Fubini and Increasing-Union Stability of Local HCP-Finiteness for I-sn-Networks

1. Introduction

Throughout, all spaces are $T_1$, 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 [18]. Lahiri and Das [22] extended it to topological spaces. Zhou, Liu and Lin studied topological spaces determined by $\mathcal{I}$-convergence [42]. 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 [38]. Liu, Lin and Zhou obtained metric-image characterizations for spaces defined by ideal convergence [30].

A set $P\subseteq X$ is an $\mathcal{I}$-sequential neighborhood of x if $x_n{\to }_{\mathcal{I}}x$ implies $\{n:x_n\notin P\}\in \mathcal{I}$ [41, 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 [39, Definition 5.1]. Lin introduced ordinary $sn$-networks in [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 in [7,8,9]. Luo [31] 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 [39, 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 [40, Definition 2.4]. They posed the following problem [40, 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 is $\sigma$-HCP if it is a countable union of HCP families. Every subfamily of an HCP family is HCP. 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], while Tanaka showed that a regular space is g-metrizable exactly when it is weakly first countable and has such a network [35]. Junnila–Yun gave a necessary and sufficient condition for a regular space with a $\sigma$-HCP k-network to be an ℵ-space [16]. Liu [29] surveys spaces with $\sigma$-HCP k-networks.

The ordinary part of Problem 5.2 follows from Ge’s $\sigma$-HCP characterization of $sn$-metrizable spaces [8, Lemma 2.2] together with the fact that every $\mathcal{I}$-sequential neighborhood is an ordinary sequential neighborhood [39, Lemma 5.2]. Thus a $\sigma$-HCP $\mathcal{I}$-$sn$-network is a $\sigma$-HCP ordinary $sn$-network. Zhou–Liu–Liu–Lin [40, Theorem 3.6] show that, once ordinary $sn$-metrizability is known, the remaining issue in Problem 5.2 is whether the space is $\mathcal{I}$-$snf$-countable.

An HCP form of the ordinary trace-finiteness argument already appears in Ge [8, Lemma 2.4]. If a convergent sequence is eventually contained in the union of an HCP family, some member contains a subsequence. Ge, Shen and Ying [11, Theorem 3.1] later used the same point-selection argument under weak hereditary closure preservation. After fixing a layer and a nontrivial convergent sequence, they showed that an infinite local trace is impossible. Local HCP-finiteness is the corresponding condition for ideal convergence.

For a fixed ideal $\mathcal{I}$, local HCP-finiteness, denoted $\mathsf{HCF}\left(\mathcal{I}\right)$, says that an HCP family of $\mathcal{I}$-sequential neighborhoods cannot remain infinite along an $\mathcal{I}$-convergent sequence whose non-x index set is $\mathcal{I}$-positive. The $P^{-}$ property implies this local condition by a finite diagonal selection argument. Problem 5.2 therefore has an affirmative answer for $P^{-}$ ideals, and hence for the stronger conditions $\left(AP\right)$ and $P^{+}$.

A Fubini product $\mathcal{K}\otimes \mathcal{J}$ has local HCP-finiteness whenever $\mathcal{K}$ has $P^{-}$ and $\mathcal{J}$ has local HCP-finiteness. Iteration gives the conclusion of Problem 5.2 for every finite Fubini product of $P^{-}$ ideals. In contrast, every Fubini product of two admissible ideals fails $P^{-}$, so the $P^{-}$ hypothesis is not necessary.

The Fubini theorem also has a heterogeneous form in which the inner ideal may vary from row to row. For a $\mathcal{K}$-Fubini sum of varying inner ideals $\left({\mathcal{J}}_i\right)$, local HCP-finiteness follows if $\mathcal{K}$ has $P^{-}$ and, outside a $\mathcal{K}$-small set of rows, each ${\mathcal{J}}_i$ has local HCP-finiteness and every positive restriction of ${\mathcal{J}}_i$ is Katětov-below row ideals on a $\mathcal{K}$-positive set of indices. Here ${\mathrm{Fin}}^{\omega }$ denotes the heterogeneous Fubini-sum ideal from Katětov’s transfinite hierarchy [17, p. 240], in the ideal formulation used by Kwela [19, Subsection 2.2], not the inductive-limit ideal ${\mathrm{Fin}}_{\omega }$ of Section 5. Applying the theorem at Katětov’s first limit stage proves the base case $\mathsf{HCF}\left({\mathrm{Fin}}^{\omega }\right)$, and constant-inner lifting gives all finite successor cases. Thus $\mathsf{HCF}\left({\mathrm{Fin}}^{\omega +r}\right)$ holds for every $r<\omega$. Every one of these ideals fails $P^{-}$.

