $\sigma$-point-discrete $\mathcal{I}$-$sn$-network. In the regular $\sigma$-HCP case, Ge’s characterization of ordinary $sn$-metrizability [8, Lemma 2.2] and Zhou–Liu–Liu–Lin’s characterization of $\mathcal{I}$-$sn$-metrizability [41, Theorem 3.6] then settle Problem 5.2 [41, Problem 5.2] affirmatively for every admissible ideal.

The countability conclusion cannot in general be strengthened to finiteness, as Theorem 3.4 gives an admissible ideal for which local HCP-finiteness fails. The Fubini theorems show that local HCP-finiteness passes from an inner ideal to a product when the outer ideal has $P^{-}$. The heterogeneous theorem allows the row ideals to vary under cross-row Katětov absorption. The increasing-union theorem gives a second permanence principle through K-uniform stages. As a result, local HCP-finiteness holds for the finite Fubini powers of $\mathrm{Fin}$, for ${\mathrm{Fin}}^{\omega +r}$ with $r<\omega$, and for the rank-$\omega$ ideals ${\mathcal{H}}_{<\omega }$ and ${\mathrm{Fin}}_{\omega }$. Among these ideals, ${\mathrm{Fin}}^{\otimes r}$ for $r\ge 2$, every ${\mathrm{Fin}}^{\omega +r}$ with $r<\omega$, ${\mathcal{H}}_{<\omega }$, and ${\mathrm{Fin}}_{\omega }$ fail $P^{-}$.

The remaining structural problem is to characterize the admissible ideals satisfying local HCP-finiteness. Two related tasks remain. One is to determine whether local HCP-finiteness admits a direct description in terms of standard selection or Katětov-theoretic properties of ideals. The second asks which hypotheses in the permanence theorems can be weakened.