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 the following are equivalent: (1) is in , (2) is CDH and (3) is homeomorphic to or to . 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 ns and Zhou, by showing that 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

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

DOI:
http://dx.doi.org/10.1090/S0002-9939-05-07858-5

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.