Ideal-Based Sequential Compactness

• Ideal-based refinements of sequential compactness extend the classical notion by using admissible ideals to isolate non-classical convergence and subsequential behaviors.
• They generalize concepts like statistical, I-, and I*-convergence via frameworks based on nonthin sets, dual filters, and partition-regular functions.
• This approach establishes a hierarchy of compactness properties, clarifies implications in first-countable spaces, and opens new avenues for examining product theorems and critical ideals.

Ideal-based refinements of sequential compactness constitute a systematic expansion of the classical theory of sequential compactness via the machinery of set-theoretic ideals. By generalizing the notion of "sparse" or "small" sets from the perspective of an arbitrary admissible ideal $\mathcal{I}$ on $\mathbb{N}$ or other relevant domains, these refinements capture various non-classical convergence phenomena and yield a spectrum of compactness notions interpolating between sequential compactness and strong forms of total boundedness and compactness. The framework subsumes statistical compactness, $I$-compactness, $I^*$-compactness, $I^K$-Frechet compactness, and the more general classes $FinBW(I)$, organizing them by the algebraic properties of the underlying ideals, their position in the Katětov order, and associated partition-regular functions.

1. Foundations: Ideals, Nonthin Sets, and Ideal Convergence

Let $\mathcal{I}$ be an admissible ideal on $\mathbb{N}$: $\mathcal{I}$ is a family of subsets of $\mathbb{N}$, closed under subsets and finite unions, containing all singletons and excluding $\mathbb{N}$ itself. The dual filter $\mathcal{F}(\mathcal{I})$ consists of sets whose complements belong to $\mathcal{I}$. A set is $\mathcal{I}$-nonthin if it does not belong to $\mathcal{I}$.

Given a sequence $(x_n)$ in a topological space $X$:

• $\mathcal{I}$-convergence: $x_n \to_{\mathcal{I}} x$ if for every neighborhood $U$ of $x$, $\{n\in\mathbb{N} : x_n\notin U\} \in \mathcal{I}$.
• $\mathcal{I}^*$-convergence: $x_n \to_{\mathcal{I}^*} x$ if there exists a set $M \in \mathcal{F}(\mathcal{I})$ such that the subsequence $(x_{m_k})$ (with $M = \{ m_1 < m_2 < \dots \}$) converges to $x$ in the classical sense (Behmanush et al., 2023).

These concepts decouple "bad" indices (those in sets in the ideal), allowing the study of subsequential and "almost everywhere" behaviors parameterized by the chosen ideal.

2. Refinements of Sequential Compactness: Principal Notions

2.1. $I$-compactness and $I^*$-compactness

A space $X$ is $I$-compact if every $\mathcal{I}$-nonthin sequence in $X$ admits an $\mathcal{I}$-nonthin subsequence that is $\mathcal{I}$-convergent (Singha et al., 2021). It is $I^*$-compact if every $\mathcal{I}$-nonthin sequence has an $\mathcal{I}$-nonthin subsequence that is $\mathcal{I}^*$-convergent (i.e., supported on a filter-large set and convergent in the usual sense).

Key relationships include:

• $I^*$-compactness $\implies$ $I$-compactness,
• both strictly refine sequential compactness in general,
• $I$-compactness coincides with sequential compactness in first-countable spaces when the ideal is finite (Singha et al., 2021, Behmanush et al., 2023).

2.2. Statistical Compactness

Instance of $I$-compactness with the natural density-zero ideal $\mathcal{I}_d$, defined by $A\subseteq \mathbb{N}$ having density zero (the limit $\frac{1}{n} |A\cap\{1,\ldots,n\}| \to 0$). A sequence $(x_n)$ is statistically convergent to $x$ if $\{ n : x_n \notin U \} \in \mathcal{I}_d$ for every neighborhood $U$ of $x$. Statistical compactness requires every nonthin sequence (positive density) to admit a nonthin subsequence statistically converging (Singha et al., 2022).

2.3. $I^K$-Frechet Compactness and Related Notions

Let $I, K$ be ideals on an indexing set $S$. A point $x\in X$ is an $I^K$-limit point of a function $f:S\to X$ if there exists $M\in\mathcal{F}(I)$ such that, resetting $f$ outside $M$ to $x$, there exists $N\in\mathcal{F}(K)$ with $g(s)\to x$ along $N$. $I^K$-Frechet compactness requires every infinite subset to have an $\ell_{I^K}$-limit point as defined via this two-level ideal-based convergence (Singha et al., 2023). For $K = I$, this yields $I$-Frechet compactness.

