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Nearly Countable Dense Homogeneous Spaces

Published online by Cambridge University Press:  20 November 2018

Michael Hrušák  and
Jan van Mill
Michael Hrušák
Affiliation:
Centro de Ciencas Matemáticas, UNAM, A.P. 61-3, Xangari, Morelia, Michoacán, 58089, México. e-mail: michael@matmor.unam.mx
Jan van Mill
Affiliation:
Faculty of Sciences, Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a , 1081 HV Amsterdam, The Netherlands. e-mail: j.van.mill@vu.nl
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Abstract

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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 $X\,\backslash \,S$ 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

54H0503E1554E50countable dense homogeneousnearly countable dense homogeneousEffros TheoremVaught's conjecture
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 4 , 01 August 2014 , pp. 743 - 758
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

*

The first author was supported by a PAPIIT grant IN 102311 and CONACyT grant 177758. The second author is pleased to thank the Centro de Ciencas Matemáticas in Morelia for generous hospitality and support.

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