On properties of h-homogeneous spaces with the Baire property

Medvedev, S. V.

doi:10.1016/j.topol.2011.10.016

Cited by 7 publications

(4 citation statements)

References 8 publications

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“…Knaster-Reichbach covers were introduced in [30] and have been successfully applied by several authors, including van Engelen, Medvedev and Ostrovskiȋ. Let us mention for example the articles [5], [22], [23], [24], [25] and [34], where one can find much more general results than the ones stated here. The first application of this technique to the theory of countable dense homogeneity was recently given by Hernández-Gutiérrez, Hrušák and van Mill in [10].…”

Section: Knaster-reichbach Coversmentioning

confidence: 54%

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Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces

Medini¹

2015

Can. math. bull.

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Abstract. We show that, for a coanalytic subspace X of 2 ω , the countable dense homogeneity of X ω is equivalent to X being Polish. This strengthens a result of Hrušák and Zamora Avilés. Then, inspired by results of Hernández-Gutiérrez, Hrušák and van Mill, using a technique of Medvedev, we construct a non-Polish subspace X of 2 ω such that X ω is countable dense homogeneous. This gives the first ZFC answer to a question of Hrušák and Zamora Avilés. Furthermore, since our example is consistently analytic, the equivalence result mentioned above is sharp. Our results also answer a question of Medini and Milovich. Finally, we show that if every countable subset of a zero-dimensional separable metrizable space X is included in a Polish subspace of X then X ω is countable dense homogeneous.

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Section: Knaster-reichbach Coversmentioning

confidence: 54%

“…The following result has been obtained independently by van Engelen (see [5,Theorem 4.4]) and Medvedev (see [23,Corollary 6]). Theorem 9.2 (van Engelen; Medvedev).…”

Section: A Sufficient Conditionmentioning

confidence: 76%

Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces

Medini¹

2015

Can. math. bull.

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“…By virtue of [5,Exercise 4.3.H], the theorem holds if X is the Cantor set or the space of irrationals. According to [10,Theorem 5.4] (see also [7,Corollary 2]) the Baire space B(k) of weight k is densely homogeneous. There is no another h-homogeneous complete spaces.…”

Section: The Main Theoremmentioning

confidence: 99%

Zero-dimensional CDH spaces with a dense completely metrizable subset

Medvedev

2019

Acta Math. Hungar.

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It is proved that for an h-homogeneous space X the following conditions are equivalent: 1) X is a densely homogeneous space with a dense complete subspace; 2) X is σ-discretely controlled.All spaces under discussion are metrizable and strongly zero-dimensional. In 2005, M. Hrušák and B. Z. Avilés [1, Theorem 2.3] proved that every analytic CDHspace is completely Baire. It is therefore of interest to construct a non-complete CDHspace with a dense complete subset. It was realized by R. Hernández-Gutiérrez, M. Hrušák, and J. van Mill [2, Corollary 4.6] and A. Medini [3, Theorem 7.3]. They shown that if a homogeneous space X with a dense complete subset can be embedded in the Cantor set 2 ω by a special way, then X is CDH. Moreover, Medini [3, Theorem 9.4] proved that X ω is CDH if X is a countably controlled space. We suggest a criterion for checking the CDHproperty for a space without its embedding into the Cantor set. A similar result is obtained for non-separable spaces. Notice that the following characterization of non-Baire densely homogeneous h-homogeneous spaces was given by Medvedev [4, Theorem 6]:Theorem 1. Let X be a non-σ-discrete h-homogeneous space. Then X is a densely homogeneous space of the first category if and only if every σ-discrete subset of X is a G δ -set in X.For all undefined terms and notations see [5]. We shall say that a space X is h-homogeneous if IndX = 0 and every non-empty clopen subset of X is homeomorphic to X (this term was first used by Ostrovsky [6]). By a complete space we mean a completely metrizable space, or an absolute G δ -space. A separable topological space X is countable dense homogeneous (briefly, CDH) if, given any two countable dense subsets A and B of X, there is a homeomorphism h : X → X such that h(A) = B. Similarly, a space X is densely homogeneous (briefly, DH) if, given any two σ-discrete dense subsets A and B of X, there is a homeomorphism h : X → X such that h(A) = B.Medini [3] says that a separable space X is countably controlled if for every countable D ⊂ X there exists a Polish subspace G of X such that D ⊂ G. Similarly, we shall say that a space X is σ-discretely controlled if for every σ-discrete D ⊂ X there exists a completely metrizable subspace G of X such that D ⊂ G. Clearly, every σ-discretely controlled space 2010 Mathematics Subject Classification. 54H05, 54E52.

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“…Next, we list a few partial results on Question 18. The following theorem is due independently to van Engelen and Medvedev (see [vE,Theorems 4.2 and 4.4] or [Mv,proof of Theorem 25]). 4 Recall that the assumption dim(X) = 0 on a metrizable space X is in general stronger than the assumption that X is zerodimensional (see [Pr]).…”

Section: A Problem Of Teradamentioning

confidence: 99%

Infinite Powers and Cohen Reals

Medini¹,

Mill²,

Zdomskyy³

2018

Can. math. bull.

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Abstract. We give a consistent example of a zero-dimensional separable metrizable space Z such that every homeomorphism of Z ω acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada. Our example Z is simply the set of ω 1 Cohen reals, viewed as a subspace of 2 ω .

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On properties of h-homogeneous spaces with the Baire property

Cited by 7 publications

References 8 publications

Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces

Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces

Zero-dimensional CDH spaces with a dense completely metrizable subset

Infinite Powers and Cohen Reals

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