297 research outputs found
On disjoint Borel uniformizations
Larman showed that any closed subset of the plane with uncountable vertical
cross-sections has aleph_1 disjoint Borel uniformizing sets. Here we show that
Larman's result is best possible: there exist closed sets with uncountable
cross-sections which do not have more than aleph_1 disjoint Borel
uniformizations, even if the continuum is much larger than aleph_1. This
negatively answers some questions of Mauldin. The proof is based on a result of
Stern, stating that certain Borel sets cannot be written as a small union of
low-level Borel sets. The proof of the latter result uses Steel's method of
forcing with tagged trees; a full presentation of this method, written in terms
of Baire category rather than forcing, is given here
Disjoint Infinity-Borel Functions
This is a followup to a paper by the author where the disjointness relation
for definable functions from to is
analyzed. In that paper, for each we defined a Baire
class one function which encoded
in a certain sense. Given , let
be the statement that is disjoint from at most countably many of
the functions . We show the consistency strength of is that of an inaccessible cardinal. We show that
implies . Finally, we show that assuming large
cardinals, holds in models of the form
where is a selective ultrafilter on
.Comment: 16 page
Foliations with few non-compact leaves
Let F be a foliation of codimension 2 on a compact manifold with at least one
non-compact leaf. We show that then F must contain uncountably many non-compact
leaves. We prove the same statement for oriented p-dimensional foliations of
arbitrary codimension if there exists a closed p form which evaluates
positively on every compact leaf. For foliations of codimension 1 on compact
manifolds it is known that the union of all non-compact leaves is an open set
[A Haefliger, Varietes feuilletes, Ann. Scuola Norm. Sup. Pisa 16 (1962)
367-397].Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-12.abs.htm
Essential countability of treeable equivalence relations
We establish a dichotomy theorem characterizing the circumstances under which
a treeable Borel equivalence relation E is essentially countable. Under
additional topological assumptions on the treeing, we in fact show that E is
essentially countable if and only if there is no continuous embedding of E1
into E. Our techniques also yield the first classical proof of the analogous
result for hypersmooth equivalence relations, and allow us to show that up to
continuous Kakutani embeddability, there is a minimum Borel function which is
not essentially countable-to-one
Analytic Colorings
We investigate the existence of perfect homogeneous sets for analytic
colorings. An analytic coloring of X is an analytic subset of [X]^N, where N>1
is a natural number. We define an absolute rank function on trees representing
analytic colorings, which gives an upper bound for possible cardinalities of
homogeneous sets and which decides whether there exists a perfect homogeneous
set. We construct universal sigma-compact colorings of any prescribed rank
gamma<omega_1. These colorings consistently contain homogeneous sets of
cardinality aleph_gamma but they do not contain perfect homogeneous sets. As an
application, we discuss the so-called defectedness coloring of subsets of
Polish linear spaces
Locally Constant Functions
Let X be a compact Hausdorff space and M a metric space. E_0(X,M) is the set
of f in C(X,M) such that there is a dense set of points x in X with f constant
on some neighborhood of x. We describe some general classes of X for which
E_0(X,M) is all of C(X,M). These include beta N - N, any nowhere separable
LOTS, and any X such that forcing with the open subsets of X does not add
reals. In the case that M is a Banach space, we discuss the properties of
E_0(X,M) as a normed linear space. We also build three first countable Eberlein
compact spaces, F,G,H, with various E_0 properties: For all metric M: E_0(F,M)
contains only the constant functions, and E_0(G,M) = C(G,M). If M is the
Hilbert cube or any infinite dimensional Banach space, E_0(H,M) is not all of
C(H,M), but E_0(H,M) = C(H,M) whenever M is a subset of RR^n for some finite n
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