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A TOPOMETRIC EFFROS THEOREM
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 02 February 2023, pp. 1-11
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Given a continuous and isometric action of a Polish group G on an adequate Polish topometric space
$(X,\tau ,\rho )$ and 
$x \in X$, we find a necessary and sufficient condition for 
$\overline {Gx}^{\rho }$ to be co-meagre; we also obtain a criterion that characterizes when such a point exists. This work completes a criterion established in earlier work of the authors.
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
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- 01 August 2014
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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, pp. 743-758
- Print publication:
- 01 August 2014
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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.
