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Nearly Countable Dense Homogeneous Spaces

  • Michael Hrušák (a1) and Jan van Mill (a2)
  • Please note a correction has been issued for this article.

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 $X\,\backslash \,S$ 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.

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References

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Nearly Countable Dense Homogeneous Spaces

  • Michael Hrušák (a1) and Jan van Mill (a2)
  • Please note a correction has been issued for this article.

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