Jan van Mill (2011)
Fundamenta Mathematicae
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We prove that if a space X is countable dense homogeneous and no set of size n-1 separates it, then X is strongly n-homogeneous. Our main result is the construction of an example of a Polish space X that is strongly n-homogeneous for every n, but not countable dense homogeneous.
Marilina Carena, Marisa Toschi (2015)
Colloquium Mathematicae
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Let (X,d,μ) be a space of homogeneous type. We study the relationship between two types of s-sets: relative to a distance and relative to a measure. We find a condition on a closed subset F of X under which F is an s-set relative to the measure μ if and only if F is an s-set relative to δ. Here δ denotes the quasi-distance defined by Macías and Segovia such that (X,δ,μ) is a normal space. In order to prove this result, we prove a covering type lemma and a type of Hausdorff measure based...
Arhangel’skii, Alexander, Choban, Mitrofan, Mihaylova, Ekaterina (2012)
Union of Bulgarian Mathematicians
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Александър В. Архангелски, Митрофан М. Чобан, Екатерина П. Михайлова - Въведени са понятията o-хомогенно пространство, lo-хомогенно пространство, do-хомогенно пространство и co-хомогенно пространство. Показано е, че ако lo-хомогенно пространство X има отворено подпространство, което е q-пълно, то и самото X е q-пълно. Показано е, че ако lo-хомогенно пространство X съдържа навсякъде гъсто екстремално несвързано подпространство, тогава X е екстремално несвързано. In this paper...
Jan van Mill (2008)
Fundamenta Mathematicae
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We show that there is a Polish space which is countable dense homogeneous but contains a dense open rigid connected subset. This answers several questions of Fitzpatrick and Zhou.
Dušan Repovš, Lyubomyr Zdomskyy, Shuguo Zhang (2014)
Fundamenta Mathematicae
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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.
Jan van Mill (2005)
Bulletin of the Polish Academy of Sciences. Mathematics
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We present an example of a connected, Polish, countable dense homogeneous space X that is not strongly locally homogeneous. In fact, a nontrivial homeomorphism of X is the identity on no nonempty open subset of X.
R. Bennett (1972)
Fundamenta Mathematicae
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Jacek Cichoń (2001)
Acta Universitatis Carolinae. Mathematica et Physica
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David H. Fremlin (2004)
Acta Universitatis Carolinae. Mathematica et Physica
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Eric K. van-Douwen (1979)
Colloquium Mathematicae
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Rodrigo Hernández-Gutiérrez, Michael Hrušák, Jan van Mill (2014)
Fundamenta Mathematicae
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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...