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1
Content available remote

The power set of ω Elementary submodels and weakenings of CH

100%
Juhász I., Kunen K.
Fundamenta Mathematicae
|
2001
|
tom 170
|
nr 3
257-265
EN
We define a new principle, SEP, which is true in all Cohen extensions of models of CH, and explore the relationship between SEP and other such principles. SEP is implied by each of CH*, the weak Freeze-Nation property of 𝓟(ω), and the (ℵ₁,ℵ₀)-ideal property. SEP implies the principle $C₂^{s}(ω₂)$, but does not follow from $C₂^{s}(ω₂)$, or even $C^{s}(ω₂)$.
2
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On the convergence and character spectra of compact spaces

100%
Juhász I., Weiss W.
Fundamenta Mathematicae
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2010
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tom 207
|
nr 2
179-196
EN
An infinite set A in a space X converges to a point p (denoted by A → p) if for every neighbourhood U of p we have |A∖U| < |A|. We call cS(p,X) = {|A|: A ⊂ X and A → p} the convergence spectrum of p in X and cS(X) = ⋃{cS(x,X): x ∈ X} the convergence spectrum of X. The character spectrum of a point p ∈ X is χS(p,X) = {χ(p,Y): p is non-isolated in Y ⊂ X}, and χS(X) = ⋃{χS(x,X): x ∈ X} is the character spectrum of X. If κ ∈ χS(p,X) for a compactum X then {κ,cf(κ)} ⊂ cS(p,X). A selection of our results (X is always a compactum):
(1) If $χ(p,X) > λ = λ^{ λ = λ^{ω}$ implies that λ ∈ χS(p,X).
(2) If $χ(X) > 2^{ω}$ then ω₁ ∈ χS(X) or ${2^{ω},(2^{ω})⁺} ⊂ χS(X)$.
(3) If χ(X) > ω then $χS(X) ∩ [ω₁,2^{ω}] ≠ ∅$.
(4) If $χ(X) > 2^{κ}$ then κ⁺ ∈ cS(X), in fact there is a converging discrete set of size κ⁺ in X.
(5) If we add λ Cohen reals to a model of GCH then in the extension for every κ ≤ λ there is X with χS(X) = {ω,κ}. In particular, it is consistent to have X with $χS(X) = {ω, ℵ_{ω}}$.
(6) If all members of χS(X) are limit cardinals then $|X| ≤ (sup{|S̅|: S ∈ [X]^{ω}})^{ω}$.
(7) It is consistent that $2^{ω}$ is as big as you wish and there are arbitrarily large X with $χS(X) ∩ (ω,2^{ω}) = ∅$.
It remains an open question if, for all X, min cS(X) ≤ ω₁ (or even min χS(X) ≤ ω₁) is provable in ZFC.
3
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Cardinal sequences of length < ω₂ under GCH

81%
Juhász I., Soukup L., Weiss W.
Fundamenta Mathematicae
|
2006
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tom 189
|
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.
4
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Cardinal sequences and Cohen real extensions

81%
Juhász I., Shelah S., Soukup L., Szentmiklóssy Z.
Fundamenta Mathematicae
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2004
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tom 181
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nr 1
75-88
EN
We show that if we add any number of Cohen reals to the ground model then, in the generic extension, a locally compact scattered space has at most $(2^{ℵ₀})^V$ levels of size ω. We also give a complete ZFC characterization of the cardinal sequences of regular scattered spaces. Although the classes of regular and of 0-dimensional scattered spaces are different, we prove that they have the same cardinal sequences.
5
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Regular spaces of small extent are ω-resolvable

81%
Juhász I., Soukup L., Szentmiklóssy Z.
Fundamenta Mathematicae
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2015
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tom 228
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nr 1
27-46
EN
We improve some results of Pavlov and Filatova, concerning a problem of Malykhin, by showing that every regular space X that satisfies Δ(X) > e(X) is ω-resolvable. Here Δ(X), the dispersion character of X, is the smallest size of a non-empty open set in X, and e(X), the extent of X, is the supremum of the sizes of all closed-and-discrete subsets of X. In particular, regular Lindelöf spaces of uncountable dispersion character are ω-resolvable. We also prove that any regular Lindelöf space X with |X| = Δ(X) = ω₁ is even ω₁-resolvable. The question whether regular Lindelöf spaces of uncountable dispersion character are maximally resolvable remains wide open.
6
Content available remote

First countable spaces without point-countable π-bases

81%
Juhász I., Soukup L., Szentmiklóssy Z.
Fundamenta Mathematicae
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2007
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tom 196
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nr 2
139-149
EN
We answer several questions of V. Tkachuk [Fund. Math. 186 (2005)] by showing that ∙ there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable π-base (in fact, the minimum order of a π-base of the space can be made arbitrarily large); ∙ if there is a κ-Suslin line then there is a first countable GO-space of cardinality κ⁺ in which the order of any π-base is at least κ; ∙ it is consistent to have a first countable, hereditarily Lindelöf regular space having uncountable π-weight and ω₁ as a caliber (of course, such a space cannot have a point-countable π-base).
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