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Another ⋄-like principle

100%

Hrušák M.

Fundamenta Mathematicae

2001

tom 167

nr 3

277-289

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.

Selections and weak orderability

63%

Hrušák M. , Martínez-Ruiz I.

Fundamenta Mathematicae

2009

tom 203

nr 1

1-20

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.

Non-meager P-filters are countable dense homogeneous

63%

Hernández-Gutiérrez R. , Hrušák M.

Colloquium Mathematicum

2013

tom 130

nr 2

281-289

We prove that if ℱ is a non-meager P-filter, then both ℱ and $^{ω}ℱ$ are countable dense homogeneous spaces.

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

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.

Countable dense homogeneity and λ-sets

51%

Hernández-Gutiérrez R. , Hrušák M. , van Mill J.

Fundamenta Mathematicae

2014

tom 226

nr 2

157-172

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.