45 research outputs found
Quasi-Polish Spaces
We investigate some basic descriptive set theory for countably based
completely quasi-metrizable topological spaces, which we refer to as
quasi-Polish spaces. These spaces naturally generalize much of the classical
descriptive set theory of Polish spaces to the non-Hausdorff setting. We show
that a subspace of a quasi-Polish space is quasi-Polish if and only if it is
level \Pi_2 in the Borel hierarchy. Quasi-Polish spaces can be characterized
within the framework of Type-2 Theory of Effectivity as precisely the countably
based spaces that have an admissible representation with a Polish domain. They
can also be characterized domain theoretically as precisely the spaces that are
homeomorphic to the subspace of all non-compact elements of an
\omega-continuous domain. Every countably based locally compact sober space is
quasi-Polish, hence every \omega-continuous domain is quasi-Polish. A
metrizable space is quasi-Polish if and only if it is Polish. We show that the
Borel hierarchy on an uncountable quasi-Polish space does not collapse, and
that the Hausdorff-Kuratowski theorem generalizes to all quasi-Polish spaces
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