Cardinal sequences of length < ω₂ under GCH

• Autorzy: István Juhász , Lajos Soukup , William Weiss
• Języki publikacji: EN

Abstrakty

EN

Let 𝓒(α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put $𝓒_{λ}(α) = {s ∈ 𝓒(α): s(0) = λ = min[s(β): β < α]}$.
We show that f ∈ 𝓒(α) iff for some natural number n there are infinite cardinals $λ₀i > λ₁ > ... > λ_{n-1}$ and ordinals $α₀,...,α_{n-1}$ such that $α = α₀ + ⋯ +α_{n-1}$ and $f = f₀⏜f₁⏜...⏜f_{n-1}$ where each $f_i ∈ 𝓒_{λ_i}(α_i)$. Under GCH we prove that if α < ω₂ then
(i) $𝓒_{ω}(α) = {s ∈ ^{α}{ω,ω₁}: s(0) = ω}$;
(ii) if λ > cf(λ) = ω,
$𝓒_{λ}(α) = {s ∈ ^{α}{λ,λ⁺}: s(0) = λ, s^{-1}{λ} is ω₁-closed in α}$;
(iii) if cf(λ) = ω₁,
$𝓒_{λ}(α) = {s ∈ ^{α}{λ,λ⁺}: s(0) = λ, s^{-1}{λ} is ω-closed and successor-closed in α}$;
(iv) if cf(λ) > ω₁, $𝓒_{λ}(α) = ^{α}{λ}$.
This yields a complete characterization of the classes 𝓒(α) for all α < ω₂, under GCH.

Kategorie tematyczne

• 06E05: Structure theory
• 54D30: Compactness
• 54A25: Cardinality properties (cardinal functions and inequalities, discrete subsets)
• 54G12: Scattered spaces
• 03E75: Applications of set theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2006
• Tom: 189
• Numer: 1
• Strony: 35-52

Daty

wydano

2006

Twórcy

autor

István Juhász

• Alfréd Rényi Institute of Mathematics, V. Reáltanoda utca, 13-15, H-1053 Budapest, Hungary

autor

Lajos Soukup

• Alfréd Rényi Institute of Mathematics, V. Reáltanoda utca, 13-15, H-1053 Budapest, Hungary

autor

William Weiss

• Mathematics Department, University of Toronto, Toronto, ON, M5S 1A1, Canada

Identyfikatory

DOI

10.4064/fm189-1-3