The increasing-union argument relies on K-uniformity. For a K-uniform ideal, every positive restriction is, by definition, Katětov-below the ideal. Zhang–Zhang [37, Theorem 2.5] used this reindexing into an $\mathcal{I}$-positive part. Zhou–Lin–Zhang later adapted the argument to the $\mathcal{I}$-$sn$ setting [39, Lemma 5.5]. The cross-ideal form shows that a sequential neighborhood for the union ideal is also a sequential neighborhood for any stage satisfying the uniformity condition. If ${\mathcal{J}}_0\subseteq {\mathcal{J}}_1\subseteq \dots$, the union ${\bigcup }_n{\mathcal{J}}_n$ is a proper ideal, and cofinally many stages have local HCP-finiteness and are K-uniform, then the union has local HCP-finiteness.

For $r\ge 1$, write ${\mathrm{Fin}}^{\otimes r}$ for the right-associated r-fold Fubini product of $\mathrm{Fin}$. These finite powers are homogeneous [21, Example 2.3 and Proposition 2.9], hence K-uniform, and have local HCP-finiteness. Pelayo Gómez constructed an increasing hierarchy $\left({\mathcal{H}}_n\right)$ with

$${\mathcal{H}}_n\cong {\mathrm{Fin}}^{\otimes (n+1)},{\mathcal{H}}_{<\omega }=\bigcup _n{\mathcal{H}}_n,$$

and proved that ${\mathcal{H}}_{<\omega }$ is a tall ${\Sigma }_{\omega }^0$ ideal of separation rank exactly $\omega$ [33, Theorem 3.5 and Proposition 3.8]. The union theorem therefore gives an affirmative answer to Problem 5.2 for ${\mathcal{H}}_{<\omega }$. A finite-dual obstruction at the stage ${\mathcal{H}}_1$ shows that ${\mathcal{H}}_{<\omega }$ does not have $P^{-}$.

The inductive limit ${\mathrm{Fin}}_{\omega }$ comes from the Debs–Saint Raymond hierarchy [4, Subsection 6.2], with the full-domain convention fixed by Kwela [19, Definition 2.1 and Subsection 2.4]. Its finite approximating stages are isomorphic to support lifts of the finite Fubini powers. Both local HCP-finiteness and K-uniformity pass to support lifts, so the increasing-union theorem gives an affirmative answer for ${\mathrm{Fin}}_{\omega }$. A finite-dual tail witness shows that ${\mathrm{Fin}}_{\omega }$ does not have $P^{-}$, and its stage presentation also recovers rank $\omega$ in the full-domain convention. Pelayo Gómez [33, Theorem 4.18] proved ${\mathrm{Fin}}_{\omega }\neg {\le }_K{\mathcal{H}}_{<\omega }$, so the two rank-$\omega$ limit ideals are not isomorphic.

These results extend the metrization conclusion beyond $P^{-}$. In particular, it holds for the finite Fubini products from Corollary 2, for ${\mathrm{Fin}}^{\omega +r}$ with $r<\omega$, and for the rank-$\omega$ ideals ${\mathcal{H}}_{<\omega }$ and ${\mathrm{Fin}}_{\omega }$. The general form of Problem 5.2 remains open, so any counterexample must lie outside these classes.

2. Preliminaries

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. 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], in the convention recalled by Kwela [19, p. 1].

Lemma 1  

([39, Lemma 5.2]). For every admissible ideal $\mathcal{I}$, every $\mathcal{I}$-sequential neighborhood of x is an ordinary sequential neighborhood of x.

