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Non-meager P-filters are countable dense homogeneous

100%

Hernández-Gutiérrez R., Hrušák M.

Colloquium Mathematicum

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2013

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tom 130

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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.

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Wijsman hyperspaces of non-separable metric spaces

100%

Hernández-Gutiérrez R., Szeptycki P.

Fundamenta Mathematicae

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2015

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tom 228

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nr 1

63-79

EN

Given a metric space ⟨X,ρ⟩, consider its hyperspace of closed sets CL(X) with the Wijsman topology $τ_{W(ρ)}$. It is known that $⟨CL(X),τ_{W(ρ)}⟩$ is metrizable if and only if X is separable, and it is an open question by Di Maio and Meccariello whether this is equivalent to $⟨CL(X),τ_{W(ρ)}⟩$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $⟨CL(X),τ_{W(ρ)}⟩$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces.

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Countable dense homogeneity and λ-sets

81%

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

Fundamenta Mathematicae

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2014

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tom 226

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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.

Ograniczanie wyników

2 Fundamenta Mathematicae

1 Colloquium Mathematicum

3 Hernández-Gutiérrez R.

2 Hrušák M.

1 Szeptycki P.

1 van Mill J.

1 2015

1 2014

1 2013

Non-meager P-filters are countable dense homogeneous

Wijsman hyperspaces of non-separable metric spaces

Countable dense homogeneity and λ-sets