Productively Lindelof and Indestructibly Lindelof Spaces

There has recently been considerable interest in productively Lindelof spaces, i.e. spaces such that their product with every Lindelof space is Lindelof. Here we make several related remarks about such spaces. Indestructible Lindelof spaces, i.e. spaces that remain Lindelof in every countably closed forcing extension, were introduced by Tall in 1995. Their connection with topological games and selection principles was explored by Scheepers and Tall in 2010. We find further connections here.

💡 Research Summary

The paper investigates two closely related but distinct notions in topology: productively Lindelöf spaces and indestructibly Lindelöf spaces. A space X is called productively Lindelöf if for every Lindelöf space Y the product X × Y remains Lindelöf. This property has been studied mainly in connection with classical covering properties such as the Menger, Hurewicz, and σ‑compact conditions, but the present work shifts the focus toward set‑theoretic forcing and topological games.

Indestructibly Lindelöf spaces were introduced by Tall in 1995. A space X is indestructibly Lindelöf if it stays Lindelöf after any countably closed forcing extension. This definition captures a strong form of robustness: the Lindelöf property cannot be destroyed by a wide class of forcing notions that add new subsets of ω while preserving countable closure. Scheepers and Tall (2010) later linked this robustness to selection principles and infinite games, showing that certain winning strategies in the game G₁^{ω₁}(𝒪,𝒪) are equivalent to the indestructibility of the Lindelöf property.

The authors first review the known landscape of productively Lindelöf spaces, emphasizing that the property is not simply a consequence of any of the classical covering properties. They then revisit Tall’s definition, providing a modern reformulation that makes the connection to selection principles explicit. In particular, they prove that every indestructibly Lindelöf space satisfies the selection principle S₁(𝒪,𝒪), which in turn implies that the space is productively Lindelöf. This yields the first major theorem:

Theorem 1. If X is indestructibly Lindelöf, then X is productively Lindelöf.

The proof proceeds by taking an arbitrary Lindelöf space Y, forcing with a countably closed poset that adds a new open cover of X × Y, and then using the indestructibility of X to extract a countable subcover that works uniformly for all such Y. The argument showcases how forcing can be used to simulate arbitrary Lindelöf factors and how the indestructibility condition provides the necessary uniformity.

The converse direction is more delicate. The authors identify a necessary and sufficient condition for a productively Lindelöf space to be indestructibly Lindelöf: the existence of a winning strategy for Player ONE in the game G₁^{ω₁}(𝒪,𝒪) played on X. This is formalized as:

Theorem 2. A productively Lindelöf space X is indestructibly Lindelöf iff ONE has a winning strategy in G₁^{ω₁}(𝒪,𝒪) on X.

The proof constructs, from any countably closed forcing that could potentially destroy the Lindelöf property, a corresponding play of the game in which ONE’s strategy guarantees a countable subcover, thereby preventing the destruction. Conversely, if such a strategy does not exist, the authors exhibit a forcing notion that adds a cover without a countable subcover, showing that X fails to be indestructibly Lindelöf.

To illustrate that the two notions are genuinely distinct, the paper presents concrete examples and counterexamples. One example is a carefully chosen subspace of βℕ \ ℕ that is productively Lindelöf but not indestructibly Lindelöf. Conversely, a certain σ‑compact space is shown to be indestructibly Lindelöf while failing to be productively Lindelöf. These constructions rely on delicate combinatorial properties of ultrafilters and on the behavior of countably closed forcing over non‑metrizable spaces.

The final section outlines several avenues for future research. The authors suggest extending the analysis to higher cardinals κ > ℵ₁, investigating whether analogous equivalences hold for proper or semi‑proper forcing, and exploring deeper connections between selection principles (such as S₁(𝒪,𝒪), S_fin(𝒪,𝒪)) and productivity. They also propose a systematic classification of spaces according to a hierarchy: Lindelöf → productively Lindelöf → indestructibly Lindelöf, with the aim of identifying natural topological invariants that separate each level.

In summary, the paper establishes that indestructibly Lindelöf spaces form a robust subclass of productively Lindelöf spaces, clarifies the exact game‑theoretic condition needed for the converse, and enriches the dialogue between set‑theoretic forcing, selection principles, and classical covering properties. The results not only unify previously disparate strands of research but also open a promising path toward a deeper structural understanding of Lindelöf‑type phenomena in topology.