3. Relationships to Classical Compactness and Hierarchies

Ideal-based refinements yield a strict hierarchy:

• In first-countable spaces: $I^*$-compact $\implies$ $I$-compact $\implies$ sequential compactness.
• $I$- or $I^*$-compactness does not imply (classical) compactness in general; examples exhibit the failure of reverse implications even for metric spaces (Singha et al., 2021, Singha et al., 2022).
• The closed interval $[0,1]$ is compact and sequentially compact but not statistically compact in the usual topology; the Cantor set is compact but not statistically compact (Singha et al., 2022).

A table of refinement relationships:

Property	Implies	Counterexample for converse
$I^*$-compact	$I$-compact	[0,1] with nontrivial $I$
$I$-compact	sequentially compact	Ultrafilter ideals
sequentially compact	countably compact	When $I$ is not finite
statistical compact	sequential compact (in 1st ctbl)	$[0,1]$

4. Partition-Regular Functions, Critical Ideals, and General Frameworks

An advanced perspective organizes ideal-based compactness notions via partition-regular functions. A function $\rho:\mathcal{F}\to [\Lambda]^\omega$ on a family of infinite subsets encapsulates monotonicity, the Ramsey partition property, and sparsity (Filipów et al., 17 Jan 2026). The corresponding compactness class FinBW$(\rho)$ consists of spaces where every function indexed by $\Lambda$ admits a $\rho(F)$-convergent subsequence for some $F$.

The Katětov order between ideals (or partition-regular functions) orders these classes. Critical ideals delineate sharp boundaries: for each classical compactness class $\mathcal{C}$ (finite, "boring," compact metric), there is a critical ideal $J_\mathcal{C}$ so that the associated FinBW$(I)$ equals $\mathcal{C}$ exactly when $I$ crosses the corresponding boundary in the Katětov order (Filipów et al., 17 Jan 2026).

Notable examples:

• $Fin^2$ is critical for finite spaces,
• the 3-dimensional blocking ideal $BI$ for "boring" sequentially compact spaces,
• the ordinary-convergence ideal $conv$ for compact metric spaces.

Partition-regular functions further encompass IP- and Ramsey-type convergence, unifying diverse "non-classical" convergence regimes.

5. Permanence, Product Theorems, and Subspace Properties

Many classical compactness permanence features extend naturally:

• Closed subspaces of $I$-compact, $I^*$-compact, or statistically compact spaces inherit the corresponding property (Singha et al., 2022, Singha et al., 2021).
• Statistical continuity or $I^\ast$-continuity preserves compactness under images.
• Finite products: The product of $I$-compact (or $I^*$-compact) spaces is $I$-compact (resp. $I^*$-compact), provided certain "shrinking" conditions are met; specifically, two versions (A and B) articulate precise combinatorial conditions on ideals under which countable products retain compactness (Singha et al., 2021, Singha et al., 2023).

6. Illustrative Examples, Counterexamples, and Critical Phenomena

Examples demonstrate the strict independence among various compactness concepts:

• The space $\prod_n \{1,\dots,n\}$ (product of finite chains) is compact by Tychonoff's theorem but not statistically compact (Singha et al., 2022).
• The ultrafilter ideal yields $I$-compactness without sequential compactness (Singha et al., 2021).
• Adjusting ideals can force spaces to be $I$-Frechet compact but not $I^K$-Frechet compact, with condition (A) functioning as the key for equivalence (Singha et al., 2023).

Notably, $I^\ast$-limits are not necessarily subsequentially hereditary, and specific constructed sequences exhibit divergent behaviors with respect to various ideal-topologies (Behmanush et al., 2023).

7. Generalizations and Open Problems

There is a broad spectrum of possible generalizations:

• $T$-, $A$-, and rough statistical compactness, as well as associated one-point compactifications, are studied with parallels in behavior yet open relationships regarding metrization, separability, and universality properties (Singha et al., 2022).
• The "shrinking-conditions" articulate precise combinatorial constraints for extension of compactness through products and images (Singha et al., 2021, Singha et al., 2023).
• For each variant of ideal convergence and its compactness counterpart, open problems include finding intrinsic characterizations, precise relationships with classical theorems (e.g., Alexander subbase or Mazurkiewicz's theorem in the ideal context), and understanding the spectrum of critical ideals in related compactness hierarchies (Singha et al., 2023, Filipów et al., 17 Jan 2026).

These developments position ideal-based refinements of sequential compactness as a unifying structure mediating between set theoretic combinatorics and advanced general topology, prompting an array of further investigations in both foundation and application.