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Countable dense homogeneity of definable spaces

Authors: Michael Hrusák and Beatriz Zamora Avilés
Journal: Proc. Amer. Math. Soc. 133 (2005), 3429-3435
MSC (2000): Primary 54E52, 54H05, 03E15
Published electronically: May 2, 2005
MathSciNet review: 2161169
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Abstract: We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zero-dimensional Borel CDH spaces. We also show that for a Borel $X\subseteq 2^{\omega }$ the following are equivalent: (1) $X$ is $G_{\delta }$ in $2^{\omega }$, (2) $X^{\omega }$ is CDH and (3) $X^{\omega }$is homeomorphic to $2^{\omega }$ or to $\omega ^{\omega }$. Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo a large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Stepr $\bar {\text{a}}$ns and Zhou, by showing that $\mathfrak{p}= \min \{\kappa : 2^{\kappa }$ is not CDH$\}$.

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Additional Information

Michael Hrusák
Affiliation: Instituto de Matemáticas, UNAM, Unidad Morelia, A. P. 61-3, Xangari, C. P. 58089, Morelia, Michoacán, México

Beatriz Zamora Avilés
Affiliation: Instituto de Matemáticas, UNAM, Unidad Morelia, A. P. 61-3, Xangari, C. P. 58089, Morelia, Michoacán, México

Keywords: Countable dense homogeneous, Borel, Baire
Received by editor(s): June 13, 2003
Received by editor(s) in revised form: June 11, 2004
Published electronically: May 2, 2005
Additional Notes: The first author’s research was supported partially by grant GAČR 201/03/0933 and by a PAPIIT grant IN108802-2 and CONACYT grant 40057-F
Communicated by: Alan Dow
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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