On productively Lindel"of spaces

The class of spaces such that their product with every Lindel"of space is Lindel"of is not well-understood. We prove a number of new results concerning such productively Lindel"of spaces with some extra property, mainly assuming the Continuum Hypothesis.

💡 Research Summary

The paper investigates topological spaces whose product with any Lindelöf space remains Lindelöf—so‑called productively Lindelöf (PL) spaces—and studies how additional structural hypotheses interact with this property, especially under the Continuum Hypothesis (CH) or related cardinal assumptions.

The authors begin by recalling the classical question of Michael (whether every PL space is “powerfully Lindelöf”, i.e. its ω‑th power is Lindelöf) and introduce the related notions of Alster spaces, Hurewicz, Menger, and D‑spaces. They also define “powerfully P” and “finitely powerfully P” for a property P, and the new concept of indestructibly productively Lindelöf (a space that stays PL in every countably closed forcing extension).

The first technical tool is the countably closed elementary submodel method. For a PL space X of a certain type (e.g., sequential T₃), one selects a countably closed elementary submodel M ⊂ H_θ of size ℵ₁ containing X and the relevant open cover. The subspace X∩M is closed in X, and the topology induced on X∩M by M (denoted X_M) has weight ≤ℵ₁. By Alster’s lemma (which under CH says that any T₃ space of weight ≤ℵ₁ is powerfully Lindelöf), (X_M)^ω is Lindelöf. Because M is countably closed, (X^ω)_M = (X_M)^ω, so X^ω itself is Lindelöf. This yields Theorem 1.4: under CH, every sequential T₃ PL space is powerfully Lindelöf. The same argument gives Corollary 1.5: under CH, any T₃ PL space of cardinality ℵ₁ is powerfully Lindelöf.

Next the paper turns to selective covering properties. An Alster space is defined via G_δ‑covers that admit countable subcovers; the authors prove Theorem 2.3 that every Alster space is Hurewicz, and consequently Menger. Using the fact that finite products of Alster spaces are Alster, they obtain Corollary 2.4: Alster spaces are finitely powerfully Hurewicz. They also exhibit, via results of Michael and of Bartoszyński–Tsaban, a finitely powerfully Hurewicz set of reals that is not PL (Theorem 2.5), showing that the Alster property is strictly stronger than Hurewicz.

A central open problem (Problem 2.7) asks whether CH implies that every sequential T₃ PL space is Alster. While the authors cannot resolve it, they prove Theorem 2.8: under CH, sequential T₃ PL spaces are finitely powerfully Hurewicz. The proof again uses elementary submodels to lift Hurewicz covers from the model to the ambient space.

The paper then introduces indestructibly productively Lindelöf (IPL) spaces. Theorem 3.2 shows that any IPL T₃ space is powerfully Lindelöf and finitely powerfully Hurewicz (hence finitely powerfully D). The proof collapses 2^{ℵ₀} and the weight of X to ℵ₁ via a countably closed forcing, making X Alster in the extension; CH then forces X to be powerfully Lindelöf there, and the Lindelöfness of X^ω descends to the ground model. The authors ask (Problem 3.3) whether IPL spaces must be Alster.

A corollary (3.4) states that IPL p‑spaces (in the sense of Arhangel’skiĭ) are σ‑compact, because such spaces are perfect images of metrizable spaces, and for metrizable spaces IPL coincides with σ‑compactness (as shown in earlier work).

Theorem 3.6 proves that if X^ω is IPL then X is compact. The argument again uses a collapse forcing to produce a Michael space (available under CH) and deduces compactness in the extension, which then reflects back to the ground model.

The final section deals with γ‑spaces and Michael spaces under the cardinal invariant b (the bounding number). Assuming b = ℵ₁, the authors construct a set X ⊆ P(ω) equipped with the Michael topology that is both a γ‑space and a Michael space (Theorem 4.6). The construction starts with an ℵ₁‑long ⊆*-decreasing unbounded family {x_α : α < ℵ₁} ⊂