Combinatorics of dense subsets of the rationals

B. Balcar ; F. Hernández-Hernández ; M. Hrušák

Fundamenta Mathematicae, Tome 184 (2004), p. 59-80 / Harvested from The Polish Digital Mathematics Library

Résumé

We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants $_{ℚ}$, $_{ℚ}$, $_{ℚ}$, $_{ℚ}$, $_{ℚ}$, $_{ℚ}$ describing properties of Dense(ℚ). These invariants satisfy $_{ℚ}$ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ$. We compare them with their analogues in the well studied Boolean algebra (ω)/fin. We show that $ℚ = p$, $ℚ = t$ and $ℚ = i$, whereas $ℚ > h$ and $ℚ > r$ are both shown to be relatively consistent with ZFC. We also investigate combinatorics of the ideal nwd of nowhere dense subsets of ,ℚ. In particular, we show that $non(M)=min||: ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ )(I ∩ D = ∅) and cof(M) = min||: ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ )(I ∩ = ∅). We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.

Détail

Publié le : 2004-01-01

Zbl 1051.03038

EUDML-ID : urn:eudml:doc:283218