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1. CJM Online first
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 conjecture Categories:54H05, 03E15, 54E50 |
2. CJM Online first
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 conjecture Categories:54H05, 03E15, 54E50 |
3. CJM 2006 (vol 58 pp. 529)
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, hyperspace Category:57S05 |