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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
Non-tame Mice from Tame Failures of the Unique Branch Hypothesis In this paper, we show that the failure of the unique branch
hypothesis (UBH) for tame trees
implies that in some homogenous generic extension of $V$ there is a
transitive model $M$ containing $Ord \cup \mathbb{R}$ such that
$M\vDash \mathsf{AD}^+ + \Theta \gt \theta_0$. In particular, this
implies the existence (in $V$) of a non-tame mouse. The results of
this paper significantly extend J. R. Steel's earlier results
for tame trees.
Keywords:mouse, inner model theory, descriptive set theory, hod mouse, core model induction, UBH Categories:03E15, 03E45, 03E60 |
3. 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 |
4. CJM 2012 (vol 64 pp. 1378)
On Weakly Tight Families Using ideas from Shelah's recent proof that a completely
separable maximal almost disjoint family exists when
$\mathfrak{c} \lt {\aleph}_{\omega}$, we construct a weakly tight family
under the hypothesis $\mathfrak{s} \leq \mathfrak{b} \lt
{\aleph}_{\omega}$.
The case when $\mathfrak{s} \lt \mathfrak{b}$
is handled in $\mathrm{ZFC}$ and does not require $\mathfrak{b} \lt {\aleph}_{\omega}$,
while an additional PCF type hypothesis, which holds when $\mathfrak{b} \lt
{\aleph}_{\omega}$ is used to treat the case $\mathfrak{s} = \mathfrak{b}$. The notion of
a weakly tight family is a natural weakening of the well studied
notion of a Cohen indestructible maximal almost disjoint family. It
was introduced by HruÅ¡Ã¡k and GarcÃa
Ferreira, who applied it to the KatÃ©tov order on almost
disjoint families.
Keywords:maximal almost disjoint family, cardinal invariants Categories:03E17, 03E15, 03E35, 03E40, 03E05, 03E50, 03E65 |
5. CJM 1999 (vol 51 pp. 309)
Symmetric sequence subspaces of $C(\alpha)$, II If $\alpha$ is an ordinal, then the space of all ordinals less than or
equal to $\alpha$ is a compact Hausdorff space when endowed with the
order topology. Let $C(\alpha)$ be the space of all continuous
real-valued functions defined on the ordinal interval $[0,
\alpha]$. We characterize the symmetric sequence spaces which embed
into $C(\alpha)$ for some countable ordinal $\alpha$. A hierarchy
$(E_\alpha)$ of symmetric sequence spaces is constructed so that, for
each countable ordinal $\alpha$, $E_\alpha$ embeds into
$C(\omega^{\omega^\alpha})$, but does not embed into
$C(\omega^{\omega^\beta})$ for any $\beta < \alpha$.
Categories:03E13, 03E15, 46B03, 46B45, 46E15, 54G12 |