Geodesic
Nearly Countable Dense Homogeneous Spaces
Canadian journal of mathematics, Tome 66 (2014) no. 4, pp. 743-758

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-2013-006-8
Mots-clés : 54H05, 03E15, 54E50, countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjecture 
Hrušák, Michael; Mill, Jan van. Nearly Countable Dense Homogeneous Spaces. Canadian journal of mathematics, Tome 66 (2014) no. 4, pp. 743-758. doi: 10.4153/CJM-2013-006-8
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