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1. CJM 2013 (vol 66 pp. 759)

Hrušák, Michael; van Mill, Jan
 Addendum to "Nearly Countable Dense Homogeneous Spaces" This paper provides an addendum to M. HruÅ¡Ã¡k and J. van Mill Nearly countable dense homogeneous spaces.'' Canad. J. Math., published online 2013-03-08 http://dx.doi.org/10.4153/CJM-2013-006-8. Keywords:countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjectureCategories:54H05, 03E15, 54E50

2. CJM 2013 (vol 66 pp. 743)

Hrušák, Michael; van Mill, Jan
 Nearly Countable Dense Homogeneous Spaces 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 $X\setminus S$ 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:countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjectureCategories:54H05, 03E15, 54E50

3. CJM 2006 (vol 58 pp. 529)

Dijkstra, Jan J.; Mill, Jan van
 On the Group of Homeomorphisms of the Real Line That Map the Pseudoboundary Onto Itself In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line $\R$, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers $\Q$ onto itself is homeomorphic to the infinite power of $\Q$ with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of $\Q$ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to $\R^n$ for $n\ge 2$, by linking the groups in question with Erd\H os space. Keywords:homeomorphism group, real line, countable dense set, pseudoboundary, Erd\H{o}s space, hyperspaceCategory:57S05
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