Hrušák, Meza-Alcántara, Thümmel and Uzcátegui [15, Theorem 3.8] give the Katětov-theoretic boundary for the standard $P^{-}$ property. The equivalent partition form in Definition 1 appears in Camargo–Uzcátegui [3, Theorem 3.1(v)] and Uzcátegui Aylwin [36, Theorem 8.2(v)].

Definition 1.  

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.

Proposition 1.  

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 1. □

Proposition 2.  

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 Proposition 1 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 Proposition 2 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 [32, 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 [15, 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.

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 $A_d=\{e\in E:(d,e)\in A\}$ [21]. 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

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

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 [14, 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 [39, Definition 5.3] use the same condition in the ideal-convergence setting. An ideal $\mathcal{A}$ is homogeneous if $\mathcal{A}\upharpoonright A\cong \mathcal{A}$ for every $A\in {\mathcal{A}}^{+}$ [21, Definition 1.3]. Every homogeneous ideal is K-uniform.

3. The Local HCP-finiteness Principle

Definition 2.  

An admissible ideal $\mathcal{I}$ has thelocal 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.

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.

Proposition 3.  

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 Proposition 1, 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. □

Proposition 4.  

Let X be a regular space with a σ-hereditarily closure-preserving $\mathcal{I}$-$sn$-network. Then X is ordinary $sn$-metrizable. Consequently,

$$X\mathrm{is}\mathcal{I}-sn-\mathrm{metrizable}⟺X\mathrm{is}\mathcal{I}-snf-\mathrm{countable}.$$

Proof. 

By Lemma 1, the given network is a $\sigma$-HCP ordinary $sn$-network. Ge’s characterization [8, Lemma 2.2] gives ordinary $sn$-metrizability. The equivalence then follows from [40, Theorem 3.6]. □

Lemma 2.  

(a)

A finite union of HCP families is HCP.

(b)

A finite intersection of $\mathcal{I}$-sequential neighborhoods of the same point is an $\mathcal{I}$-sequential neighborhood of that point.

Proof. 

For (a), it suffices to consider two families. After arbitrary subpieces have been chosen, the closure-preserving property of each family and the identity

$$cl(A\cup B)=clA\cup clB$$

show that their union is closure-preserving. Induction gives the finite case.

For (b), if $x_n{\to }_{\mathcal{I}}x$ and $P_0,\dots ,P_r$ are $\mathcal{I}$-sequential neighborhoods of x, then

$$\{n:x_n\notin \bigcap _{i\le r}{P}_i\}=\bigcup _{i\le r}\{n:x_n\notin P_i\}\in \mathcal{I}.$$

□

Theorem 1.  

If $\mathsf{HCF}\left(\mathcal{I}\right)$ holds, then every regular space with a σ-hereditarily closure-preserving $\mathcal{I}$-$sn$-network is $\mathcal{I}$-$sn$-metrizable.

Proof. 

Let

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

be a $\sigma$-HCP $\mathcal{I}$-$sn$-network on X, with each ${\mathcal{P}}_m$ HCP. After replacing ${\mathcal{P}}_m$ by ${\mathcal{P}}_0\cup \dots \cup {\mathcal{P}}_m$ and using Lemma 2(a), the layers may be assumed to satisfy

$${\mathcal{P}}_m\subseteq {\mathcal{P}}_{m+1}.$$

Fix $x\in X$ and put

$${\mathcal{P}}_{x,m}={\mathcal{P}}_m\cap {\mathcal{P}}_x,R_{x,m}=\bigcap {\mathcal{P}}_{x,m},$$

with the empty intersection interpreted as X.

