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Addendum to “Nearly Countable Dense Homogeneous Spaces”
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- Journal:
- Canadian Journal of Mathematics / Volume 66 / Issue 4 / 01 August 2014
- Published online by Cambridge University Press:
- 20 November 2018, p. 759
- Print publication:
- 01 August 2014
-
- Article
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- You have access
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Nearly Countable Dense Homogeneous Spaces
-
- Journal:
- Canadian Journal of Mathematics / Volume 66 / Issue 4 / 01 August 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 743-758
- Print publication:
- 01 August 2014
-
- Article
-
- You have access
- Export citation
-
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.
