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Nearly Countable Dense Homogeneous Spaces

  Published:2013-03-08
 Printed: Aug 2014
  • Michael Hrušák,
    Centro de Ciencas Matemáticas, UNAM, A.P. 61-3, Xangari, Morelia, Michoacán, 58089, México
  • Jan van Mill,
    Faculty of Sciences, Department of Mathematics, VU University Amsterdam, De Boelelaan 1081${}^a$, 1081 HV Amsterdam, The Netherlands
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Abstract

We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely $n$ types of countable dense sets: such a space contains a subset $S$ of size at most $n{-}1$ such that $S$ is invariant under all homeomorphisms of $X$ and $X\setminus S$ is countable dense homogeneous. We prove that every Borel space having fewer than $\mathfrak{c}$ types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or $\mathfrak{c}$ many types of countable dense sets, is shown to be closely related to Topological Vaught's Conjecture.
Keywords: countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjecture countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjecture
MSC Classifications: 54H05, 03E15, 54E50 show english descriptions Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05]
Descriptive set theory [See also 28A05, 54H05]
Complete metric spaces
54H05 - Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05]
03E15 - Descriptive set theory [See also 28A05, 54H05]
54E50 - Complete metric spaces
 

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