The set $R_{x,m}$ is an $\mathcal{I}$-sequential neighborhood of x. If ${\mathcal{P}}_{x,m}$ is finite, this follows from Lemma 2(b), with the empty case giving $R_{x,m}=X$. Suppose instead that ${\mathcal{P}}_{x,m}$ is infinite. Then $\mathsf{HCF}\left(\mathcal{I}\right)$ implies that every sequence $y_n{\to }_{\mathcal{I}}x$ has $S=\{n:y_n\ne x\}\in \mathcal{I}$, since otherwise the fixed HCP family ${\mathcal{P}}_{x,m}$ and such a sequence would contradict $\mathsf{HCF}\left(\mathcal{I}\right)$. As every member of ${\mathcal{P}}_x$ contains x,

$$\{n:y_n\notin R_{x,m}\}\subseteq S\in \mathcal{I}.$$

Thus $R_{x,m}$ is $\mathcal{I}$-sequential in all cases.

The countable family $\{R_{x,m}:m\in \mathbb{N}\}$ is decreasing and is a local network at x. Given a neighborhood U of x, choose $P\in {\mathcal{P}}_x$ with $P\subseteq U$, and then choose m with $P\in {\mathcal{P}}_m$. Thus $R_{x,m}\subseteq P\subseteq U$. Therefore X is $\mathcal{I}$-$snf$-countable. Proposition 4 finishes the proof. □

Corollary 1.  

If $\mathcal{I}$ has $P^{-}$, then every regular space with a σ-hereditarily closure-preserving $\mathcal{I}$-$sn$-network is $\mathcal{I}$-$sn$-metrizable. The same conclusion holds if $\mathcal{I}$ has $\left(AP\right)$ or $P^{+}$.

Proof. 

The first assertion follows from Proposition 3 and Theorem 1. For the final assertion, apply the preceding implications $\left(AP\right)\Rightarrow P^{-}$ and $P^{+}\Rightarrow P^{-}$. □

4. Fubini Stability beyond $P^{-}$

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

Lemma 3.  

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 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 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 Proposition 1 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. □

Proposition 5.  

If $\mathcal{K}$ and $\mathcal{J}$ are admissible ideals, then $\mathcal{K}\otimes \mathcal{J}$ does not have $P^{-}$.

Proof. 

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 any set meeting each $C_i$ in a finite set has all sections in $\mathcal{J}$. Proposition 2 applies. □

The Katětov boundary for $P^{-}$ gives an alternative argument. After reindexing the countable carriers, $\mathrm{Fin}\otimes \mathrm{Fin}\subseteq \mathcal{K}\otimes \mathcal{J}$, so $\mathrm{Fin}\otimes \mathrm{Fin}{\le }_K\mathcal{K}\otimes \mathcal{J}$. Taking the whole carrier as the positive restriction in [15, Theorem 3.8(4)] then gives the failure of $P^{-}$.

Corollary 2.  

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. Consequently, every regular space with a σ-hereditarily closure-preserving $({\mathcal{I}}_1\otimes \dots \otimes {\mathcal{I}}_r)$-$sn$-network is $({\mathcal{I}}_1\otimes \dots \otimes {\mathcal{I}}_r)$-$sn$-metrizable. 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 2 inductively from the innermost factor outward. The metrization conclusion follows from Theorem 1, and Proposition 5 proves the last assertion. □

Corollary 3.  

For every $r\ge 1$, $\mathsf{HCF}\left({\mathrm{Fin}}^{\otimes r}\right)$ holds, and Problem 5.2 has an affirmative answer for ${\mathrm{Fin}}^{\otimes r}$. 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 $P^{-}$. Proposition 3 gives $\mathsf{HCF}\left(\mathrm{Fin}\right)$, so Corollary 2 applies. □

Thus the $P^{-}$ hypothesis in Corollary 1 is not necessary.

4.1. Heterogeneous Fubini Sums

The heterogeneous case allows the inner ideal to vary from row to row. The Fubini-sum notation follows [19, 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

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

(1)

For fixed ideals $\mathcal{I}$ and $\mathcal{J}$, Kwela–Lesner–Tryba [20, 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. Thus the all-positive-restriction form of the relation in (1) is equivalent to preservation of $\mathcal{I}$-convergence by every $\mathcal{J}$-continuous function. In the constant-row case $D_i=D$ and ${\mathcal{J}}_i=\mathcal{J}$ for all i, condition (1) reduces to K-uniformity of $\mathcal{J}$. Theorem 2 does not require this additional uniformity assumption. Theorem 3 requires this absorption condition only for rows in some $W\in {\mathcal{K}}^{★}$.

