176 research outputs found
On the structure of finite level and \omega-decomposable Borel functions
We give a full description of the structure under inclusion of all finite
level Borel classes of functions, and provide an elementary proof of the
well-known fact that not every Borel function can be written as a countable
union of \Sigma^0_\alpha-measurable functions (for every fixed 1 \leq \alpha <
\omega_1). Moreover, we present some results concerning those Borel functions
which are \omega-decomposable into continuous functions (also called countably
continuous functions in the literature): such results should be viewed as a
contribution towards the goal of generalizing a remarkable theorem of Jayne and
Rogers to all finite levels, and in fact they allow us to prove some restricted
forms of such generalizations. We also analyze finite level Borel functions in
terms of composition of simpler functions, and we finally present an
application to Banach space theory.Comment: 31 pages, 2 figures, revised version, accepted for publication on the
Journal of Symbolic Logi
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
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