Combinatorics of dense subsets of the rationals

• Autorzy: B. Balcar , F. Hernández-Hernández , M. Hrušák
• Języki publikacji: EN

Abstrakty

EN

We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants $𝔭_{ℚ}$, $𝔱_{ℚ}$, $𝔥_{ℚ}$, $𝔰_{ℚ}$, $𝔯_{ℚ}$, $𝔦_{ℚ}$ describing properties of Dense(ℚ). These invariants satisfy $𝔭_{ℚ}$ ≤ 𝔱_{ℚ} ≤ 𝔥_{ℚ} ≤ 𝔰_{ℚ} ≤ 𝔯_{ℚ} ≤ 𝔦_{ℚ}$. We compare them with their analogues in the well studied Boolean algebra 𝒫(ω)/fin. We show that $𝔭_{ℚ} = p$, $𝔱_{ℚ} = t$ and $𝔦_{ℚ} = i$, whereas $𝔥_{ℚ} > h$ and $𝔯_{ℚ} > r$ 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{|𝒟|: 𝒟 ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ 𝒟)(I ∩ D = ∅)} and cof(M) = min{|𝒟|: 𝒟 ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ 𝒟)(I ∩ 𝒟 = ∅)}.
We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.

Kategorie tematyczne

• 06E15: Stone spaces (Boolean spaces) and related structures
• 03E17: Cardinal characteristics of the continuum
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2004
• Tom: 183
• Numer: 1
• Strony: 59-80

Daty

wydano

2004

Twórcy

autor

B. Balcar

• Mathematical Institute, of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic

autor

F. Hernández-Hernández

• Instituto de Matemáticas, UNAM Unidad Morelia, A. P. 61-3, Xangari, C.P. 58089, Morelia, Mich., México

autor

M. Hrušák

• Instituto de Matemáticas, UNAM Unidad Morelia, A. P. 61-3, Xangari, C.P. 58089, Morelia, Mich., México

Identyfikatory

DOI

10.4064/fm183-1-4