Countable dense homogeneity of definable spaces
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- by Michael Hrušák and Beatriz Zamora Avilés
- Proc. Amer. Math. Soc. 133 (2005), 3429-3435
- DOI: https://doi.org/10.1090/S0002-9939-05-07858-5
- Published electronically: May 2, 2005
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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$\}$.References
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Bibliographic Information
- Michael Hrušák
- Affiliation: Instituto de Matemáticas, UNAM, Unidad Morelia, A. P. 61-3, Xangari, C. P. 58089, Morelia, Michoacán, México
- MR Author ID: 602083
- ORCID: 0000-0002-1692-2216
- Email: michael@matmor.unam.mx
- Beatriz Zamora Avilés
- Affiliation: Instituto de Matemáticas, UNAM, Unidad Morelia, A. P. 61-3, Xangari, C. P. 58089, Morelia, Michoacán, México
- Email: bzamora@matmor.unam.mx
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 3429-3435
- MSC (2000): Primary 54E52, 54H05, 03E15
- DOI: https://doi.org/10.1090/S0002-9939-05-07858-5
- MathSciNet review: 2161169