209 research outputs found
Lebesgue's Density Theorem and definable selectors for ideals
We introduce a notion of density point and prove results analogous to
Lebesgue's density theorem for various well-known ideals on Cantor space and
Baire space. In fact, we isolate a class of ideals for which our results hold.
In contrast to these results, we show that there is no reasonably definable
selector that chooses representatives for the equivalence relation on the Borel
sets of having countable symmetric difference. In other words, there is no
notion of density which makes the ideal of countable sets satisfy an analogue
to the density theorem. The proofs of the positive results use only elementary
combinatorics of trees, while the negative results rely on forcing arguments.Comment: 28 pages; minor corrections and a new introductio
A Special Class of Almost Disjoint Families
The collection of branches (maximal linearly ordered sets of nodes) of the
tree (ordered by inclusion) forms an almost disjoint
family (of sets of nodes). This family is not maximal -- for example, any level
of the tree is almost disjoint from all of the branches. How many sets must be
added to the family of branches to make it maximal? This question leads to a
series of definitions and results: a set of nodes is {\it off-branch} if it is
almost disjoint from every branch in the tree; an {\it off-branch family} is an
almost disjoint family of off-branch sets; {\frak o}=\min\{|{\Cal O}|: {\Cal
O} is a maximal off-branch family. Results concerning include:
(in ZFC) , and (consistent with ZFC) is not
equal to any of the standard small cardinal invariants , ,
, or . Most of these consistency results use
standard forcing notions -- for example, comes from starting with a model of and
adding -many Cohen reals. Many interesting open questions remain,
though -- for example,
Perfect subsets of generalized Baire spaces and long games
We extend Solovay's theorem about definable subsets of the Baire space to the
generalized Baire space , where is an uncountable
cardinal with . In the first main theorem, we show
that that the perfect set property for all subsets of
that are definable from elements of is consistent
relative to the existence of an inaccessible cardinal above . In the
second main theorem, we introduce a Banach-Mazur type game of length
and show that the determinacy of this game, for all subsets of
that are definable from elements of
as winning conditions, is consistent relative to the
existence of an inaccessible cardinal above . We further obtain some
related results about definable functions on and
consequences of resurrection axioms for definable subsets of
On critical cardinalities related to -sets
In this note we collect some known information and prove new results about
the small uncountable cardinal . The cardinal is
defined as the smallest cardinality of a subset
which is not a -set (a subspace is called a -set if
each subset is of type in ). We present a simple
proof of a folklore fact that , and also establish the
consistency of a number of strict inequalities between the cardinal and other standard small uncountable cardinals. This is done by combining
some known forcing results. A new result of the paper is the consistency of
, where denotes
the linear refinement number. Another new result is the upper bound holding for any -flexible cccc
-ideal on .Comment: 8 page
- …