Lemma 4.  

Let $W\subseteq I$ and suppose $R(i,E)\in {\mathcal{K}}^{+}$ for $i\in W$ and $E\in {\mathcal{J}}_i^{+}$. 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 3.  

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 $i\in W$, $E\in {\mathcal{J}}_i^{+}$, 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, 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

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

(2)

then the absorption hypothesis of Theorem 3 holds by transitivity of the Katětov order.

This criterion applies to the ideal form of Katětov’s transfinite Fubini hierarchy. Katětov’s original filter recursion appears in [17, p. 240]. Kwela [19, 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,

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

(3)

For $1\le n<\omega$, ${\mathrm{Fin}}^n\cong {\mathrm{Fin}}^{\otimes n}$. Corollary 3 gives local HCP-finiteness. Kwela–Tryba [21, 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 ${\mathrm{Fin}}^n{\le }_K{\mathrm{Fin}}^m$ [19, 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 (2) holds for the outer ideal $\mathrm{Fin}$. Since $\mathrm{Fin}$ has $P^{-}$, Theorem 3 applies at the limit stage.

Corollary 4.  

For every $r<\omega$,

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

holds. Consequently, every regular space with a σ-hereditarily closure-preserving ${\mathrm{Fin}}^{\omega +r}$-$sn$-network is ${\mathrm{Fin}}^{\omega +r}$-$sn$-metrizable. Moreover, ${\mathrm{Fin}}^{\omega +r}$ does not have $P^{-}$.

Proof. 

Equation (3) and Theorem 3 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 2 applies at each finite successor. The metrization conclusion follows from Theorem 1.

For the failure of $P^{-}$ at $r=0$, reindex the outer rows of (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 ${\mathrm{Fin}}^{\omega }$. Proposition 2 applies. For each $s<\omega$, Proposition 5 applies to the displayed product ${\mathrm{Fin}}^{\omega +(s+1)}$. □

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

The increasing-union theorem uses a cross-ideal version of the K-uniform reindexing. Zhang–Zhang [37, Theorem 2.5] used the same-ideal form, and Zhou–Lin–Zhang [39, Definition 5.3 and Lemma 5.5] later applied the reindexing in the $\mathcal{I}$-$sn$ setting. In Lemma 5, the convergence ideal and the K-uniform stage ideal need not be the same.

Lemma 5.  

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 [21], 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 4.  

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 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 convergence must occur at some finite stage. 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, 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. □

Pelayo Gómez introduced the tree-derived hierarchy [33]. For the auxiliary ideal $\mathrm{Fin}$, let ${\left({\mathcal{H}}_n\right)}_{n<\omega }$ denote its finite derived levels. Pelayo Gómez [33, 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

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

By [33, Proposition 3.8], ${\mathcal{H}}_{<\omega }$ is a tall ${\Sigma }_{\omega }^0$ ideal of separation rank exactly $\omega$. Its properness follows from [33, Theorem 3.4].

Theorem 5.  

The ideal ${\mathcal{H}}_{<\omega }$ has local HCP-finiteness and does not have $P^{-}$. Consequently, every regular space with a σ-hereditarily closure-preserving ${\mathcal{H}}_{<\omega }$-$sn$-network is ${\mathcal{H}}_{<\omega }$-$sn$-metrizable.

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 3 and ideal-isomorphism invariance give local HCP-finiteness of each ${\mathcal{H}}_n$. By Kwela–Tryba [21, 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 4 gives $\mathsf{HCF}\left({\mathcal{H}}_{<\omega }\right)$, and the metrization conclusion follows from Theorem 1.

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 $B\in {\mathcal{H}}_1\subseteq {\mathcal{H}}_{<\omega }$. Proposition 2 shows that ${\mathcal{H}}_{<\omega }$ does not have $P^{-}$. □

For an admissible ideal $\mathcal{I}$ on a countably infinite set D, call the ideal $\mathcal{I}\times 0$ from the standard Fubini-sum notation [1, Section 2.1] its support lift. 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 6.  

