Topology Atlas Document # paao-38
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 subset or equal 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 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¯ans and Zhou by showing that \mathfrakp = min{\kappa: 2\kappa is not CDH}.
Mathematics Subject Classification: 54E52, 54H05, 03E15
Keywords: Countable dense homogeneous, Borel, Baire
Date received: June 17, 2003.
Copyright © 2003 by the authors. Distributed by Topology Atlas with permission of the authors.