47 research outputs found
Levels of discontinuity, limit-computability, and jump operators
We develop a general theory of jump operators, which is intended to provide
an abstraction of the notion of "limit-computability" on represented spaces.
Jump operators also provide a framework with a strong categorical flavor for
investigating degrees of discontinuity of functions and hierarchies of sets on
represented spaces. We will provide a thorough investigation within this
framework of a hierarchy of -measurable functions between arbitrary
countably based -spaces, which captures the notion of computing with
ordinal mind-change bounds. Our abstract approach not only raises new questions
but also sheds new light on previous results. For example, we introduce a
notion of "higher order" descriptive set theoretical objects, we generalize a
recent characterization of the computability theoretic notion of "lowness" in
terms of adjoint functors, and we show that our framework encompasses ordinal
quantifications of the non-constructiveness of Hilbert's finite basis theorem
A generalization of a theorem of Hurewicz for quasi-Polish spaces
We identify four countable topological spaces , , , and
which serve as canonical examples of topological spaces which fail to be
quasi-Polish. These four spaces respectively correspond to the , ,
, and -separation axioms. is the space of rationals, is
the natural numbers with the cofinite topology, is an infinite chain
without a top element, and is the set of finite sequences of natural
numbers with the lower topology induced by the prefix ordering. Our main result
is a generalization of Hurewicz's theorem showing that a co-analytic subset of
a quasi-Polish space is either quasi-Polish or else contains a countable
-subset homeomorphic to one of these four spaces
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
Constructing the Space of Valuations of a Quasi-Polish Space as a Space of Ideals
We construct the space of valuations on a quasi-Polish space in terms of the characterization of quasi-Polish spaces as spaces of ideals of a countable transitive relation. Our construction is closely related to domain theoretical work on the probabilistic powerdomain, and helps illustrate the connections between domain theory and quasi-Polish spaces. Our approach is consistent with previous work on computable measures, and can be formalized within weak formal systems, such as subsystems of second order arithmetic
Ideal presentations and numberings of some classes of effective quasi-Polish spaces
The well known ideal presentations of countably based domains were recently
extended to (effective) quasi-Polish spaces. Continuing these investigations,
we explore some classes of effective quasi-Polish spaces. In particular, we
prove an effective version of the domain-characterization of quasi-Polish
spaces, describe effective extensions of quasi-Polish topologies, discover
natural numberings of classes of effective quasi-Polish spaces, estimate the
complexity of the (effective) homeomorphism relation and of some classes of
spaces w.r.t. these numberings, and investigate degree spectra of continuous
domains