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 $y_{d,r}{\to }_{\widehat{\mathcal{I}}}x$ and suppose

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

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 an HCP family of $\widehat{\mathcal{I}}$-sequential neighborhoods is an $\mathcal{I}$-sequential neighborhood. Applying $\mathsf{HCF}\left(\mathcal{I}\right)$ to $\left(z_d\right)$ proves (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 [19, Subsection 2.4]. 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. With the full-domain convention in [19, 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 ${\mathrm{Fin}}_{\omega }={\bigcup }_{k\ge 1}{\mathcal{L}}_k$. This limit is an admissible ideal by [19, Subsection 2.1 and Corollary 3.5].

Lemma 6.  

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 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 6.  

The ideal ${\mathrm{Fin}}_{\omega }$ has local HCP-finiteness, does not have $P^{-}$, and has separation rank ω. Consequently, every regular space with a σ-hereditarily closure-preserving ${\mathrm{Fin}}_{\omega }$-$sn$-network is ${\mathrm{Fin}}_{\omega }$-$sn$-metrizable.

Proof. 

By Corollary 3, every ${\mathrm{Fin}}^{\otimes k}$ has local HCP-finiteness. Kwela–Tryba [21, Example 2.3 and Proposition 2.9] show that every finite Fubini power of $\mathrm{Fin}$ is homogeneous and hence K-uniform. Proposition 6 therefore shows that $\widehat{{\mathrm{Fin}}^{\otimes k}}$ is admissible, has local HCP-finiteness, and is K-uniform. Lemma 6 and ideal-isomorphism invariance transfer these properties to every ${\mathcal{L}}_k$. Theorem 4 gives

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

and the metrization conclusion follows from Theorem 1.

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 $B\in {\mathcal{L}}_2\subseteq {\mathrm{Fin}}_{\omega }$. Proposition 2 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]. Because the original limit-stage presentation has a domain ambiguity, as discussed by Kwela [19, Subsection 2.4], the rank is verified directly for the full-domain convention used here. 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 [19, 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 6 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, Kwela [19, proof of Corollary 5.2] states that ${\mathrm{Fin}}_{n+1}$ embeds isomorphically into ${\mathrm{Fin}}_{\omega }$ for every $n\in \mathbb{N}$. Debs–Saint Raymond’s finite-stage rank computation and monotonicity lemma [4, Theorem 6.5 and Lemma 7.2] then give $rk\left({\mathrm{Fin}}_{\omega }\right)\ge n+1$ for every n, hence $rk\left({\mathrm{Fin}}_{\omega }\right)\ge \omega$, and equality follows. □

6. The Remaining Problem

These results show that $P^{-}$ is not the boundary of the original Problem 5.2. In Theorem 3, cross-row Katětov absorption descends sequential neighborhoods to row ideals, while $P^{-}$ selects finite row sets with positive union. By contrast, Theorem 4 gives convergence with respect to some stage ideal, and the K-uniformity of that stage implies that every sequential neighborhood for the union ideal is a sequential neighborhood for the same stage. Thus the metrization conclusion holds for the non-$P^{-}$ finite Fubini products ${\mathrm{Fin}}^{\otimes r}$ for $r\ge 2$, for ${\mathrm{Fin}}^{\omega +r}$ with $r<\omega$, and, by Theorems 5 and 6, for the rank-$\omega$ limit ideals ${\mathcal{H}}_{<\omega }$ and ${\mathrm{Fin}}_{\omega }$.

The general question of Zhou–Liu–Liu–Lin remains open.

Question 7  

([40, Problem 5.2]). Let $\mathcal{I}$ be an arbitrary admissible ideal. If X is a regular space with a σ-hereditarily closure-preserving $\mathcal{I}$-$sn$-network, must X be $\mathcal{I}$-$sn$-metrizable?

The results here do not determine whether $\mathsf{HCF}\left(\mathcal{I}\right)$ can fail for an admissible ideal, nor whether local HCP-finiteness is necessary for the metrization conclusion in Question 7.

