Simultaneous Sequential Compactness

A set of sequences is said to converge simultaneously if there exists an infinite subset $H$ of the index set $ω$ such that all sequences converge when restricted to $H$. We discuss simultaneous convergence of sequences in the same or in different sequentially compact spaces; we link the results for different spaces to ones for the same space; we show that simultaneous convergence happens for less than $\mathfrak s$ sequences in spaces with weight bounded by $\mathfrak s$ and for less than $\mathfrak h$ sequences in general; we show a slight generalisation of these results in the context of Hausdorff spaces; and finally we investigate their optimality.

💡 Research Summary

The paper introduces the notion of simultaneous sequential compactness by defining a space X to be λ‑simultaneously sequentially compact (λ‑ssc) if for any family of λ sequences taking values in X there exists an infinite subset H⊆ω such that every sequence converges when restricted to H. This generalises ordinary sequential compactness (the case λ=ℵ₀) and allows one to ask how large λ can be while the property still holds, depending on the topological characteristics of X.

Section 1 reviews the basic definitions of convergence along a subset of ω, the equivalence between ℵ₀‑ssc and ordinary sequential compactness, and establishes a diagonal construction that shows any sequentially compact space is automatically ℵ₀‑ssc.

Section 2 brings in the splitting number s, the smallest cardinality of a family of subsets of ω that splits every infinite subset. Lemma 2.3 shows that any non‑trivial space fails to be s‑ssc, while Lemma 2.4 proves that the discrete two‑point space {0,1} is λ‑ssc for every λ < s. The central result, Theorem 2.7, states that if λ < s and the weight w(X) of a sequentially compact space X satisfies w(X) < s, then X is λ‑ssc. The proof translates each original sequence into a family of binary sequences indexed by a basis of size < s, applies Lemma 2.4 to obtain a common infinite H, and then shows that the original sequences converge along H. A similar argument yields Theorem 2.8, which removes the sequential compactness hypothesis and only assumes compactness together with w(X) < s.

From these, classical corollaries follow: any compact space of weight < s is sequentially compact (Corollary 2.9), and the unit ball of the dual of a separable normed space is λ‑simultaneously weak‑* sequentially compact for λ < s (Corollary 2.10), extending the Banach–Alaoglu theorem to families of functionals.

Section 3 focuses on Hausdorff spaces and introduces a finer cardinal invariant, the countable Hausdorff weight cHw(X). This invariant measures, for each countable subset Y⊆X, the smallest size of a family of open sets that “sequentially separates” limit points of Y. Lemma 3.3 shows cHw(X) ≤ w(X), while Lemma 3.4 proves cHw(X) ≤ |X| for Hausdorff spaces. The main Hausdorff result, Theorem 3.5, asserts that if λ < s and cHw(X) < s, then any sequentially compact Hausdorff space X is λ‑ssc. The proof again uses binary sequences derived from the separating families, applies Lemma 2.4, and then reconstructs convergence of the original sequences via a standard Urysohn‑type argument.

A refinement, Theorem 3.7, shows that when cHw(X)=s but X does not “reach” its countable Hausdorff weight, the space is λ‑ssc for all λ < cof(s), linking the result to the cofinality of s. Proposition 3.8 provides an explicit example (an Aleksandrov compactification of a discrete space) where cHw(X)=ℵ₀ while w(X) can be arbitrarily large, demonstrating that the Hausdorff improvement is genuine.

Section 4 generalises the discussion to families of sequences that live in different topological spaces. Definition 4.1 extends λ‑ssc to a family 𝔽 of spaces: 𝔽 is λ‑ssc if any λ sequences, each taken from some member of 𝔽, admit a common infinite H on which they all converge. By embedding each space into a common compact space (e.g., via Tychonoff products or by considering compact subsets of ℝ), the authors show that the results of Sections 2 and 3 automatically transfer to this more general setting.

Section 5 removes any weight restriction and works solely with the distributivity number h, the smallest cardinal κ such that the Boolean algebra ℘(ω)/fin is not κ‑distributive. Theorem 5.2 proves that for any sequentially compact space X, if λ < h then X is λ‑ssc. Since h ≥ s, this theorem subsumes the earlier s‑based results and shows that the bound λ < h is optimal in ZFC: it cannot be improved without additional set‑theoretic assumptions.

The paper concludes with a discussion of optimality. It notes that the bounds λ < s (or λ < h) are sharp: for λ = s one can construct a family of s sequences in a non‑trivial sequentially compact space that fail to have a common convergent subsequence, using the splitting family itself. Likewise, the improvement from w(X) to cHw(X) in Hausdorff spaces is shown to be genuine via the Aleksandrov example. Open problems are posed concerning the exact relationship between s, h, and other cardinal characteristics (e.g., 𝔟, 𝔡) in determining the maximal λ for which a given space is λ‑ssc.

Overall, the work systematically develops a hierarchy of simultaneous convergence properties, ties them to well‑studied cardinal invariants of the continuum, and provides both general theorems and concrete examples that illuminate the delicate interplay between topology and set theory.