Products of Hurewicz Spaces in the Laver Model

Abstract

Abstract. This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We illustrate this interplay by proving that in the Laver model for the consistency of the Borel's conjecture, the product of any two metrizable spaces with the Hurewicz property has the Menger property.

Products of H-separable spaces in the Laver model

“…However, this paper concentrates on set-theoretic and combinatorial aspects of the property of Menger and its variations. Theorem 1.1 is closely related to the main result of [13] asserting that in the Laver model the product of any two Hurewicz metrizable spaces has the Menger property. Let us note that our proof in [13] is conceptually different, even though both proofs are based on the same main technical lemma of [9].…”

“…Theorem 1.1 is closely related to the main result of [13] asserting that in the Laver model the product of any two Hurewicz metrizable spaces has the Menger property. Let us note that our proof in [13] is conceptually different, even though both proofs are based on the same main technical lemma of [9]. Regarding the relation between Theorem 1.1 and the main result of [13], each of them implies a weak form of the other one via the following duality results: For a metrizable space X, C p (X) is M -separable (resp.…”

“…Let us note that our proof in [13] is conceptually different, even though both proofs are based on the same main technical lemma of [9]. Regarding the relation between Theorem 1.1 and the main result of [13], each of them implies a weak form of the other one via the following duality results: For a metrizable space X, C p (X) is M -separable (resp. H-separable) if and only if all finite powers of X are Menger (resp.…”

Preservation of 𝛾-spaces and covering properties of products

“…One of the motivations behind this question comes from spaces of continuous functions, see [8,Theorem 21]. In the case of general topological spaces there are ZFC examples of Hurewicz spaces whose product is not even Menger, see [18, §3] and the discussion in the introduction of [14]. That is why we concentrate in what follows on subspaces of the Cantor space 2 ω .…”

“…On the other hand, the product of any two Hurewicz subspaces of 2 ω is Menger in the Laver and Miller models, see [14] and [19], respectively. In the Miller model we actually know that the product of finitely many Hurewicz subspaces of 2 ω is Menger (for the Laver model this is unknown even for three Hurewicz subspaces), because in this model the Menger property is preserved by products of subspaces of 2 ω , see [19].…”