Preservation of 𝛾-spaces and covering properties of products

Abstract

We prove that the Hurewicz property is not preserved by finite products in the Miller model. This is a consequence of the fact that Miller forcing preserves ground model γ-spaces.2010 Mathematics Subject Classification. Primary: 03E35, 54D20. Secondary: 54C50, 03E05.

Selection Principles in the Laver, Miller, and Sacks models

“…The product of finitely many Hurewicz subspaces of 2 ω is Menger by Theorem 1.5, and thus the Miller model seemed for a while to be the best candidate for a model where the Hurewicz property is preserved by finite products of metrizable spaces. The next theorem proved in [41] refutes this expectation.…”

“…Unlike the Laver and Miller models, in the Sacks model introduced in [45] we have the best possible non-preservation by products results: Countable support iterations of the Sacks forcing preserve γ-subspaces of R [41], while being non-Menger is obviously preserved by any forcing which does not add unbounded reals. Thus in the Sacks model there exist γ-spaces X, Y ⊂ 2 ω with non-Menger product.…”