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1

Definable Davies' theorem

100%

Törnquist A., Weiss W.

Fundamenta Mathematicae

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2009

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tom 205

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nr 1

77-89

EN

We prove the following descriptive set-theoretic analogue of a theorem of R. O. Davies: Every Σ¹₂ function f:ℝ × ℝ → ℝ can be represented as a sum of rectangular Σ¹₂ functions if and only if all reals are constructible.

2

Cardinal sequences of length < ω₂ under GCH

81%

Juhász I., Soukup L., Weiss W.

Fundamenta Mathematicae

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2006

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tom 189

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nr 1

35-52

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.

3

On a problem of Sikorski

41%

Juhász I., Weiss W.

Fundamenta Mathematicae

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1978

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tom 100

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nr 3

223-227

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Weiss W.

2 Juhász I.

1 Soukup L.

1 Törnquist A.

1 2009

1 2006

1 1978

Definable Davies' theorem

Cardinal sequences of length < ω₂ under GCH

On a problem of Sikorski