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Arhangel'skiĭ sheaf amalgamations in topological groups

100%
Tsaban B., Zdomskyy L.
Fundamenta Mathematicae
|
2016
|
tom 232
|
nr 3
281-293
EN
We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos's property $α_{1.5}$ is equivalent to Arhangel'skiĭ's formally stronger property α₁. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space X such that the space $C_{p}(X)$ of continuous real-valued functions on X with the topology of pointwise convergence has Arhangel'skiĭ's property α₁ but is not countably tight. This follows from results of Arhangel'skiĭ-Pytkeev, Moore and Todorčević, and provides a new solution, with stronger properties than the earlier solution, of a problem of Averbukh and Smolyanov (1968) concerning topological vector spaces.
2
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Measurable cardinals and the cofinality of the symmetric group

100%
Friedman S.-D., Zdomskyy L.
Fundamenta Mathematicae
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2010
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tom 207
|
nr 2
101-122
EN
Assuming the existence of a P₂κ-hypermeasurable cardinal, we construct a model of Set Theory with a measurable cardinal κ such that $2^{κ} = κ⁺⁺$ and the group Sym(κ) of all permutations of κ cannot be written as the union of a chain of proper subgroups of length < κ⁺⁺. The proof involves iteration of a suitably defined uncountable version of the Miller forcing poset as well as the "tuning fork" argument introduced by the first author and K. Thompson [J. Symbolic Logic 73 (2008)].
3
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Countable dense homogeneous filters and the Menger covering property

81%
Repovš D., Zdomskyy L., Zhang S.
Fundamenta Mathematicae
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2014
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tom 224
|
nr 3
233-240
EN
We present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hernández-Gutiérrez and Hrušák. The method of the proof also allows us to obtain for any n ∈ ω ∪ {∞} an n-dimensional metrizable Baire topological group which is strongly locally homogeneous but not countable dense homogeneous.
4
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Δ₁-Definability of the non-stationary ideal at successor cardinals

81%
Friedman S.-D., Wu L., Zdomskyy L.
Fundamenta Mathematicae
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2015
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tom 229
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nr 3
231-254
EN
Assuming V = L, for every successor cardinal κ we construct a GCH and cardinal preserving forcing poset ℙ ∈ L such that in $L^{ℙ}$ the ideal of all non-stationary subsets of κ is Δ₁-definable over H(κ⁺).
5
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Seven characterizations of non-meager 𝖯-filters

81%
Kunen K., Medini A., Zdomskyy L.
Fundamenta Mathematicae
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2015
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tom 231
|
nr 2
189-208
EN
We give several topological/combinatorial conditions that, for a filter on ω, are equivalent to being a non-meager 𝖯-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager 𝖯-filter. Here, we identify a filter with a subspace of $2^{ω}$ through characteristic functions. Along the way, we generalize to non-meager 𝖯-filters a result of Miller (1984) about 𝖯-points, and we employ and give a new proof of results of Marciszewski (1998). We also employ a theorem of Hernández-Gutiérrez and Hrušák (2013), and answer two questions that they posed. Our result also resolves several issues raised by Medini and Milovich (2012), and proves false one "theorem" of theirs. Furthermore, we show that the statement "Every non-meager filter contains a non-meager 𝖯-subfilter" is independent of 𝖹𝖥𝖢 (more precisely, it is a consequence of 𝔲 < 𝔤 and its negation is a consequence of ⋄). It follows from results of Hrušák and van Mill (2014) that, under 𝔲 < 𝔤, a filter has less than 𝔠 types of countable dense subsets if and only if it is a non-meager 𝖯-filter. In particular, under 𝔲 < 𝔤, there exists an ultrafilter with 𝔠 types of countable dense subsets. We also show that such an ultrafilter exists under 𝖬 𝖠(countable).
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