Covering properties of $$\omega $$-mad families

Abstract

We prove that CH implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while b = c is consistent with the negation of this statement as witnessed by the Laver model for the consistency of Borel's conjecture.2010 Mathematics Subject Classification. Primary: 03E35, 54D20. Secondary: 03E05.

Selection Principles in the Laver, Miller, and Sacks models

“…Theorem 1.3 improves our earlier result proved in [3] asserting, under the same premises, that M F (A)↾X keeps the ground model unbounded. In our proof of Theorem 1.3 we shall not work with the Mathias forcing directly, but rather use the following characterization obtained in [17]: For a filter F on ω the poset M F adds no dominating reals (resp.…”

“…In what follows we concentrate on the case P being the Cohen forcing C. Mad families A which remain maximal in V C will be called Cohen-indestructible. The following theorem has been proved in [3] Theorem 1.2. p = cov (N ) = c implies the existence of a Cohen-indestructible mad family A such that M F (A) adds a dominating real.…”

“…Hurewicz) covering property when considered with the topology inherited from P(ω), which is identified with the Cantor space 2 ω via characteristic functions. Thus Theorem 1.3 says that if A is an ω-mad family, then F (A) is Hurewicz, whereas its predecessor from [3] gave only the Menger property. Let us note that there are ZFC examples of Menger non-Hurewicz filters on ω, see [42].…”

“…Replacing p ′′ with a stronger condition p (3) obtained by throwing away for every t n ∈ S ′′ finitely many of its successors (as well as all their extensions in p ′′ (0)), we can get ( 5)…”

“…However, there are many other negative results stating that under certain equality among cardinal characteristics (e.g., cov (N ) = cof (N ), b = d, etc. 3 ) there are spaces X, Y ⊂ R with some combinatorial covering property such that X × Y is not Menger, see, e.g., [4,39,54].…”