Abstract view
Nearly Countable Dense Homogeneous Spaces


Published:20130308
Printed: Aug 2014
Michael Hrušák,
Centro de Ciencas Matemáticas, UNAM, A.P. 613, Xangari, Morelia, Michoacán, 58089, México
Jan van Mill,
Faculty of Sciences, Department of Mathematics, VU University Amsterdam, De Boelelaan 1081${}^a$, 1081 HV Amsterdam, The Netherlands
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 $X\setminus S$ 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.
MSC Classifications: 
54H05, 03E15, 54E50 show english descriptions
Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05] Descriptive set theory [See also 28A05, 54H05] Complete metric spaces
54H05  Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05] 03E15  Descriptive set theory [See also 28A05, 54H05] 54E50  Complete metric spaces
