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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
 MSC Classifications: 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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