Products of Menger spaces in the Miller model

Abstract

We prove that in the Miller model the Menger property is preserved by finite products of metrizable spaces. This answers several open questions and gives another instance of the interplay between classical forcing posets with fusion and combinatorial covering properties in topology.2010 Mathematics Subject Classification. Primary: 03E35, 54D20. Secondary: 54C50, 03E05.

Preservation of 𝛾-spaces and covering properties of products

“…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]. This is why the Miller model seemed to be the best candidate for a model where the Hurewicz property is preserved by finite products of metrizable spaces.…”

Finite powers and products of Menger sets

“…Since the set (2X) ⊕ Y is a continuous image of the product space X × Y , the latter is not Menger. [25]. We show that the inequality d ≤ r is not necessary to prove that the Menger property is not productive.…”

“…We restrict our consideration to the realm of sets of reals. The three properties are consistently equivalent [25]. In other words, for these properties to differ, special set-theoretic hypotheses are necessary.…”

Products of Hurewicz Spaces in the Laver Model

“…For instance, the core of the proof of Theorem 1.1 is that Hurewicz subspaces of the real line are concentrated in a sense around their "simpler" subspaces in the Laver model, see Lemma 2.2. As a consequence of corresponding structural results we have proved [33] that the Menger property is preserved by finite products in the Miller model constructed in [17], and there are only c many Menger subspaces of R in the Sacks model constructed in [23], see [10].…”