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1
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Another ⋄-like principle

100%
Hrušák M.
Fundamenta Mathematicae
|
2001
|
tom 167
|
nr 3
277-289
EN
A new ⋄-like principle $⋄_{𝔡}$ consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that $¬ ⋄_{𝔡}$ is consistent with CH and that in many models of 𝔡 = ω₁ the principle $⋄_{𝔡}$ holds. As $⋄_{𝔡}$ implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether 𝔡 = ω₁ implies 𝔞 = ω₁. It is proved that $⋄_{𝔡}$ holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether 𝔞 = ω₁ in such models.
2
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Non-meager P-filters are countable dense homogeneous

64%
Hernández-Gutiérrez R., Hrušák M.
Colloquium Mathematicum
|
2013
|
tom 130
|
nr 2
281-289
EN
We prove that if ℱ is a non-meager P-filter, then both ℱ and $^{ω}ℱ$ are countable dense homogeneous spaces.
3
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Selections and weak orderability

64%
Hrušák M., Martínez-Ruiz I.
Fundamenta Mathematicae
|
2009
|
tom 203
|
nr 1
1-20
EN
We answer a question of van Mill and Wattel by showing that there is a separable locally compact space which admits a continuous weak selection but is not weakly orderable. Furthermore, we show that a separable space which admits a continuous weak selection can be covered by two weakly orderable spaces. Finally, we give a partial answer to a question of Gutev and Nogura by showing that a separable space which admits a continuous weak selection admits a continuous selection for all finite sets.
4
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A countable dense homogeneous set of reals of size ℵ₁

51%
Farah I., Hrušák M., Ranero C.
Fundamenta Mathematicae
|
2005
|
tom 186
|
nr 1
71-77
EN
We prove there is a countable dense homogeneous subspace of ℝ of size ℵ₁. The proof involves an absoluteness argument using an extension of the $L_{ω₁ω}(Q)$ logic obtained by adding predicates for Borel sets.
5
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Countable dense homogeneity and λ-sets

51%
Hernández-Gutiérrez R., Hrušák M., van Mill J.
Fundamenta Mathematicae
|
2014
|
tom 226
|
nr 2
157-172
EN
We show that all sufficiently nice λ-sets are countable dense homogeneous (𝖢𝖣𝖧). From this fact we conclude that for every uncountable cardinal κ ≤ 𝔟 there is a countable dense homogeneous metric space of size κ. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size κ is equivalent to the existence of a λ-set of size κ. On the other hand, it is consistent with the continuum arbitrarily large that every 𝖢𝖣𝖧 metric space has size either ω₁ or 𝔠. An example of a Baire 𝖢𝖣𝖧 metric space which is not completely metrizable is presented. Finally, answering a question of Arhangel'skii and van Mill we show that that there is a compact non-metrizable 𝖢𝖣𝖧 space in ZFC.
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