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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Countable dense homogeneity of definable spaces
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by Michael Hrušák and Beatriz Zamora Avilés PDF
Proc. Amer. Math. Soc. 133 (2005), 3429-3435 Request permission

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 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