The HCP contradiction arguments in the preceding proofs have the following common form. Let $y_n{\to }_{\mathcal{I}}x$ with $S=\{n:y_n\ne x\}\in {\mathcal{I}}^{+}$, and let $P_0,P_1,\dots$ be distinct members of an HCP family of $\mathcal{I}$-sequential neighborhoods of x. If there are sets $F_j\subseteq S\cap \{n:y_n\in P_j\}$ such that $x\notin cl\{y_n:n\in F_j\}$ for every j and ${\bigcup }_j{F}_j\in {\mathcal{I}}^{+}$, then hereditary closure preservation fails. The individual sets have closures avoiding x, whereas $\mathcal{I}$-convergence places x in the closure of their union. Thus any counterexample to $\mathsf{HCF}\left(\mathcal{I}\right)$ must rule out every such selection.

Let $N_{\mathcal{I}}=\mathbb{N}\cup \{\infty \}$ be the prime space considered by Sleziak [34, Section 3]. Its points in $\mathbb{N}$ are isolated, and its neighborhoods of ∞ are the sets $\{\infty \}\cup G$ with $G\in {\mathcal{I}}^{★}$. For admissible $\mathcal{I}$, the space $N_{\mathcal{I}}$ is $T_1$. The point ∞ is a limit point of $\mathbb{N}$, and

$$\{\infty \}=\bigcap _{m\in \mathbb{N}}\left(\{\infty \}\cup \{n\in \mathbb{N}:n\ge m\}\right)$$

is a $G_{\delta }$ subset of $N_{\mathcal{I}}$ because every tail $\{n:n\ge m\}$ belongs to ${\mathcal{I}}^{★}$. The result of Burke–Engelking–Lutzer [2, Lemma 4] therefore implies that every HCP family of neighborhoods of ∞ is finite.

The $\mathcal{I}$-sequential neighborhoods of ∞ are exactly its neighborhoods. The constant sequence at ∞ and $y_n=n{\to }_{\mathcal{I}}\infty$ show that every $\mathcal{I}$-sequential neighborhood P contains ∞ and satisfies $P\cap \mathbb{N}\in {\mathcal{I}}^{★}$, so P is a neighborhood. Conversely, every neighborhood is $\mathcal{I}$-sequential by the definition of $\mathcal{I}$-convergence. At an isolated point x, every sequence $y_n{\to }_{\mathcal{I}}x$ satisfies $\{n:y_n\ne x\}\in \mathcal{I}$. Hence $N_{\mathcal{I}}$ cannot serve as a witness to $\neg \mathsf{HCF}\left(\mathcal{I}\right)$.

At the next limit stage $\lambda =\omega ·2$, the outer bounded ideal ${\mathcal{I}}_{\lambda }$ has $P^{-}$. If an unbounded S is partitioned into bounded sets $E_n$, choose $a_n\in S$ above $sup{\bigcup }_{i\le n}{E}_i$. These suprema are cofinal in $\lambda$ because every $s\in S$ belongs to some $E_i$ and hence $s\le sup{\bigcup }_{j\le i}{E}_j$. Therefore $\{a_n:n\in \mathbb{N}\}$ is unbounded. For fixed i, the implication $n\ge i\Rightarrow a_n\notin E_i$ shows that $E_i$ meets this set only finitely. The Katětov hierarchy is increasing, ${\mathrm{Fin}}^{\alpha }{\le }_K{\mathrm{Fin}}^{\beta }$ for $0<\alpha \le \beta <{\omega }_1$ [13, Proposition 1]. Corollaries 3 and 4 give $\mathsf{HCF}\left({\mathrm{Fin}}^{\alpha }\right)$ for every $0<\alpha <\lambda$. Consequently, taking W to be a co-bounded tail, the localized criterion (2) would apply at $\lambda$ once ${\mathrm{Fin}}^{\omega +r}$ is K-uniform for all sufficiently large r. Beyond $\omega ·2$, it remains to determine how far the absorption hypothesis in Theorem 3 extends through countable ordinals. These results do not establish a transfinite induction through arbitrary countable ordinals.